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+{"text":"\\section*{Introduction}\n\nCoarse, or large-scale, geometry has long been studied in various guises, but \nmost notably in the context of metric spaces. Most generically, a \\emph{coarse \nspace} is a space together with some kind of large-scale structure (e.g., a \nmetric modulo \\emph{quasi-isometry}; see Remark~\\ref{rmk:lsLip-qisom}). A \n\\emph{coarse map} between coarse spaces is then a map which respects this \nstructure (e.g., a large-scale Lipschitz map). Since the small-scale (i.e., the \ntopology) is ignored, one can typically take coarse spaces to be \n\\emph{discrete}, replacing any nondiscrete space by some ``coarsely dense'' \nsubset.\n\nIn recent decades, coarse ideas have played an important role in the study of \ninfinite discrete groups using the methods of geometric group theory, \nespecially in the work of Gromov and his followers (see, e.g., \n\\cite{MR1253544}). The most basic example here is that if $\\Gamma$ is a \nfinitely generated group, then the word length metric on $\\Gamma$ is modulo \nquasi-isometry independent of the finite set of generators used in defining it.\n\nCoarse ideas have also arisen in geometric topology, and more specifically \ncontrolled topology which primarily concerns itself with problems on the \nstructure of manifolds. (We refer the reader to \\cite{MR1308714}*{Ch.~9} for a \nsurvey of the topic and for references.) In this setting, one is interested in \n``operations'' (e.g., homotopies, surgeries) on spaces which respect some \nlarge-scale structure, i.e., are \\emph{controlled}. As before, one may take the \nlarge-scale structure to be given by a metric (i.e., \\emph{bounded control}). \nHowever, it is often more convenient to work with a coarser large-scale \nstructure which is defined in purely topological terms (i.e., \\emph{continuous \ncontrol}; see \\S\\ref{subsect:cts-ctl}).\n\nControlled topology parallels the more classical theory for compact manifolds \nwhich relies on the use of algebraic invariants (e.g., algebraic $K$-theory). \nIn controlled topology, one gets controlled versions of those invariants (in, \ne.g., \\emph{bounded} and \\emph{continuously controlled $K$-theory} \n\\cites{MR1000388, MR802790, MR1277522}; see also \\cite{MR1208729}). By \nconsidering the fundamental group of a space, a key object of study in the \nstudy of homotopy invariants (e.g., the Novikov Conjecture on higher \nsignatures), many of the problems of geometric topology are related back to \ngeometric group theory.\n\nOn the analytic side, to any coarse space $X$, Roe has associated a \n$C^*$-algebra $C^*(X)$ (the \\emph{Roe algebra} of $X$), as well as various \n``(co)homology'' groups, e.g., \\emph{coarse $K$-homology} $KX_\\grstar(X)$. (For \na good overview of this and the following, see \\cite{MR1399087}.) On the other \nhand, one can also take the $K$-theory of $C^*(X)$; the Coarse Baum--Connes \nConjecture is that a certain assembly map $KX_\\grstar(X) \\to K_\\grstar(C^*(X))$ \nis an isomorphism, at least for suitably nice $X$.\n\nThe $K$-theory of Roe algebras arises in the index theory of elliptic operators \non noncompact manifolds (on compact manifolds, the Roe algebra is just the \ncompact operators and the results specialize to classical index theory). \nIndeed, historically it was the study of index theory on noncompact manifolds \nwhich led Roe to coarse geometry (see \\cites{MR918459, MR1399087}), and not the \nother way around. In this way, the analytic approaches to the Novikov \nConjecture (starting with the work of Lusztig \\cite{MR0322889}) are again \nrelated to coarse geometry. (See \\cite{MR1388295} for a nice survey of the \ndifferent approaches to the Novikov Conjecture.)\n\n\n\\subsection*{Roe's coarse geometry}\n\nAfter originally developing coarse geometry in the metric context \n\\cite{MR1147350}, Roe (and his collaborators) realized that one can define an \nabstract notion of \\emph{coarse space}, just as in small-scale geometry one has \nabstract \\emph{topological spaces}. Just as the passage from metric space to \ntopological space forgets large-scale (metric) information, the passage from \nmetric space to coarse space should forget small-scale information. But an \nabstract coarse space retains enough structure to perform the large-scale \nconstructions which were previously done in the metric context (e.g., construct \nthe Roe algebras, coarse $K$-homology, etc.).\n\nA \\emph{coarse space} is a set $X$ together with a \\emph{coarse structure}, \nwhich is a collection $\\calE_X$ of subsets of $X^{\\cross 2} \\defeq X \\cross X$ \n(called the \\emph{entourages} of $X$) satisfying various axioms. When $X$ is a \n(proper) metric space, $\\calE_X$ consists of the subsets $E \\subseteq X^{\\cross \n2}$ such that\n\\[\n    \\sup \\set{d_X(x,x') \\suchthat (x,x') \\in E} < \\infty.\n\\]\nA subset $K \\subseteq X$ is \\emph{bounded} if and only if $K^{\\cross 2}$ is an \nentourage of $X$; when $X$ is a metric space, $K$ is bounded if and only if it \nis metrically bounded. If $X$ is a discrete set, one typically axiomatically \ninsists that the bounded subsets of $X$ be finite (we call this the \n\\emph{properness axiom}; see Definition~\\ref{def:prop-ax}); more generally, if \n$X$ is a topological space, the bounded subsets are required to be compact.\n\nA set map $f \\from Y \\to X$ is a \\emph{coarse map} if $f$ is \\emph{proper} in \nthe sense that the inverse image of any bounded subset of $X$ is a bounded \nsubset of $Y$ and if $f$ \\emph{preserves entourages} in the sense that \n$f^{\\cross 2}(F) \\defeq (f \\cross f)(F)$ is an entourage of $X$. In the metric \ncase, $f$ is a coarse map if it is metrically proper and ``nonexpansive''.\n\nThere is an obvious notion of closeness for maps into a metric space: maps $f, \nf' \\from Y \\to X$ are \\emph{close} if\n\\[\n    \\sup \\set{d_X(f(y),f'(y)) \\suchthat y \\in Y} < \\infty.\n\\]\nThis generalizes to the case when $X$ is a general coarse space: $f$, $f'$ are \nclose if $(f \\cross f')(1_Y)$ is an entourage of $X$, where $1_Y$ is the \ndiagonal set $\\set{(y,y) \\suchthat y \\in Y}$.\n\nRoe's \\emph{coarse category} has coarse spaces as objects, and closeness \nclasses of coarse maps as morphisms. (A coarse map is a \\emph{coarse \nequivalence} if it represents an isomorphism in the coarse category.) Coarse \ninvariants are defined on this category, either as functions on the isomorphism \nclasses of the coarse category (e.g., \\emph{asymptotic dimension}) or as \nfunctors from the coarse category to some other category (e.g., \\emph{coarse \n$K$-homology}). Though coarse invariants are the primary object of study in \ncoarse geometry, the coarse category is rarely analyzed directly, and there is \nsome confusion in the literature about what the coarse category is (some \nauthors take its arrows to be actual coarse maps; we will call this the \n\\emph{precoarse category}).\n\nThere is an obvious ``product coarse structure'' on the cartesian (set) product \n$X \\cross Y$. The entourages are exactly the subsets of $(X \\cross Y)^{\\cross \n2}$ which project to entourages of $X$ and $Y$ in the obvious way. However, \nthis is not (usually) a product in the coarse category: the projection maps are \nnot proper, unless both $X$ and $Y$ are finite\/compact. This problem already \narises in the category of proper metric spaces and proper maps (modulo \ncloseness).\n\n\\begin{UNremark}\nThe above does \\emph{not} prove that $X$ and $Y$ do not have a product in the \ncoarse category. Certain products (of infinite\/noncompact coarse spaces) \n\\emph{do} exist in the coarse category; indeed, there is an infinite space $X$\n(namely the continuously controlled ray, or equivalently a countable set \nequipped with the \\emph{terminal}, i.e., ``indiscrete'', coarse structure) such \nthat the product of $X$ with every countable coarse space exists \n(Remark~\\ref{rmk:term-unital-prod}). The above does not even prove that the \n``product coarse space'' $X \\cross Y$, as defined above, is not a product of \n$X$ and $Y$ if equipped with suitable maps $X \\cross Y \\to X$ and $X \\cross Y \n\\to Y$ (not the set projections).\n\\end{UNremark}\n\n\n\\subsection*{Nonunital coarse spaces and locally proper maps}\n\nMetric spaces always yield \\emph{unital} coarse spaces, i.e., coarse spaces $X$ \nsuch that $1_X \\defeq \\set{(x,x) \\suchthat x \\in X}$ is an entourage. Though \nRoe defines nonunital coarse spaces, unitality is usually a standing \nassumption, presumably since nonunital coarse spaces have no obvious use.\n\n\\emph{The} major innovation of this paper is the following: We relax the \nrequirement that coarse maps be proper, to a requirement that we call \n\\emph{locally properness}. Local properness is not new: it is actually included \nin Bartels's definition of ``coarse map'' \\cite{MR1988817}*{Def.~3.3}. When the \ndomain is a unital coarse space, local properness is equivalent to (``global'') \nproperness (Corollary~\\ref{cor:loc-prop-uni}). However, when the domain is \nnonunital, we get many more coarse maps. Consequently, using nonunital coarse \nspaces, it becomes extremely easy to construct (nonzero) categorical products \nin the coarse category. Indeed, we can do much more.\n\n\\begin{UNexample}\nSuppose $X'$ is a (closed) subspace of a proper metric space $X$, so that $X'$ \nis itself a coarse space. There is an obvious \\emph{ideal} $\\lAngle 1_{X'} \n\\rAngle_X$ of $\\calE_X$ generated by $1_{X'}$ (see Definition~\\ref{def:ideal}). \nThe coarse space $|X|_{\\lAngle 1_{X'} \\rAngle_X}$ with underlying set $X$ and \ncoarse structure $\\lAngle 1_{X'} \\rAngle_X$ is nonunital, unless $X'$ is \n``coarsely dense'' in $X$.\n\nDefine a (set) map $p \\from X \\to X'$ which sends each $x \\in X$ to a point \n$p(x)$ in $X'$ closest to $x$. Then $p$ is usually not proper, hence is not \ncoarse as a map $X \\to X'$. However, it \\emph{is} locally proper and coarse (in \nour generalized sense) as a map $|X|_{\\lAngle 1_{X'} \\rAngle_X} \\to X'$, and is \nactually a coarse equivalence. (We leave it to the reader to verify this, after \nlocating the required definitions.)\n\\end{UNexample}\n\nFor simplicity as well as for philosophical reasons, we only consider \n\\emph{discrete} coarse spaces; hence for us a map is (globally) proper if and \nonly if the inverse image of any point is a finite set. If a map $f \\from Y \\to \nX$ between coarse spaces is proper, then $f^{\\cross 2}$ is a proper map, and \nhence the restriction of $f^{\\cross 2}$ to any entourage $F \\subseteq Y^{\\cross \n2}$ of $Y$ is proper. We take the latter as the definition of local properness: \nA map $f \\from Y \\to X$ between coarse spaces (not necessarily unital) is \n\\emph{locally proper} if, for all entourages $F$ of $Y$, the restriction \n$f^{\\cross 2} |_F \\from F \\to X^{\\cross 2}$ is a proper map. There are a number \nof equivalent ways of defining local properness, the most intuitive of which is \nthe following. For a nonunital coarse space, there is an obvious notion of \n\\emph{unital subspace}; a map is locally proper if and only if the restriction \nto every unital subspace of its domain is a proper map \n(Corollary~\\ref{cor:loc-prop-uni}).\n\nWhen $X$ is nonunital, we must modify the the definition of closeness, lest the \nidentity map on $X$ not be close to itself. We modify it in a simple way, now \nrequiring that the domain also be a coarse space: Coarse maps $f, f' \\from Y \n\\to X$ (between possibly nonunital coarse spaces) are \\emph{close} if $(f \n\\cross f')(F)$ is an entourage of $X$ for every entourage $F$ of $Y$. After \nchecking the usual things, we get our nonunital \\emph{coarse category}, whose \nobjects are (possibly nonunital) coarse spaces and whose arrows are closeness \nclasses of (locally proper) coarse maps.\n\n\\begin{UNremark}\nEmerson--Meyer have defined a notion of \\emph{$\\sigma$-coarse spaces}, coarse \nmaps between such spaces, and an appropriate notion of closeness \n\\cite{MR2225040}. A $\\sigma$-coarse space is just the colimit of an increasing \nsequence of unital coarse spaces. In fact, we show that the (pre)coarse \ncategory of discrete $\\sigma$-coarse spaces is equivalent to a subcategory of \nour (pre)coarse category consisting of the \\emph{$\\sigma$-unital coarse spaces} \n(we do not examine the situation when one allows $\\sigma$-coarse spaces to have \nnontrivial topologies).\n\\end{UNremark}\n\n\n\\subsection*{Products, limits, etc.}\n\nLet us see how to construct the product of coarse spaces $X$ and $Y$ in this \ncategory. We do so by putting a \\emph{nonunital} coarse structure on the set $X \n\\cross Y$. The entourages of the \\emph{coarse product} $X \\cross Y$ are the $G \n\\subseteq (X \\cross Y)^{\\cross 2}$ such that:\n\\begin{enumerate}\n\\item the restricted projections $\\pi_1 |_G, \\pi_2 |_G \\from G \\to X \\cross Y$ \n    are proper maps (this is the aforementioned properness axiom);\n\\item $\\pi_X |_G \\from G \\to X^{\\cross 2}$ and $\\pi_Y |_G \\from G \\to Y^{\\cross \n    2}$ are proper maps; and\n\\item $(\\pi_X)^{\\cross 2}(G)$ is an entourage of $X$ and $(\\pi_Y)^{\\cross \n    2}(G)$ is an entourage of $Y$.\n\\end{enumerate}\nOne can then check that this is a product in our nonunital coarse category \n(indeed, it is a product in our nonunital \\emph{precoarse category}). We must \nemphasize that the coarse structure on the set product is crucial: If $\\ast$ is \na one-point coarse space, then $X \\cross \\ast \\cong X$ as a set, but unless $X$ \nis bounded the coarse product $X \\cross \\ast$ is \\emph{not} coarsely equivalent \nto $X$.\n\nThe above construction generalizes to all nonzero products (by nonzero product, \nwe mean a product of a nonempty collection of spaces), including infinite \nproducts (Theorem~\\ref{thm:PCrs-lim} and Proposition~\\ref{prop:Crs-prod}). We \nwill then proceed to examine equalizers in the nonunital coarse category, and \ndiscover that it has all equalizers of pairs of maps \n(Proposition~\\ref{prop:Crs-equal}). A standard categorical corollary is that \nthe nonunital coarse category has all nonzero (projective) limits \n(Theorem~\\ref{thm:Crs-lim}). One can similarly analyze coproducts (i.e., sums \nor ``disjoint unions'') and coequalizers, and get that the nonunital coarse \ncategory has all colimits, i.e., inductive limits \n(Theorem~\\ref{thm:Crs-colim}).\n\n\n\\subsection*{Terminal objects and quotients}\n\nFor set theoretic reasons, the coarse category does not have a terminal object. \n(As we shall see in \\S\\ref{subsect:rest-Crs}, one way of obtaining a terminal \nobject is to restrict the cardinality of coarse spaces, though there is a \nbetter way to proceed. For most purposes, no such restriction is needed.) \nHowever, there is a plethora of coarse spaces which behave like terminal \nobjects. The \\emph{terminal coarse structure} on a set $X$ consists of the \nsubsets of $X^{\\cross 2}$ which are both ``row- and column-finite''; denote the \nresulting coarse space by $|X|_1$. A rather underappreciated fact about such \ncoarse spaces is that, for any coarse space $Y$, \\emph{all} coarse maps $Y \\to \n|X|_1$ are close to one another. An immediate categorical consequence of this \nis that, assuming that any such coarse map exists, the product of $|X|_1$ and \n$Y$ in the (unital or nonunital) coarse category is just $Y$ itself \n(Proposition~\\ref{prop:term-id}).\n\nIn the unital coarse category, $X \\mapsto |X|_1$ is a functor. In the nonunital \ncoarse category, one must replace $|X|_1$ with a different coarse space, \ndenoted $\\Terminate(X)$ (with $\\Terminate(X) = |X|_1$ for $X$ unital), to \nobtain a functor. In an abelian category, one can define a quotient $X\/f(Y)$ \n(for $f \\from Y \\to X$) as push-out $X \\copro_Y 0$. This generalizes to any \ncategory with zero objects and push-out squares. We will see that in fact we \ncan generalize this to the coarse setting, defining $X\/[f](Y)$ to be the \npush-out $X \\copro_Y \\Terminate(Y)$ in the (nonunital) coarse category. \n(Indeed, one can do the same in the category of topological spaces, noting that \nthere are two cases: ``$\\Terminate(X)$'' is a one-point space if $X \\neq \n\\emptyset$ and the empty space otherwise.)\n\n\n\\subsection*{Applications}\n\nOur development of coarse geometry is a strict generalization of Roe's, despite \nour assumption of discreteness (see \\S\\ref{sect:top-crs}). Most of the standard \nconstructions in Roe's coarse geometry (such as those mentioned above) \ngeneralize easily to our nonunital, locally proper version. (Note, however, \nthat our theory does not encompass what one may call, following the language of \n\\cite{MR1817560}*{Ch.~12}, the ``uniform category'' in which both the coarse \nstructure and the topology are important. For example, Roe's $C^*$-algebras \n$D^*(X)$, which are functorial for uniform maps, require a notion of \n\\emph{topological coarse space}. We defer this task to \\cite{crscat-III}; see \nRemark~\\ref{rmk:top-crs-sp}.) However, we will refrain from fully developing \nthese applications in this paper. For the purposes of this paper, we briefly \nexamine some things enabled by our generalization.\n\nHaving examined the coarse category from the categorical point of view, many \nstandard constructions from topology transfer easily over to the coarse \nsetting. For example, one obtains a notion of coarse simplicial complex. Of \ncourse, it is easy to deal with finite complexes explicitly in the unital \ncoarse category. However, one result of having \\emph{all} colimits, including \ninfinite ones, is that we actually obtain infinite coarse simplicial complexes. \nThis should enable one to apply simplicial methods in coarse geometry.\n\n\\begin{comment}\nAnother application is to coarse homotopy. There are various notions of \nhomotopy used in coarse geometry, e.g., the coarse homotopy of Higson and Roe \n\\cite{MR1243611}. However, the standard description of coarse homotopy is not \n``categorical'', for the obvious reason that the standard, unital coarse \ncategory does not seem to have products in general. We rectify this, and \nreformulate coarse homotopy in much more familiar categorical terms: We find a \ncoarse space $I$ such that, for (at most) countable coarse spaces $X$ and $Y$, \na coarse homotopy of maps $Y \\to X$ is exactly given by a coarse map $(h_t) \n\\defeq Y \\cross I \\to X$. This $I$ comes equipped with coarse maps $\\delta_j \n\\from P \\defeq |\\setN|_1 \\to I$, $j = 0, 1$ (or, indeed, $j \\in \\ccitvl{0,1}$), \nwhich allows one to recover the coarse maps ``at the endpoints'' via the \ncompositions\n\\[\n    Y \\isoto Y \\cross P \\nameto{\\smash{\\id \\cross \\delta_j}} Y \\cross I \\to X.\n\\]\nAs a historical note, we mention that this description is motivated by the \ncontinuously controlled case in which the connection to topology is much more \nobvious.\n\\end{comment}\n\n\n\\subsection*{Notes on history and references}\n\nThe framework and terminology we use are essentially due to Roe and his \ncollaborators (see \\cites{MR1147350, MR1451755}, in particular). Since our \ndevelopment differs in various details and in the crucial concept of local \nproperness, and for the sake of completeness, we provide a complete exposition \nfrom basic principles; other, more standard, expositions include \n\\cites{MR1451755, MR1399087, MR1817560, MR2007488}. In the basics, we do not \nclaim much originality and most of the results will be known to those familiar \nwith coarse geometry. However, in the context of locally proper maps, we have \nfound certain methods of proof (in particular, the use of \nProposition~\\ref{prop:prop}) to be particularly effective, and have emphasized \nthe use of these methods. Thus our proofs of standard results may differ from \nthe usual proofs.\n\nWe have endeavoured to provide reasonably thorough references. However, it is \noften unwieldy to provide complete data for things which have been generalized \nand refined over the years. In such cases, rather than providing references to \nthe original definition and all the subsequent generalizations, we simply \nreference a work (often expository in nature) which provides the current \nstandard definition; often, such definitions can be found in a number of \nplaces, such as the aforementioned standard expositions.\n\n\n\\subsection*{Organization}\n\nThe rest of this paper is organized into five (very unequal) sections:\n\\begin{description}\n\\item[\\S\\ref{sect:crs-geom}] We define our basic framework of coarse \n    structures, coarse spaces, and coarse maps, together with important results \n    on local properness, and push-forward and pull-back coarse structures.\n\\item[\\S\\ref{sect:Crs}] We consider the precoarse categories (and $\\CATPCrs$ in \n    particular) and their properties; the arrows in these categories are actual \n    coarse maps. We discuss limits and colimits in these categories, as well as \n    the relation between the general category $\\CATPCrs$ and the subcategories \n    of unital and\/or connected coarse spaces.\n\\item[\\S\\ref{sect:Crs}] We discuss the relation of closeness on coarse maps, \n    establish basic properties of closeness, and consider the quotient coarse \n    categories ($\\CATCrs$ in particular). We show that $\\CATCrs$ has all \n    nonzero products and all equalizers, hence all nonzero limits. Similarly, \n    it has all coproducts and all coequalizers, hence all colimits. We define \n    the termination functor $\\Terminate$, and examine some of its properties; \n    in particular, it provides ``identities'' for the product. We characterize \n    the monic arrows of $\\CATCrs$ and show that $\\CATCrs$ has categorical \n    images, and dually we do the same for epi arrows and coimages. We apply \n    $\\Terminate$, together with push-outs, to define quotient coarse spaces. \n    Finally, we discuss ways to ``restrict'' $\\CATCrs$ to obtain subcategories \n    with terminal objects.\n\\item[\\S\\ref{sect:top-crs}] We examine Roe's formalization of coarse geometry, \n    which allows coarse spaces to carry topologies, and the relation \n    between the Roe coarse category and ours. In particular, we discuss how, \n    given a ``proper coarse space'' (in the sense of Roe), one can functorially \n    obtain a (discrete) coarse space (in our sense). We show that this gives a \n    fully faithful functor from the Roe coarse category to $\\CATCrs$.\n\\item[\\S\\ref{sect:ex-appl}] We give the basic examples of coarse spaces: those \n    which come from proper metric spaces, and those which come from \n    compactifications (i.e., continuously controlled coarse spaces). We define \n    corresponding metric and continuously controlled coarse simplices, and \n    indicate how one might then develop coarse simplicial theory.\n    We compare Emerson--Meyer's $\\sigma$-coarse spaces to our nonunital coarse \n    spaces (in the discrete case). Finally, we briefly examine the relation \n    between quotients of coarse spaces, the $K$-theory of Roe algebras, and \n    Kasparov $K$-homology.\n\\end{description}\n\n\n\\subsection*{Acknowledgements}\n\nThis work has been greatly influenced by many people, too many to enumerate. \nHowever, I would like to specifically thank my thesis advisor John Roe for his \nguidance over the years, as well as Heath Emerson and Nick Wright for helpful \ndiscussions. I would also like to thank Marcelo Laca, John Phillips, and Ian \nPutnam for their support at the University of Victoria.\n\n\n\n\n\\section{Foundations of coarse geometry}\\label{sect:crs-geom}\n\nThroughout this section, $X$, $Y$, and $Z$ will be sets (sometimes with extra \nstructure), and $f \\from Y \\to X$ and $g \\from Z \\to Y$ will be (set) maps. We \ndenote the restriction of $f$ to $T \\subseteq Y$ by $f |_T \\from T \\to X$. When \n$f(Y) \\subseteq S \\subseteq X$, we denote the range restriction of $f$ to $S$ \nby $f |^S \\from Y \\to S$. Thus if $T \\subseteq Y$ and $f(T) \\subseteq S \n\\subseteq X$, we have a restriction $f |_T^S \\from T \\to S$.\n\n\n\\subsection{\\pdfalt{\\maybeboldmath Operations on subsets of $X \\cross X$}%\n        {Operations on subsets of X x X}}\n\nMuch of the following can be developed in the more abstract context of \ngroupoids \\cite{MR1451755}, but we will refrain from doing so. The basic object \nin question is the pair groupoid $X^{\\cross 2} \\defeq X \\cross X$. Recall that \n$X^{\\cross 2}$ has object set $X$ and set of arrows $X \\cross X$. The map $X \n\\injto X^{\\cross 2}$ is $x \\mapsto (x,x) \\eqdef 1_x$ for $x \\in X$. The target \nand source maps are the projections $\\pi_1, \\pi_2 \\from X^{\\cross 2} \\to X$, \nrespectively. For $x,x',x'' \\in X$, composition is given by $(x,x') \\circ \n(x',x'') \\defeq (x,x'')$ and the inverse by $(x,x')^{-1} \\defeq (x',x)$. Any \nset map $f \\from Y \\to X$ induces a groupoid morphism\n\\[\n    f^{\\cross 2} \\defeq f \\cross f \\from Y^{\\cross 2} \\to X^{\\cross 2}\n\\]\nwhich in turn induces a map $\\powerset(Y^{\\cross 2}) \\to \\powerset(X^{\\cross \n2})$, again denoted $f^{\\cross 2}$, between power sets.\n\n\\begin{definition}[\\maybeboldmath operations on $\\powerset(X^{\\cross 2})$]\nFor $E, E' \\in \\powerset(X^{\\cross 2})$:\n\\begin{enumerate}\n\\item (\\emph{addition}) $E + E' \\defeq E \\union E'$;\n\\item (\\emph{multiplication}) $E \\circ E' \\defeq \\set{e \\circ e' \\suchthat \n    \\text{$e \\in E$, $e' \\in E'$, and $\\pi_2(e) = \\pi_1(e')$}}$; and\n\\item (\\emph{transpose}) $E^\\transpose \\defeq \\set{e^{-1} \\suchthat e \\in E}$.\n\\end{enumerate}\nFor $S \\subseteq X$, put $1_S \\defeq \\set{1_x \\suchthat x \\in S}$ (the \n\\emph{local unit} over $S$, or simply \\emph{unit} if $S = X$).\n\\end{definition}\n\n\\begin{proposition}\nFor all $E \\in \\powerset(X^{\\cross 2})$,\n\\[\n    E \\circ 1_S = (\\pi_2 |_E)^{-1}(S)\n\\quad\\text{and}\\quad\n    1_S \\circ E = (\\pi_1 |_E)^{-1}(S)\n\\]\n\\end{proposition}\n\n\\begin{remark}\nWe will refrain from calling $E \\circ E'$ a ``product'' to avoid confusion with \ncartesian\/categorical products (e.g., $X \\cross Y$). The transpose \n$E^\\transpose$ is often called the ``inverse'' and denoted $E^{-1}$; we avoid \nthis terminology and notation since it is somewhat deceptive (though, \nadmittedly, also rather suggestive). Our units $1_X$ are usually denoted \n$\\Delta_X$ (and called the diagonal, for obvious reasons); our terminology is \nmore representative of the ``algebraic'' role played by the unit (and the local \nunits) and avoids confusion with the (related) diagonal map $\\Delta_X \\from X \n\\to X \\cross X$ (where $X \\cross X$ is the cartesian\/categorical product).\n\\end{remark}\n\nThe operations of addition and composition make $\\powerset(X^{\\cross 2})$ into \na semiring: addition is commutative with identity $\\emptyset$, multiplication \nis associative with identity $1_X$, multiplication distributes over addition, \nand $\\emptyset \\circ E = \\emptyset = E \\circ \\emptyset$ for all $E$. Addition \nis idempotent in that $E + E = E$ for all $E$. Each $1_S$ is idempotent with \nrespect to multiplication, i.e., $1_S \\circ 1_S = 1_S$ for all $S$. The \ntranspose is involutive, i.e., $(E^\\transpose)^\\transpose = E$ for all $E$, \nand, moreover, $(E + E')^\\transpose = E^\\transpose + (E')^\\transpose$, $(E \n\\circ E')^\\transpose = (E')^\\transpose \\circ E^\\transpose$, and \n$(1_S)^\\transpose = 1_S$, for all $E$, $E'$, and $S$.\n\n\\begin{definition}[neighbourhoods and supports]\nFor any $S \\subseteq X$ and $E \\in \\powerset(X^{\\cross 2})$, put\n\\begin{align*}\n    E \\cdot S & \\defeq \\pi_1(E \\circ 1_S) = \\pi_1( (\\pi_2 |_E)^{-1}(S) )\n        && \\text{(\\emph{left $E$-neighbourhood of $S$})}\n\\shortintertext{and}\n    S \\cdot E & \\defeq \\pi_2(1_S \\circ E) = \\pi_2( (\\pi_1 |_E)^{-1}(S) )\n        && \\text{(\\emph{right $E$-neighbourhood of $S$})}.\n\\end{align*}\nWe also call $E \\cdot X = \\pi_1(E)$ the \\emph{left support} of $E$ and $X \\cdot \nE = \\pi_2(E)$ the \\emph{right support} of $E$.\n\\end{definition}\n\n\\begin{remark}\nThe notations $N_E(S) \\defeq E_S \\defeq E[S] \\defeq E \\cdot S$ and $E^S \\defeq \nS \\cdot E$ are common, though our notation is hopefully more suggestive of the \nrelation between $E \\cdot S$, $S \\cdot E$ and the previously defined \noperations.\n\\end{remark}\n\n\\begin{proposition\nFor all $E$, $E'$, and $S$:\n\\begin{align*}\n    (E + E') \\cdot S & = E \\cdot S \\union E' \\cdot S &\n    & \\text{and} &\n    S \\cdot (E + E') & = S \\cdot E \\union S \\cdot E'; \\\\\n    E \\circ 1_{E' \\cdot S} & = E \\circ E' \\circ 1_S &\n    & \\text{and} &\n    1_{S \\cdot E} \\circ E' & = 1_S \\circ E \\circ E'; \\\\\n    (E \\circ E') \\cdot S & = E \\cdot (E' \\cdot S) &\n    & \\text{and} &\n    S \\cdot (E \\circ E') & = (S \\cdot E) \\cdot E'; \\quad\\text{and} \\\\\n    E^\\transpose \\cdot S & = S \\cdot E &\n    & \\text{and} &\n    S \\cdot E^\\transpose & = E \\cdot S.\n\\end{align*}\n\\end{proposition}\n\n$\\powerset(X^{\\cross 2})$ and $\\powerset(X)$ are partially ordered by \ninclusion. All of the above ``operations'' are monotonic with respect to these \npartial orders.\n\n\\begin{proposition\nIf $E_1, E_2, E'_1, E'_2 \\in \\powerset(X^{\\cross 2})$ with $E_1 \\subseteq E_2$ \nand $E'_1 \\subseteq E'_2$, and $S_1, S_2 \\subseteq X$ with $S_1 \\subseteq S_2$, \nthen:\n\\begin{align*}\n    E_1 + E'_1 & \\subseteq E_2 + E'_2, & &&\n    E_1 \\circ E'_1 & \\subseteq E_2 \\circ E'_2, \\\\\n    (E_1)^\\transpose & \\subseteq (E_2)^\\transpose, & &&\n    1_{S_1} & \\subseteq 1_{S_2}, \\\\\n    E_1 \\cdot S_1 & \\subseteq E_2 \\cdot S_2, &\n    & \\quad\\text{and} &\n    S_1 \\cdot E_1 & \\subseteq S_2 \\cdot E_2.\n\\end{align*}\n\\end{proposition}\n\n\n\\subsection{Discrete properness}\n\nSince our coarse spaces are essentially discrete, for now we only discuss \nproperness for maps between discrete sets.\n\n\\begin{definition}\\label{def:proper}\nA set map $f \\from Y \\to X$ is \\emph{proper} if the inverse image $f^{-1}(K)$ \nof every finite subset $K \\subseteq X$ is again finite.\n\\end{definition}\n\nIf $Y$ is itself a finite set, then any $f \\from Y \\to X$ is automatically \nproper. We will use the following facts extensively (compare \n\\cite{MR979294}*{\\S{}10.1 Prop.~5}).\n\n\\begin{proposition}\\label{prop:prop}\nConsider the composition of (set) maps $Z \\nameto{\\smash{g}} Y \n\\nameto{\\smash{f}} X$:\n\\begin{enumerate}\n\\item\\label{prop:prop:I} If $f$ and $g$ are proper, then $f \\circ g$ is proper.\n\\item\\label{prop:prop:II} If $f \\circ g$ is proper, then $g$ is proper.\n\\item\\label{prop:prop:III} If $f \\circ g$ is proper and $g$ is surjective, then \n    $f$ (and $g$) are proper.\n\\end{enumerate}\nNote that injectivity implies properness.\n\\end{proposition}\n\nIn \\enumref{prop:prop:III} above, the hypothesis that $g$ be surjective can be \nweakened to the requirement that $Y \\setminus g(Z)$ be a finite set. All \nrestrictions (including range restrictions) of proper maps are again proper.\n\n\n\\subsection{The properness axiom and coarse structures}\n\n\\begin{definition}\\label{def:prop-ax}\nA set $E \\in \\powerset(X^{\\cross 2})$ satisfies the \\emph{properness axiom} if \nthe restricted target and source maps (i.e., projections) $\\pi_1 |_E, \\pi_2 |_E \n\\from E \\to X$ (or, also restricting the ranges, $\\pi_1 |_E^{E \\cdot X}$, \n$\\pi_2 |_E^{X \\cdot E}$) are proper set maps.\n\\end{definition}\n\n\\begin{proposition}\\label{prop:prop-ax}\nFor $E \\in \\powerset(X^{\\cross 2})$, the following are equivalent:\n\\begin{enumerate}\n\\item\\label{prop:prop-ax:I} $E$ satisfies the properness axiom;\n\\item\\label{prop:prop-ax:II} $E \\circ 1_S$ and $1_S \\circ E$ are finite for all \n    finite $S \\subseteq X$; and\n\\item\\label{prop:prop-ax:III} $E \\circ E'$ and $E' \\circ E$ are finite for all \n    finite $E' \\in \\powerset(X^{\\cross 2})$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\\enumref{prop:prop-ax:I} $\\iff$ \\enumref{prop:prop-ax:II}: Immediate from $E \n\\circ 1_S = (\\pi_2 |_E)^{-1}(S)$ (and symmetrically).\n\n\\enumref{prop:prop-ax:II} $\\iff$ \\enumref{prop:prop-ax:III}: The reverse \nimplication is clear. For the forward implication, if $E'$ is finite then $E' \n\\cdot X$ is finite, and hence so too is\n\\[\n    E \\circ E' = E \\circ E' \\circ 1_X = E \\circ 1_{E' \\cdot X}\n\\]\n(and symmetrically).\n\\end{proof}\n\n\\begin{corollary}\\label{cor:prop-ax}\nIf $E \\in \\powerset(X^{\\cross 2})$ satisfies the properness axiom, then $E \n\\cdot S$ and $S \\cdot E$ are finite for all finite $S \\subseteq X$.\n\\end{corollary}\n\n\\begin{proof}\nUse $E \\cdot S \\defeq \\pi_1(E \\circ 1_S)$ (and similarly symmetrically).\n\\end{proof}\n\n\\begin{remark}\nThe converse of the above Corollary holds since we are only considering pair \ngroupoids: observe that\n\\[\n    (\\pi_1 |_E)^{-1}(S) \\subseteq S \\cross S \\cdot E\n\\]\n(and similarly symmetrically). However, the converse does not hold in general \nfor coarse structures on groupoids.\n\\end{remark}\n\n\\begin{proposition}[``algebraic'' operations and the properness axiom]%\n        \\label{prop:prop-ax-alg}\nIf $E, E' \\in \\powerset(X^{\\cross 2})$ satisfy the properness axiom, then $E + \nE'$, $E \\circ E'$, $E^\\transpose$, and all subsets of $E$ satisfy the \nproperness axiom. Also, all singletons $\\set{e}$, $e \\in X^{\\cross 2}$, and \nhence all finite subsets of $X^{\\cross 2}$ satisfy the properness axiom, as \ndoes the unit $1_X$.\n\\end{proposition}\n\n\\begin{proof}\nClear, except possibly for $E \\circ E'$; for this, use \nProposition~\\ref{prop:prop-ax}\\enumref{prop:prop-ax:III} (and associativity of \nmultiplication).\n\\end{proof}\n\nIf $T$, $T'$ are matrices over $X^{\\cross 2}$ (with values in some ring) are \nsupported on $E, E' \\in \\powerset(X^{\\cross 2})$ satisfying the properness \naxiom, then the product $TT'$ is defined and has support contained in $E \\circ \nE'$. The passage to rings of matrices motivates the following.\n\n\\begin{definition}\\label{def:crs-sp}\nA \\emph{coarse structure} on $X$ is a subset $\\calE_X \\subseteq\n\\powerset(X^{\\cross 2})$ such that:\n\\begin{enumerate}\n\\item each $E \\in \\calE_X$ satisfies the properness axiom;\n\\item $\\calE_X$ is closed under the operations of addition, multiplication, \n    transpose, and the taking of subsets (i.e., if $E \\subseteq E'$ and $E' \\in \n    \\calE_X$, then $E \\in \\calE_X$); and\n\\item for all $x \\in X$, the singleton $\\set{1_x}$ is in $\\calE_X$.\n\\end{enumerate}\nA \\emph{coarse space} is a set $X$ equipped with a coarse structure $\\calE_X$ \non $X$. We denote such a coarse space by $|X|_{\\calE_X}$ or simply $X$. The \nelements of $\\calE_X$ are called \\emph{entourages} (of $\\calE_X$ or of $X$).\n\\end{definition}\n\n\\begin{example}[finite sets]\nIf $X$ is a finite set, then any coarse structure on $X$ must be unital. \nMoreover, there is only one connected coarse structure on $X$, namely the power \nset of $X$.\n\\end{example}\n\nHere are two natural coarse structures which exist on any set.\n\n\\begin{definition}\nThe \\emph{initial coarse structure} $\\calE_{|X|_0}$ on $X$ is the minimum \ncoarse structure on $X$. The \\emph{terminal coarse structure} $\\calE_{|X|_1}$ \non a set $X$ is the maximum coarse structure. (We denote the corresponding \ncoarse spaces by $|X|_0$ and $|X|_1$, respectively.)\n\\end{definition}\n\nBy Proposition~\\ref{prop:prop-ax-alg}, $\\calE_{|X|_1}$ simply consists of all \nthe $E \\in \\powerset(X^{\\cross 2})$ which satisfy the properness axiom. (Thus \n``$E \\in \\calE_{|X|_1}$'' is a convenient abbreviation for ``$E \\in \n\\powerset(X^{\\cross 2})$ satisfies the properness axiom''.) Any coarse \nstructure on $X$ is a subset of the terminal coarse structure (and obviously \ncontains the initial coarse structure). More generally, we have the following.\n\n\\begin{proposition}\nThe intersection of any collection of coarse structures on $X$ (possibly \ninfinite) is again a coarse structure on $X$.\n\\end{proposition}\n\n\\begin{definition}\nThe coarse structure $\\langle \\calE' \\rangle_X$ on $X$ \\emph{generated} by a \nsubset $\\calE' \\subseteq \\calE_{|X|_1}$ is the minimum coarse structure on $X$ \nwhich contains $\\calE'$.\n\\end{definition}\n\nOf course, $\\langle \\calE' \\rangle_X$  is just the intersection of all the \ncoarse structures on $X$ containing $\\calE'$. Note that $\\calE_{|X|_0} = \n\\langle \\emptyset \\rangle_X$; more concretely, $\\calE_{|X|_0}$ consists of all \nthe finite local units $1_S$, $S \\subseteq X$ finite.\n\nGiven two subsets $\\calE', \\calE'' \\subseteq \\calE_{|X|_1}$ (e.g., coarse \nstructures on $X$), denote\n\\[\n    \\langle \\calE', \\calE'' \\rangle_X\n        \\defeq \\langle \\calE' \\union \\calE'' \\rangle_X.\n\\]\nObserve that $\\langle \\calE', \\calE'' \\rangle_X$ contains both $\\langle \\calE' \n\\rangle_X$ and $\\langle \\calE'' \\rangle_X$. We use similar notation given three \nor more subsets of $\\calE_{|X|_1}$ and, more generally, if $\\set{\\calE'_j \n\\suchthat j \\in J}$ ($J$ some index set) is a collection of subsets of \n$\\calE_{|X|_1}$,\n\\[\n    \\langle \\calE'_j \\suchthat j \\in J \\rangle_X\n        \\defeq \\bigl\\langle \\textstyle\\bigunion_{j \\in J} \\calE'_j\n                    \\bigr\\rangle_X.\n\\]\n\nOne can describe the coarse structure generated by $\\calE'$ rather more \nconcretely.\n\n\\begin{proposition}\nIf $\\calE' \\subseteq \\calE_{|X|_1}$ contains all the singletons $\\set{1_x}$, $x \n\\in X$, and is closed under the ``algebraic'' operations of addition, \nmultiplication, and transpose, then\n\\[\n    \\langle \\calE' \\rangle_X = \\set{E \\subseteq E' \\suchthat E' \\in \\calE'}.\n\\]\n\\end{proposition}\n\n\\begin{corollary}\\label{cor:crs-struct-gen}\nFor any $\\calE' \\subseteq \\calE_{|X|_1}$, $\\langle \\calE' \\rangle_X$ consists \nof the all subsets of the ``algebraic closure'' of the union\n\\[\n    \\calE' \\union \\set{\\set{1_x} \\suchthat x \\in X}.\n\\]\n\\end{corollary}\n\nSubsets of coarse spaces are naturally coarse spaces.\n\n\\begin{definition}\nSuppose $X$ is a coarse space and $X' \\subseteq X$ is a subset. Then\n\\[\n    \\calE_{X'} \\defeq \\calE_X |_{X'}\n        \\defeq \\calE_X \\intersect \\powerset((X')^{\\cross 2})\n\\]\nis a coarse structure on $X'$, called the \\emph{subspace coarse structure}. \nCall $X' \\subseteq X$ equipped with the subspace coarse structure a (coarse) \n\\emph{subspace} of $X$.\n\\end{definition}\n\n\\begin{example}[discrete metric spaces]\\label{ex:disc-met}\nLet $(X,d)$ be a discrete, proper metric space. ($X$ is metrically \n\\emph{proper} if closed balls of $X$ are compact; thus $X$ is discrete and \nproper if and only if every metrically bounded subset is finite.) The \n\\emph{($d$-)metric coarse space} $|X|_d$ (or just $|X|$ for short) has as \nentourages the $E \\in \\calE_{|X|_1} \\subseteq \\powerset(X^{\\cross 2})$ such \nthat\n\\begin{equation}\\label{ex:disc-met:eq}\n    \\sup \\set{d(x,x') \\suchthat (x,x') \\in E} < \\infty.\n\\end{equation}\nWe may also allow $d(x,x') = \\infty$ (for $x \\neq x'$). In the senses defined \nbelow, $|X|_d$ is always unital but is connected if and only if $d(x,x') < \n\\infty$ always. If $X' \\subseteq X$, then the metric coarse structure on $X'$ \nobtained from the restriction of the metric $d$ is just the subspace coarse \nstructure $\\calE_{|X|_d} |_{X'}$.\n\\end{example}\n\n\n\\subsection{Unitality and connectedness}\n\n\\begin{definition}\\label{def:uni-conn}\nA coarse structure $\\calE_X$ on $X$ is \\emph{unital} if $1_X \\in \\calE_X$. \n$\\calE_X$ is \\emph{connected} if every singleton $\\set{e}$, $e \\in X^{\\cross \n2}$, is an entourage of $\\calE_X$. A pair of points $x, x' \\in X$ are \n\\emph{connected} (with respect to $\\calE_X$, or in the coarse space $X$) if \n$\\set{(x,x')} \\in \\calE_X$.\n\\end{definition}\n\nMost treatments of coarse geometry assume both unitality and connectedness, but \nwe will assume neither. Connectedness is a relatively benign assumption (see \n\\S\\ref{subsect:PCrs-conn}), but \\emph{not} assuming unitality will be \nparticularly crucial.\n\n\\begin{remark}\\label{rmk:gpd-conn}\nConnectedness in the general coarse groupoid case is more complicated, since \nthere may be multiple arrows having the same target and source, and since a \ngroupoid itself may not be connected. Let $\\calE_\\calG$ be a coarse structure \non a groupoid $\\calG$. There are several possible notions of connectedness:\n\\begin{enumerate}\n\\item The obvious translation of the above to groupoids is to say that \n    $\\calE_\\calG$ is (locally) \\emph{connected} if all singletons $\\set{e}$ \n    ($e$ an arrow in the groupoid) are entourages of $\\calE_\\calG$.\n\\item $\\calE_\\calG$ is \\emph{globally connected} if it is (locally) connected \n    and $\\calG$ is connected as a groupoid.\n\\setcounter{tempcounter}{\\value{enumi}}\n\\end{enumerate}\nObjects $x$, $x'$ are \\emph{connected} if \\emph{all} arrows $e$ with target $x$ \nand source $x'$ yield entourages $\\set{e}$. Then $\\calE_\\calG$ is (locally) \nconnected if and only if \\emph{all groupoid-connected} pairs of objects are \nconnected, and globally connected if and only if \\emph{all} pairs of objects \nare connected. But there is also a weaker notion of connectedness: $x$, $x'$ \nare \\emph{weakly connected} if there is \\emph{some} arrow $e$ with target $x$ \nand source $x'$ such that $\\set{e}$ is an entourage.\n\\begin{enumerate}\n\\setcounter{enumi}{\\value{tempcounter}}\n\\item $\\calE_\\calG$ is (locally) \\emph{weakly connected} if all \n    groupoid-connected objects $x$, $x'$ are weakly connected.\n\\item $\\calE_\\calG$ is \\emph{globally weakly connected} if it is (locally) \n    weakly connected and $\\calG$ is connected as a groupoid.\n\\end{enumerate}\nWhen $\\calG$ is a pair groupoid (i.e., in our case), all the above notions \ncoincide.\n\\end{remark}\n\n\\begin{proposition}\nThe terminal structure on any set $X$ is always unital and connected.\n\\end{proposition}\n\nThe intersection of unital coarse structures on a set $X$ is again unital, and \nsimilarly for connected coarse structures. Thus, for any $\\calE' \\subseteq \n\\calE_{|X|_1}$, there are \\emph{unital}, \\emph{connected}, and \\emph{connected \nunital} coarse structures on $X$ generated by $\\calE'$. These can be described \nrather simply as\n\\begin{align*}\n    \\langle \\calE' \\rangle_X^\\TXTuni\n        & \\defeq \\bigl\\langle \\calE', \\set{1_X} \\bigr\\rangle_X, \\\\\n    \\langle \\calE' \\rangle_X^\\TXTconn\n        & \\defeq \\bigl\\langle \\calE', \\set{\\set{e}\n                \\suchthat e \\in X^{\\cross 2}} \\bigr\\rangle_X,\n\\quad\\text{and} \\\\\n    \\langle \\calE' \\rangle_X^\\TXTconnuni\n        & \\defeq \\bigl\\langle \\calE', \\set{\\set{e}\n                \\suchthat e \\in X^{\\cross 2}}, \\set{1_X} \\bigr\\rangle_X,\n\\end{align*}\nrespectively.\n\n\\begin{definition}\nThe \\emph{initial unital}, \\emph{initial connected}, or \\emph{initial connected \nunital coarse structure} on a set $X$ is the minimum coarse structure having \nthe given property or properties, respectively. Denote the resulting coarse \nspaces by $|X|_0^\\TXTuni$, $|X|_0^\\TXTconn$, and $|X|_0^\\TXTconnuni$, \nrespectively.\n\\end{definition}\n\nClearly, $\\calE_{|X|_0^\\TXTuni} = \\langle \\set{1_X} \\rangle_X$, so a coarse \nstructure on $X$ is unital if and only if it contains $\\calE_{|X|_0^\\TXTuni}$. \nSimilarly for the other properties. Note in particular that \n$\\calE_{|X|_0^\\TXTconn}$ consists of all the finite subsets of $X^{\\cross 2}$.\n\n\\begin{remark}\nIn the groupoid case, the intersection of (locally) connected coarse structures \non a given groupoid (in the sense of Remark~\\ref{rmk:gpd-conn}) is again \n(locally) connected, and so all of the above holds. However, the intersection \nof weakly connected coarse structures (see the same Remark) may not be weakly \nconnected, so there may not be a minimum weakly connected coarse structure on a \ngiven groupoid.\n\\end{remark}\n\nWe get an obvious notion of \\emph{unital subspace} of any coarse space $X$. \nClearly, $X' \\subseteq X$ is a unital subspace if and only if $1_{X'}$ is an \nentourage of $X$. (Bartels calls the set of unital subspaces of $X$ the \n``domain of $\\calE_X$'' \\cite{MR1988817}*{Def.~3.2}.) Slightly more is true.\n\n\\begin{proposition}\\label{prop:uni-subsp}\nA subspace $X' \\subseteq X$ of a coarse space $X$ is unital if and only if it \noccurs as the left (or right) support of some entourage of $X$.\n\\end{proposition}\n\n\\begin{proof}\nIf $X'$ is a unital subspace, then $X' = 1_{X'} \\cdot X$. Conversely, if $X' = \nE \\cdot X$ for some $E \\in \\calE_X$, then $1_{X'} \\subseteq E \\circ \nE^\\transpose$ must be an entourage of $X$.\n\\end{proof}\n\nSimilarly, we get a notion of \\emph{connected subspace} of $X$.\n\n\\begin{definition}\nA (connected) \\emph{component} of a coarse space $X$ is a maximal connected \nsubspace of $X$.\n\\end{definition}\n\n\\begin{proposition}\nAny coarse space $X$ is partitioned, as a set, into (a disjoint union of) its \nconnected components.\n\\end{proposition}\n\nWe caution this decomposition is not necessarily a coproduct (in the coarse or \nprecoarse category); see Corollary~\\ref{cor:PCrs-fin-components}.\n\n\n\\subsection{Local properness, preservation, and coarse maps}\n\nRecall that any (set) map $f \\from Y \\to X$ induces a map (indeed, a groupoid \nmorphism) $f^{\\cross 2} \\from Y^{\\cross 2} \\to X^{\\cross 2}$. Insisting that \n$f$ be proper is too strong a requirement when $Y$ is a nonunital coarse space. \nWe thus introduce the following weaker requirement.\n\n\\begin{definition}\\label{def:loc-prop}\nA map $f \\from Y \\to X$ is \\emph{locally proper for $F \\in \\calE_{|Y|_1}$} if \n$E \\defeq f^{\\cross 2}(F) \\in \\calE_{|X|_1}$ and the restriction $f^{\\cross 2} \n|_F \\from F \\to X^{\\cross 2}$ (or $f^{\\cross 2} |_F^E$) is a proper (set) map. \nIf $Y$ is a coarse space, then $f$ is \\emph{locally proper} (for $\\calE_Y$) if \nit is locally proper for all $F \\in \\calE_Y$.\n\\end{definition}\n\nLocal properness only requires a coarse structure on the domain, so we cannot \nsay that the composition of locally proper maps is again locally proper. \nNonetheless, separating local properness from the following will be useful when \nwe define push-forward coarse structures (below).\n\n\\begin{definition}\nSuppose $X$ is a coarse space. A map $f \\from Y \\to X$ \\emph{preserves $F \\in \n\\calE_{|Y|_1}$} (with respect to $\\calE_X$) if $E \\defeq f^{\\cross 2}(F) \\in \n\\calE_X$. If $Y$ is also a coarse space, then $f$ \\emph{preserves entourages} \n(of $\\calE_Y$, with respect to $\\calE_X$) if $f$ preserves every $F \\in \n\\calE_Y$.\n\\end{definition}\n\n\\begin{definition}\nSuppose $X$ is a coarse space. A map $f \\from Y \\to X$ is \\emph{coarse for $F \n\\in \\calE_{|Y|_1}$} if $f$ is locally proper for $F$ and if $f$ preserves $F$. \nIf $Y$ is also a coarse space, then $f$ is \\emph{coarse} (or is a \\emph{coarse \nmap}) if $f$ is coarse for every $F \\in \\calE_Y$, i.e., if $f$ is locally \nproper and preserves entourages.\n\\end{definition}\n\n\\begin{remark}\nThe definition of ``coarse map'' is slightly redundant: If $f$ preserves \nentourages, then $f^{\\cross 2}(F) \\in \\calE_X \\subseteq \\calE_{|X|_1}$ (which \nis one of the stipulations of local properness).\n\\end{remark}\n\n\\begin{proposition}\\label{prop:crs-map-comp}\nConsider a composition of $Z \\nameto{\\smash{g}} Y \\nameto{\\smash{f}} X$, where \n$X$ and $Y$ are coarse spaces. If $f$, $g$ are locally proper and $g$ preserves \nentourages, then $f \\circ g$ is locally proper.\n\\end{proposition}\n\n\\begin{corollary}\nA composition of coarse maps is again a coarse map.\n\\end{corollary}\n\n\n\\subsection{Basic properties of maps}\n\nWe first concentrate on local properness.\n\n\\begin{proposition}\\label{prop:loc-prop}\nSuppose $f \\from Y \\to X$ is a set map, $F \\in \\calE_{|Y|_1}$, and $E \\defeq \nf^{\\cross 2}(F)$. The following are equivalent:\n\\begin{enumerate}\n\\item\\label{prop:loc-prop:I} $f$ is locally proper for $F$;\n\\item\\label{prop:loc-prop:II} the restrictions $f |_{F \\cdot Y}$ and $f |_{Y \n    \\cdot F}$ (or $f |_{F \\cdot Y}^{E \\cdot X}$ and $f |_{Y \\cdot F}^{X \\cdot \n    E}$) of $f$ to the left and right supports of $F$ are proper; and\n\\item\\label{prop:loc-prop:III} $f^{-1}(S) \\cdot F$ and $F \\cdot f^{-1}(S)$ are \n    finite for all finite $S \\subseteq X$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n(We omit proofs of the symmetric cases.) Consider the diagram\n\\[\\begin{CD}\n    F @>{f^{\\cross 2} |_F^E}>> E \\\\\n    @V{\\pi_1 |_F^{F \\cdot Y}}VV @V{\\pi_1 |_E^{E \\cdot X}}VV \\\\\n    F \\cdot Y @>{f |_{F \\cdot Y}^{E \\cdot X}}>> E \\cdot X\n\\end{CD}\\quad.\\]\nObserve the following: the above diagram commutes, i.e.,\n\\[\n    \\pi_1 |_E^{E \\cdot X} \\circ f^{\\cross 2} |_F^E\n        = f |_{F \\cdot Y}^{E \\cdot X} \\circ \\pi_1 |_F^{F \\cdot Y};\n\\]\nthe two maps emanating from $F$ are surjections; and $\\pi_1 |_F^{F \\cdot Y}$ is \nproper. We now apply Proposition~\\ref{prop:prop} several times.\n\n\\enumref{prop:loc-prop:I} \\textimplies{} \\enumref{prop:loc-prop:II}: $f^{\\cross \n2} |_F^E$ and $\\pi_1 |_E^{E \\cdot X}$ are proper, so their composition is \nproper. Since $\\pi_1 |_F^{F \\cdot Y}$ is surjective, $f |_{F \\cdot Y}$ is \nproper.\n\n\\enumref{prop:loc-prop:II} \\textimplies{} \\enumref{prop:loc-prop:III}: $f |_{F \n\\cdot Y}$ and $\\pi_1 |_F^{F \\cdot Y}$ are proper, so their composition is \nproper. Then\n\\begin{equation}\\label{prop:loc-prop:pf:eq}\\begin{split}\n    f^{-1}(S) \\cdot F & = \\pi_2( (\\pi_1 |_F)^{-1}(f^{-1}(S)) ) \\\\\n        & = \\pi_2( (f_{F \\cdot Y} \\circ \\pi_1 |_F^{F \\cdot Y})^{-1}(S) )\n\\end{split}\\end{equation}\nis finite if $S \\subseteq X$ is finite.\n\n\\enumref{prop:loc-prop:III} \\textimplies{} \\enumref{prop:loc-prop:I}: By \n\\eqref{prop:loc-prop:pf:eq} and since $\\pi_2 |_F$ is proper, the composition \n$f_{F \\cdot Y}^{E \\cdot X} \\circ \\pi_1 |_F^{F \\cdot Y}$ is proper. Hence \n$f^{\\cross 2} |_F^E$ is proper and, since $f^{\\cross 2} |_F^E$ is surjective, \nso is $\\pi_1 |_E^{E \\cdot X}$.\n\\end{proof}\n\n\\begin{corollary}\nIf a set map $f \\from Y \\to X$ is globally proper, then $f$ is locally proper \nfor any $F \\in \\calE_{|Y|_1}$ (so $f$ is locally proper for any coarse \nstructure on $Y$).\n\\end{corollary}\n\n\\begin{proof}\nThis follows from \\enumref{prop:loc-prop:III} and Corollary~\\ref{cor:prop-ax}.\n\\end{proof}\n\n\\begin{corollary}\nIf $X$ is a coarse space and $X' \\subseteq X$ is a subspace, then the inclusion \nof $X'$ into $X$ is a coarse map. Thus the restriction of any coarse map to a \nsubspace is a coarse map.\n\\end{corollary}\n\n\\begin{proof}\nBy definition of the subspace coarse structure, the inclusion map preserves \nentourages. The inclusion map is injective, hence (globally) proper, hence \nlocally proper.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:loc-prop-uni}\nSuppose $Y$ is a coarse space. A map $f \\from Y \\to X$ is locally proper if and \nonly if the restriction of $f$ to every unital subspace of $Y$ is proper. Thus, \nfor $Y$ unital, $f$ is locally proper if and only if $f$ is globally proper.\n\\end{corollary}\n\n\\begin{proof}\nThis follows from \\enumref{prop:loc-prop:II} and \nProposition~\\ref{prop:uni-subsp}.\n\\end{proof}\n\nFor (discrete) unital coarse spaces, our notion of ``coarse map'' is just the \nclassical notion. It also follows that local properness of a map $f \\from Y \\to \nX$ is a property which can be defined in terms of the unital subspaces of the \ncoarse structure on $Y$. In particular, if $f$ is locally proper, then $f$ \nwould also be locally proper for any coarse structure on $Y$ (possibly larger \nthan $\\calE_Y$) with the same unital subspaces.\n\n\\begin{remark}\nOne may take the \\emph{definition} of local properness to be the \ncharacterization of the above Corollary, i.e., define $f \\from Y \\to X$ to be \nlocally proper if $f$ is proper on every unital subspace of $Y$ (perhaps \n``unital properness'' would be a more apt term). Indeed, this is the form in \nwhich local properness appears in Bartels's definition of ``coarse map'' \n\\cite{MR1988817}*{Def.~3.3}, and hence (modulo our coarse spaces not carrying \ntopologies) our definition of ``coarse map'' is the same as Bartels's. More \ngenerally, one could remove coarse structures entirely, and define local \nproperness for sets equipped with families of supports (i.e., of unital \nsubspaces). However, we will not do so since we are mainly concerned with \ncoarse maps, for which Definition~\\ref{def:loc-prop} is most convenient.\n\\end{remark}\n\n\\begin{corollary}\nCoarse maps send unital subspaces to unital subspaces, i.e., if $f \\from Y \\to \nX$ is a coarse map and $Y' \\subseteq Y$ is a unital subspace, then the image \n$f(Y') \\subseteq X$ is a unital subspace.\n\\end{corollary}\n\n\\begin{proposition}[``algebraic'' operations and local properness]%\n        \\label{prop:loc-prop-alg}\nIf $f \\from Y \\to X$ is locally proper for $F, F' \\in \\calE_{|Y|_1}$, then $f$ \nis locally proper for $F + F'$, $F \\circ F'$, $F^\\transpose$, and all subsets \nof $F$. Also, $f$ is locally proper for all singletons $\\set{e}$, $e \\in \nY^{\\cross 2}$, hence is locally proper for $\\calE_{|Y|_0^\\TXTconn} \\supseteq \n\\calE_{|Y|_0}$. (However, $f$ is locally proper for the unit $1_Y$ if and only \nif $f$ is globally proper.)\n\\end{proposition}\n\n\\begin{proof}\nThe only nontrivial assertion is that $f$ is locally proper for $F \\circ F'$. \nBy assumption, $f^{\\cross 2}(F), f^{\\cross 2}(F') \\in \\calE_{|X|_1}$ and, since\n\\[\n    f^{\\cross 2}(F \\circ F') \\subseteq f^{\\cross 2}(F) \\circ f^{\\cross 2}(F'),\n\\]\n$f^{\\cross 2}(F \\circ F')$ also satisfies the properness axiom, by \nProposition~\\ref{prop:prop-ax-alg}. We have a commutative diagram\n\\[\\begin{CD}\n    F \\circ F' @>{f^{\\cross 2} |_{F \\circ F'}}>> X^{\\cross 2} \\\\\n    @V{\\pi_1 |_{F \\circ F'}^{(F \\circ F') \\cdot Y}}VV @V{\\pi_1}VV \\\\\n    (F \\circ F') \\cdot Y @>{f |_{(F \\circ F') \\cdot Y}}>> X\n\\end{CD}\\quad.\\]\nBy the same Proposition, $F \\circ F' \\in \\calE_{|Y|_1}$, so $\\pi_1 |_{F \\circ \nF'}^{(F \\circ F') \\cdot Y}$ is proper. Since $(F \\circ F') \\cdot Y \\subseteq F \n\\cdot Y$ and $f |_{F \\cdot Y}$ is proper by \nProposition~\\ref{prop:loc-prop}\\enumref{prop:loc-prop:II}, $f |_{(F \\circ F') \n\\cdot Y}$ is proper. Hence the composition\n\\[\n    f |_{(F \\circ F') \\cdot Y} \\circ \\pi_1 |_{F \\circ F'}^{(F \\circ F') \\cdot Y}\n        = \\pi_1 \\circ f^{\\cross 2} |_{F \\circ F'}\n\\]\nis proper, so $f^{\\cross 2} |_{F \\circ F'}$ is proper by \nProposition~\\ref{prop:prop}\\enumref{prop:prop:II}.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:loc-prop-gen}\nIf $f \\from Y \\to X$ is locally proper for all $F \\in \\calE' \\subseteq \n\\calE_{|Y|_1}$, then $f$ is locally proper for the coarse structure $\\langle \n\\calE' \\rangle_Y$ on $Y$ generated by $\\calE'$ (and for the connected coarse \nstructure $\\langle \\calE' \\rangle_Y^\\TXTconn$ generated by $\\calE'$).\n\\end{corollary}\n\n\\begin{proof}\nThis follows immediately from the Proposition and \nCorollary~\\ref{cor:crs-struct-gen}.\n\\end{proof}\n\nThe same evidently does not hold for the unital (or connected unital) coarse \nstructure generated by $\\calE'$.\n\nWe now state some parallel results for preservation of entourages. Combining \nthese with the above results for local properness, we get parallel results for \ncoarseness of maps.\n\n\\begin{proposition}\\label{``algebraic'' operations and preservation}%\n       \nSuppose $X$ is a coarse space. If $f \\from Y \\to X$ preserves $F, F' \\in \n\\calE_{|Y|_1}$, then $f$ preserves $F + F'$, $F \\circ F'$, $F^\\transpose$, and \nall subsets of $F$. Also, $f$ preserves all singletons $\\set{1_y}$, $y \\in Y$ \n(hence preserves $\\calE_{|Y|_0}$); if $X$ is connected, $f$ preserves all \nsingletons $\\set{e}$, $e \\in Y^{\\cross 2}$ (hence preserves \n$\\calE_{|Y|_0^\\TXTconn}$); and if $X$ is unital, $f$ preserves $1_Y$.\n\\end{proposition}\n\n\\begin{proof}\nThe only (slightly) nontrivial one is $F \\circ F'$, for which ones uses\n\\[\n    f^{\\cross 2}(F \\circ F') \\subseteq f^{\\cross 2}(F) \\circ f^{\\cross 2}(F').\n\\]\n\\end{proof}\n\n\\begin{corollary\nSuppose $X$ is a coarse space. If $f \\from Y \\to X$ preserves $\\calE' \\subseteq \n\\calE_{|Y|_1}$, then $f$ preserves the coarse structure $\\langle \\calE' \n\\rangle_Y$ on $Y$ generated by $\\calE'$. (If $X$ is also connected, then $f$ \npreserves $\\langle \\calE' \\rangle_Y^\\TXTconn$; if $X$ is unital, then $f$ \npreserves $\\langle \\calE' \\rangle_Y^\\TXTuni$; if $X$ is both, then $f$ \npreserves $\\langle \\calE' \\rangle_Y^\\TXTconnuni$.)\n\\end{corollary}\n\n\\begin{proposition}[``algebraic'' operations and coarseness]%\n        \\label{prop:coarse-alg}\nSuppose $X$ is a coarse space. If $f \\from Y \\to X$ is coarse for $F, F' \\in \n\\calE_{|Y|_1}$, then $f$ is coarse for $F + F'$, $F \\circ F'$, $F^\\transpose$, \nand all subsets of $F$. Also, $f$ is coarse for all singletons $\\set{1_y}$, $y \n\\in Y$ (hence is coarse for $\\calE_{|Y|_0}$); if $X$ is connected, $f$ is \ncoarse for all singletons $\\set{e}$, $e \\in Y^{\\cross 2}$ (hence is coarse for \n$\\calE_{|Y|_0^\\TXTconn}$); and if $X$ is unital and $f$ is proper, $f$ is \ncoarse for $1_Y$.\n\\end{proposition}\n\n\\begin{corollary}\\label{cor:coarse-gen}\nSuppose $X$ and $Y$ are coarse spaces, $\\calE' \\subseteq \\calE_{|Y|_1}$, and $f \n\\from Y \\to X$ is a set map.\n\\begin{enumerate}\n\\item If $\\calE_Y = \\langle \\calE' \\rangle_Y$, then $f$ is a coarse map if and \n    only if $f$ is coarse for all $F \\in \\calE'$.\n\\item If $\\calE_Y = \\langle \\calE' \\rangle_Y^\\TXTconn$, then $f$ is a coarse \n    map if and only if $f$ is coarse for all $F \\in \\calE'$ and all $\\set{e}$, \n    $e \\in Y^{\\cross 2}$.\n\\item If $\\calE_Y = \\langle \\calE' \\rangle_Y^\\TXTuni$, then $f$ is a coarse map \n    if and only if $f$ is proper and $f$ is coarse for (or preserves) all $F \n    \\in \\calE'$.\n\\item If $\\calE_Y = \\langle \\calE' \\rangle_Y^\\TXTconnuni$, then $f$ is a coarse \n    map if and only if $f$ is proper and $f$ is coarse for (or preserves) all \n    $F \\in \\calE'$ and all $\\set{e}$, $e \\in Y^{\\cross 2}$.\n\\end{enumerate}\nNote that requiring that $f$ be coarse for all $\\set{e}$, $e \\in Y^{\\cross 2}$, \nis equivalent to requiring $f(y)$, $f(y')$ be connected for all $y, y' \\in Y$.\n\\end{corollary}\n\nIf $f, f' \\from Y \\to X$ are (globally) proper maps, then certainly $f \\cross \nf' \\from Y \\cross Y \\to X \\cross X$ is proper. The same also holds locally, and \nthis will be essential later.\n\n\\begin{proposition}\\label{prop:loc-prop-prod}\nIf (set) maps $f, f' \\from Y \\to X$ are locally proper for $F \\in\n\\calE_{|Y|_1}$, then:\n\\begin{enumerate}\n\\item $E \\defeq (f \\cross f')(F) \\subseteq X^{\\cross 2}$ satisfies the \n    properness axiom; and\n\\item the restriction $(f \\cross f') |_F^E \\from F \\to E$ is a proper map.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nFix $F \\in \\calE_{|Y|_1}$, put $E \\defeq (f \\cross f')(F)$, and consider the \ncommutative diagram\n\\[\\begin{CD}\n    F @>{(f \\cross f') |_F^E}>> E \\\\\n    @V{\\pi_1 |_F^{F \\cdot Y}}VV @V{\\pi_1 |_E}VV \\\\\n    F \\cdot Y @>{f |_{F \\cdot Y}}>> X\n\\end{CD}\\quad.\\]\nThe composition along the left and bottom is proper, and thus so is composition \nalong the top and right. Consequently, $(f \\cross f') |_F^E$ is proper. Since \n$(f \\cross f') |_F^E$ is surjective, $\\pi_1 |_E$ is proper and similarly for \n$\\pi_2 |_E$.\n\\end{proof}\n\nWe have the following ``very'' local analogue of Proposition~\\ref{prop:prop}. \nFor a more general analogue, we will need push-forward coarse structures.\n\n\\begin{proposition}\\label{prop:loc-prop-for-comp}\nConsider the composition of (set) maps $Z \\nameto{\\smash{g}} Y \n\\nameto{\\smash{f}} X$, supposing that $G \\in \\calE_{|Z|_1}$ and putting $F \n\\defeq g^{\\cross 2}(G)$:\n\\begin{enumerate}\n\\item\\label{prop:loc-prop-for-comp-I} If $g$ is locally proper for $G$ and $f$ \n    is locally proper for $F$, then $f \\circ g$ is locally proper for $G$.\n\\item\\label{prop:loc-prop-for-comp-II} If $f \\circ g$ is locally proper for \n    $G$, then $g$ is locally proper for $G$.\n\\item\\label{prop:loc-prop-for-comp-III} If $f \\circ g$ is locally proper for \n    $G$, then $f$ is locally proper for $F$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nPut $E \\defeq f^{\\cross 2}(F)$. We apply Proposition~\\ref{prop:prop} to the \ncommutative diagram\n\\[\\begin{CD}\nG @>{g^{\\cross 2} |_G^F}>> F @>{f^{\\cross 2} |_F^E}>> E \\\\\n@V{\\pi_1 |_G}VV @V{\\pi_1 |_F}VV @V{\\pi_1 |_E}VV \\\\\nZ @>{g}>> Y @>{F}>> X\n\\end{CD}\\quad.\\]\n\\enumref{prop:loc-prop-for-comp-I} is clear. For \n\\enumref{prop:loc-prop-for-comp-II} and \\enumref{prop:loc-prop-for-comp-III}: \nIf $f \\circ g$ is locally proper for $G$, then $\\pi_1 |_E$ and\n\\[\n    (f \\circ g)^{\\cross 2} |_G^E\n        = f^{\\cross 2} |_F^E \\circ g^{\\cross 2} |_G^F\n\\]\nare proper. By the latter, $g^{\\cross 2} |_G^F$ is proper. $g^{\\cross 2} |_G^F$ \nis surjective, so $f^{\\cross 2} |_F^E$ is also proper. Then\n\\[\n    \\pi_1 |_E \\circ f^{\\cross 2} |_F^E = f \\circ \\pi_1 |_F\n\\]\nis proper, so $\\pi_1 |_F$ is proper.\n\\end{proof}\n\n\n\\subsection{Pull-back and push-forward coarse structures}\n\n\\begin{definition}\nSuppose $X$ is a coarse space. The \\emph{pull-back coarse structure} (of \n$\\calE_X$) on $Y$ along (a set map) $f \\from Y \\to X$ is\n\\[\n    f^* \\calE_X \\defeq \\set{F \\in \\calE_{|Y|_1} \\suchthat\n            \\text{$f$ is coarse for $F$}}.\n\\]\n\\end{definition}\n\nBy Proposition~\\ref{prop:coarse-alg}, $f^* \\calE_X$ is actually a coarse \nstructure. If $X$ is connected, then $f^* \\calE_X$ is connected. If $X$ is \nunital and $f$ is (globally) proper, then $f^* \\calE_X$ is unital. The \nfollowing are clear.\n\n\\begin{proposition}\nIf $X$ is a coarse space and $f \\from Y \\to X$ is a set map, then $f^* \\calE_X$ \nis the maximum coarse structure on $Y$ which makes $f$ into a coarse map.\n\\end{proposition}\n\n\\begin{corollary}\\label{cor:crs-factor-I}\nIf $f \\from Y \\to X$ is a coarse map, then $f$ factors as a composition of \ncoarse maps\n\\[\n    Y \\nameto{\\smash{\\beta}} |Y|_{f^* \\calE_X} \\nameto{\\smash{\\utilde{f}}} X,\n\\]\nwhere $\\beta = \\id_Y$ and $\\utilde{f} = f$ as set maps.\n\\end{corollary}\n\nMore generally, if $\\set{X_j \\suchthat j \\in J}$ ($J$ some index set) is a \ncollection of coarse spaces and $\\set{f_j \\from Y \\to X_j}$ is a collection of \nset maps, then\n\\[\n    \\calE \\defeq \\bigintersect_{j \\in J} (f_j)^* \\calE_{X_j}\n\\]\nis the maximum coarse structure on $Y$ which makes all the $f_j$ into coarse \nmaps. If $Y$ is a coarse space and the $f_j \\from Y \\to X_j$ are all coarse \nmaps, then each $f_j$ factors as a composition of coarse maps $\\utilde{f}_j \n\\circ \\beta$ in the obvious way. Moreover, if all the $X_j$ are connected, then \n$\\calE$ is connected; if all the $X_j$ are unital and all the $f_j$ are \n(globally) proper, then $\\calE$ is unital.\n\n\\begin{definition}\nSuppose $Y$ is a coarse space. The \\emph{push-forward coarse structure} (of \n$\\calE_Y$) on $X$ along a \\emph{locally proper} map $f \\from Y \\to X$ is\n\\[\n    f_* \\calE_Y \\defeq \\bigl\\langle \\set{f^{\\cross 2}(F) \\suchthat F \\in\n            \\calE_Y} \\bigr\\rangle.\n\\]\nWe similarly define \\emph{unital}, \\emph{connected}, and \\emph{connected unital \npush-forward coarse structures}.\n\\end{definition}\n\nIf $Y$ is connected and $f$ is surjective, then $f_* \\calE_Y$ is connected. \nSimilarly, if $Y$ is unital (hence $f$ globally proper) and $f$ is surjective, \nthen $f_* \\calE_Y$ is unital.\n\n\\begin{proposition}\nIf $Y$ is a coarse space and $f \\from Y \\to X$ is a locally proper map, then \n$f_* \\calE_Y$ is the minimum coarse structure on $X$ which makes $f$ into a \ncoarse map.\n\\end{proposition}\n\n\\begin{corollary}\\label{cor:crs-factor-II}\nIf $f \\from Y \\to X$ is a coarse map, then $f$ factors as a composition of \ncoarse maps\n\\[\n    Y \\nameto{\\smash{\\tilde{f}}} |X|_{f_* \\calE_Y} \\nameto{\\smash{\\alpha}} X\n\\]\nwhere $\\tilde{f} = f$ and $\\alpha = \\id_X$ as set maps.\n\\end{corollary}\n\nOf course, there are obvious unital, connected, and connected unital versions \nof the above. For the unital versions one needs $f$ to be proper and $Y$ should \nprobably be unital; for the connected versions, $Y$ should probably be \nconnected.\n\nMore generally, if $\\set{Y_j \\suchthat j \\in J}$ ($J$ some index set) is a \ncollection of coarse spaces and $\\set{f_j \\from Y_j \\to X}$ is a collection of \nlocally proper maps, then\n\\[\n    \\calE \\defeq \\bigl\\langle (f_j)_* \\calE_{Y_j} \\bigr\\rangle\n\\]\nis the minimum coarse structure on $X$ which makes all the $f_j$ into coarse \nmaps. If $X$ is a coarse space and the $f_j \\from Y_j \\to X$ are all coarse \nmaps, then each $f_j$ factors as $\\alpha \\circ \\tilde{f}_j$. Again, there are \nunital, connected, and connected unital versions of this.\n\n\\begin{remark}\nWe emphasize that whereas one can pull back coarse structures along \\emph{any} \nset map (or collection of set maps), one can only push forward coarse \nstructures along \\emph{locally proper} maps. If one wants all the coarse \nstructures to be unital (and take unital, possibly connected, push-forwards), \nthen one evidently requires all maps to be (globally) proper.\n\\end{remark}\n\nIt is easy to see what happens when one pushes a coarse structure forward and \nthen pulls it back along the same map (or vice versa).\n\n\\begin{proposition}\nIf $Y$ is a coarse space and $f \\from Y \\to X$ is a locally proper map, then \n$\\calE_Y \\subseteq f^* f_* \\calE_Y$.\n\\end{proposition}\n\n\\begin{proof}\n$f$ is coarse as a map $Y \\to |X|_{f_* \\calE_Y}$. Applying \nCorollary~\\ref{cor:crs-factor-I}, this map factors as $Y \\nameto{\\smash{\\beta}} \n|Y|_{f^* f_* \\calE_Y} \\to |X|_{f_* \\calE_Y}$ where $\\beta$ is the identity as a \nset map.\n\\end{proof}\n\n\\begin{proposition}\nIf $X$ is a coarse space and $f \\from Y \\to X$ is any set map, then $f_* f^* \n\\calE_X \\subseteq \\calE_X$.\n\\end{proposition}\n\n\\begin{proof}\nNow $f$ is coarse as a map $|Y|_{f^* \\calE_X} \\to X$, to which we apply \nCorollary~\\ref{cor:crs-factor-II}.\n\\end{proof}\n\nUsing push-forward coarse structures (and Corollary~\\ref{cor:loc-prop-gen}), we \ncan ``restate'' Proposition~\\ref{prop:loc-prop-for-comp} as follows.\n\n\\begin{proposition}\\label{prop:loc-prop-comp}\nConsider the composition of (set) maps $Z \\nameto{\\smash{g}} Y \n\\nameto{\\smash{f}} X$, where $Z$ is a coarse space:\n\\begin{enumerate}\n\\item\\label{prop:loc-prop-comp:I} If $g$ is locally proper and $f$ is locally \n    proper for the push-forward coarse structure $g_* \\calE_Z$ on $Y$, then $f \n    \\circ g$ is locally proper.\n\\item\\label{prop:loc-prop-comp:II} If $f \\circ g$ is locally proper, then $g$ \n    is locally proper.\n\\item\\label{prop:loc-prop-comp:III} If $f \\circ g$ is locally proper, then $f$ \n    is locally proper for the push-forward coarse structure $g_* \\calE_Z$ on \n    $Y$.\n\\end{enumerate}\nThe above also hold with connected push-forward coarse structures in place of \npush-forward coarse structures. Also, that injectivity implies global \nproperness implies local properness.\n\\end{proposition}\n\n\\begin{remark}\nApplying the above Proposition with $Z \\defeq |Z|_1$ having the terminal coarse \nstructure, we get \\enumref{prop:prop:I} and \\enumref{prop:prop:II} of \nProposition~\\ref{prop:prop}. If $g$ is surjective, then the push-forward coarse \nstructure $g_* \\calE_{|Z|_1}$ is the terminal coarse structure $\\calE_{|Y|_1}$ \nand we get \\enumref{prop:prop:III} as well.\n\\end{remark}\n\n\n\n\n\\section{The precoarse categories}\\label{sect:PCrs}\n\nWe now define several categories of coarse spaces, whose arrows are coarse \nmaps, and examine their properties. These \\emph{precoarse categories} differ \nfrom the coarse categories, which are quotients of these categories (see \n\\S\\ref{sect:Crs}).\n\n\n\\subsection{Set and category theory}\\label{subsect:set-cat}\n\nWe will be unusually careful with our set and category theoretic constructions. \nThe following can mostly be ignored safely, though will be needed eventually \nfor rigorous, ``canonical'' constructions (e.g., when we consider sets of \n``all'' modules over a coarse space).\n\nAssuming the Grothendieck axiom that any set is contained in some universe, we \nfirst fix a universe $\\calU$ (containing $\\omega$). \\emph{Small} (or \n$\\calU$-small) objects are elements of $\\calU$. A \\emph{$\\calU$-category} is \none whose object set is a subset of $\\calU$. A ($\\calU$-)small category is one \nwhose object set (hence morphism set and composition law) is in $\\calU$. A \nsmall category is necessarily a $\\calU$-category, but not vice versa. A \n$\\calU$-category in turn is $\\calU^+$-small, where $\\calU^+$ denotes the \nsmallest universe having $\\calU$ as an element. A \\emph{locally small} \n$\\calU$-category is a $\\calU$-category whose $\\Hom$-sets $\\Hom(\\cdot,\\cdot)$ \nare all small.\n\nRecall the notion of quotient categories (from, e.g., \\cite{MR1712872}*{Ch.~II \n\\S{}8}): Given a category $\\calC$ and an equivalence relation $\\sim$ on each \n$\\Hom$-set of $\\calC$, there is a \\emph{quotient category} $\\calC\/{\\sim}$ and a \n\\emph{quotient functor} $\\Quotient \\from \\calC \\to \\calC\/{\\sim}$ satisfying the \nfollowing universal property: For all functors $F \\from \\calC \\to \\calC'$ \n($\\calC'$ any category, which can be taken to be $\\calU$-small if $\\calC$ is \n$\\calU$-small) such that $f \\sim f'$ ($f$, $f'$ in some $\\Hom$-set of $\\calC$) \nimplies $F(f) \\sim F(f')$, there is a unique functor $F' \\from \\calC\/{\\sim} \\to \n\\calC'$ such that $F = F' \\circ \\Quotient$. Moreover, if the equivalence \nrelation $\\sim$ is preserved under composition then, for all objects $X$, $Y$ \nof $\\calC$, the set $\\Hom_{\\calC\/{\\sim}}(\\Quotient(Y),\\Quotient(X))$ is in \nnatural bijection with the set of $\\sim$-equivalence classes of \n$\\Hom_{\\calC}(Y,X)$.\n\nAs usual, $\\CATSet$ denotes the category of small sets (and set maps). \n$\\CATTop$ is the category of small topological spaces and continuous maps. \nForgetful functors will be denoted by $\\Forget$, with the source and target \ncategories (the latter often being $\\CATSet$) implied by context. For a \ncategory $\\calC$ equipped with a forgetful functor to $\\CATSet$, we denote the \nfull subcategory of $\\calC$ of nonempty objects (i.e., those $X$ with \n$\\Forget(X) \\neq \\emptyset$) by $\\CATne{\\calC}$.\n\nFor the most part, henceforth $X$, $Y$, and $Z$ will be (small) coarse spaces, \nand $f \\from Y \\to X$ and $g \\from Z \\to Y$ coarse maps. $\\setZplus \\defeq \n\\set{n \\in \\setZ \\suchthat n \\geq 0}$ is the set of nonnegative integers and \nsimilarly $\\setRplus \\defeq \\coitvl{0,\\infty}$ is the set of nonnegative real \nnumbers.\n\n\n\\subsection{The precoarse categories}\n\n\\begin{definition}\nThe \\emph{precoarse category} $\\CATPCrs$ has as objects all (small) coarse \nspaces and as arrows coarse maps. The \\emph{connected precoarse category} \n$\\CATConnPCrs$ is full subcategory of $\\CATPCrs$ consisting of the connected \ncoarse spaces. Similarly define the \\emph{unital precoarse category} \n$\\CATUniPCrs$ and the \\emph{connected unital precoarse category} \n$\\CATConnUniPCrs$.\n\\end{definition}\n\n\\begin{remarks}\nIn many ways, the category $\\CATne{\\CATConnPCrs}$ of nonempty connected coarse \nspaces, i.e., coarse spaces with exactly one connected component, is more \nnatural. Observe that that $\\CATConnUniPCrs = \\CATConnPCrs \\intersect \n\\CATUniPCrs$ is a full subcategory of the other three categories. (One might \nargue that the unital categories above are not the ``correct'' ones and further \ninsist that the arrows in the unital categories should be ``unit \npreserving'', i.e., surjective as set maps. However, the above unital \ncategories are the usual ones used in coarse geometry; see \\S\\ref{sect:top-crs} \nand especially Corollary~\\ref{cor:UniCrs-RoeCrs-equiv}.)\n\\end{remarks}\n\nWe will analyze various properties of the categories $\\CATPCrs$ and \n$\\CATConnPCrs$ (which are better behaved than the others). In particular, we \nexamine limits and colimits in these categories, which include as special cases \nproducts and coproducts, equalizers and coequalizers, and terminal and initial \nobjects. (We use the standard terminology from category theory, topology, etc.: \nlimits are also called ``projective limits'' or ``inverse limits'' and colimits \nare called ``inductive limits'' or ``direct limits'', though ``direct limits'' \nare often more specifically filtered colimits.)\n\nLet us first recall some standard terminology (see, e.g., \\cite{MR1712872}). \nLet $\\calC$ be a category and suppose $\\calF_X \\from \\calJ \\to \\calC$ ($\\calJ$ \na small, often finite, category) is a functor. A \\emph{cone $\\nu \\from X \\to \n\\calF_X$ to $\\calF_X$} consists of an $X \\in \\Obj(\\calC)$ and arrows $\\nu_j \n\\from X \\to X_j \\defeq \\calF_X(j)$, $j \\in \\Obj(\\calJ)$, such that the \ntriangles emanating from $X$ commute. A \\emph{limit} in $\\calC$ for $\\calF_X$ \nis given by a cone $X \\to \\calF_X$ which is universal, i.e., a \\emph{limiting \ncone}. Limits of $\\calF_X$ in $\\calC$ are unique up to natural isomorphism. \nThus we will sometimes follow the customary abuses of referring to \\emph{the} \nlimit of $\\calF_X$ and of referring to the object $X$ (often denoted $\\OBJlim \n\\calF_X$) as the limit with the $\\nu_j$ understood. A functor $F \\from \\calC \n\\to \\calC'$ \\emph{preserves limits} if whenever $\\nu \\from X \\to \\calF_X$ is a \nlimiting cone in $\\calC$, $F \\circ \\nu \\from F(X) \\to F \\circ \\calF_X$ is \nlimiting in $\\calC'$. Dually, one has \\emph{cones from $\\calF_Y$}, \n\\emph{colimits}, \\emph{colimiting cones}, and functors which \\emph{preserve \ncolimits}. All limits and colimits considered will be small. In particular, the \ncategory $\\calJ$ and functors $\\calF_X$ and $\\calF_Y$ will be small.\n\nFirst, we examine the relation between $\\CATPCrs$ and $\\CATConnPCrs$.\n\n\n\\subsection{\\pdfalt{\\maybeboldmath $\\CATPCrs$ versus $\\CATConnPCrs$}%\n        {PCrs versus CPCrs}}\\label{subsect:PCrs-conn}\n\nBelow, $I$ will always denote the inclusion $\\CATConnPCrs \\injto \\CATPCrs$. \nNote that $I$ is fully faithful.\n\n\\begin{definition}\n$\\Connect \\from \\CATPCrs \\to \\CATConnPCrs$ is the functor defined as follows:\n\\begin{enumerate}\n\\item For a coarse space $X$, $\\Connect(X)$ is just $X$ as a set, but with the \n    \\emph{connected} coarse structure $\\langle \\calE_X \\rangle_X^\\TXTconn$ \n    generated by $\\calE_X$.\n\\item For a coarse map $f \\from Y \\to X$, $\\Connect(f) \\from \\Connect(Y) \\to \n    \\Connect(X)$ is the same as a set map as $f$ (which is coarse by \n    Corollary~\\ref{cor:coarse-gen}).\n\\end{enumerate}\n\\end{definition}\n\nThe following is clear.\n\n\\begin{proposition}\n$\\Connect \\circ I$ is the identity functor on $\\CATConnPCrs$.\n\\end{proposition}\n\n\\begin{proposition\n$\\Connect \\from \\CATPCrs \\to \\CATConnPCrs$ is left adjoint to the inclusion \nfunctor.\n\\end{proposition}\n\nThe counit maps $Y \\to I(\\Connect(Y))$, $Y \\in \\Obj(\\CATPCrs)$, of the above \nadjunction are just the identities as set maps. The unit maps $X = \n\\Connect(I(X)) \\to X$, $X \\in \\Obj(\\CATConnPCrs)$, are the identity maps.\n\n\\begin{proof}\nSince $\\Connect \\circ I$ is the identity, $\\Connect$ induces natural maps\n\\[\n    \\Hom_{\\CATPCrs}(Y, I(X)) \\to \\Hom_{\\CATConnPCrs}(\\Connect(Y), X)\n\\]\n(for $Y$ possibly disconnected and $X$ connected), which are clearly bijections.\n\\end{proof}\n\n\\begin{corollary}\\label{cor:PCrs-ConnPCrs-preserve}\n$I \\from \\CATConnPCrs \\injto \\CATPCrs$ preserves limits and $\\Connect \\from \n\\CATPCrs \\to \\CATConnPCrs$ preserves colimits. Moreover, if $\\calF \\from \\calJ \n\\to \\CATConnPCrs$ is a functor and $\\nu$ is a limiting cone to (or colimiting \ncone from) $I \\circ \\calF$ in $\\CATPCrs$, then $\\Connect \\circ \\nu$ is a \nlimiting cone to (or colimiting cone from, respectively) $\\calF = \\Connect \n\\circ I \\circ \\calF$ in $\\CATConnPCrs$.\n\\end{corollary}\n\n\\begin{proof}\nSee, e.g., \\cite{MR1712872}*{Ch.~V \\S{}5} or \\cite{MR0349793}*{16.4.6} for the \nfirst statement, and \\cite{MR0349793}*{16.6.1} for the second.\n\\end{proof}\n\n\n\\subsection{Limits in the precoarse categories}\n\n\\begin{theorem}\\label{thm:PCrs-lim}\n$\\CATPCrs$ has all nonzero limits (i.e., limits of functors $\\calJ \\to \n\\CATPCrs$ for $\\calJ$ nonempty). Moreover, the forgetful functor $\\Forget \\from \n\\CATPCrs \\to \\CATSet$ preserves limits, and the limits of connected coarse \nspaces are connected. Consequently, the same hold with $\\CATConnPCrs$ in place \nof $\\CATPCrs$.\n\\end{theorem}\n\nIt is actually easy to see that $\\Forget \\from \\CATPCrs \\to \\CATSet$ preserves \nlimits: $\\Forget$ is naturally equivalent to the covariant $\\Hom$-functor \n$\\Hom_{\\CATPCrs}(\\ast,\\cdot) \\from \\CATPCrs \\to \\CATSet$, where $\\ast$ is any \none-point coarse space, and thus preserves limits (see, e.g., \n\\cite{MR1712872}*{Ch.~V \\S{}4 Thm.~1}). Since I do not know a similar argument \nfor colimits, let us proceed in ignorance of this.\n\n\\begin{proof}\nRecall that $\\CATSet$ has all limits. Given $\\calF_X \\from \\calJ \\to \\CATPCrs$, \nfix a limiting set cone $\\nu \\from X \\to \\Forget \\circ \\calF_X$, so that $X$ is \na set and $\\nu_j \\from X \\to X_j \\defeq \\calF_X(j)$, $j \\in \\Obj(\\calJ)$, are \nset maps. It suffices to put a coarse structure on $X$ so that we get a \nlimiting cone $\\nu \\from X \\to \\calF_X$ in $\\CATPCrs$ (with $X$ connected if \nall the $X_j$ are connected).\n\nWe need all the $\\nu_j \\from X \\to X_j$ to become coarse maps. Taking the \ncoarse structure on $X$ to be the intersection\n\\[\n    \\calE_X \\defeq \\bigintersect_{j \\in \\Obj(\\calJ)} (\\nu_j)^* \\calE_{X_j}\n\\]\nof pull-back coarse structures clearly makes this so. (Since pull-backs of \nconnected coarse structures are connected and intersections of connected coarse \nstructures are connected, $\\calE_X$ is connected if all the $X_j$ are.) Since \n$\\Forget$ is faithful, $\\nu \\from X \\to \\calF_X$ is a cone in $\\CATPCrs$. We \nmust show that it is universal.\n\nSuppose $\\mu \\from Y \\to \\calF_X$ is another cone in $\\CATPCrs$. Applying \n$\\Forget$, we get a cone $\\mu \\from Y \\to \\Forget \\circ \\calF_X$ in $\\CATSet$ \n(properly written $\\Forget \\circ \\mu \\from \\Forget(Y) \\to \\Forget \\circ \n\\calF_X$). Since $\\nu$ is universal in $\\CATSet$, there is a set map $t \\from Y \n\\to X$ such that $\\mu = \\nu \\circ t$ as cones in $\\CATSet$. We must show that \n$t$ is actually a coarse map (uniqueness is clear).\n\nFirst, since $\\calJ$ is nonzero, there is some object $j_0 \\in \\Obj(\\calJ)$; \nthen $\\mu_{j_0} = \\nu_{j_0} \\circ t$ (as set maps) is locally proper, so $t$ is \nlocally proper \n(Proposition~\\ref{prop:loc-prop-comp}\\enumref{prop:loc-prop-comp:II}). Next, \nfor each $j \\in \\Obj(\\calJ)$ and $F \\in \\calE_Y$, $\\nu_j$ is coarse for $E \n\\defeq t^{\\cross 2}(F)$ (which is in $\\calE_{|X|_1}$, by local properness) and \nhence $E \\in (\\nu_j)^* \\calE_{X_j}$: Since $\\mu_j = \\nu_j \\circ t$ is locally \nproper for $F$, $\\nu_j$ is locally proper for $E$ \n(Proposition~\\ref{prop:loc-prop-for-comp}\\enumref{prop:loc-prop-for-comp-II}), \nand also $\\nu_j$ clearly preserves $E$.\n\nFor $\\CATConnPCrs$, the assertions follow from \nCorollary~\\ref{cor:PCrs-ConnPCrs-preserve}.\n\\end{proof}\n\nThe above proof gives a rather concrete description of limits in $\\CATPCrs$ \n(and $\\CATConnPCrs$), and in particular of products. The product \n$\\pfx{\\CATPCrs}\\prod_{j \\in J} X_j$ in $\\CATPCrs$ (or in $\\CATConnPCrs$) is \njust the set product (i.e., cartesian product) $X \\defeq \\pfx{\\CATSet}\\prod_{j \n\\in J} X_j$ together with the entourages of $|X|_1$ which project properly to \nentourages of all the $X_j$.\n\nThe ``nonzero'' stipulation in Theorem~\\ref{thm:PCrs-lim} is necessary.\n\n\\begin{proposition}\\label{prop:PCrs-no-map-to}\nFor each coarse space $X$, there exists a (nonempty) connected, unital coarse \nspace $Y$ such that there is no coarse map $Y \\to X$.\n\\end{proposition}\n\n\\begin{proof}\nGiven $X$, take $Y \\defeq |Y|_1$ to be an infinite set with cardinality \nstrictly greater than the cardinality of $X$, equipped with the terminal coarse \nstructure (which is connected and unital), e.g., $Y \\defeq |\\powerset(X) \n\\disjtunion \\setN|_1$. Then no locally proper map $Y \\to X$ exists, since no \nglobally proper map $Y \\to X$ exists and $Y$ is unital (see \nCorollary~\\ref{cor:loc-prop-uni}). (Note that the cardinality of sets in our \nuniverse $\\calU$ is bounded above by some cardinal, namely by $\\# \\calU$, but \nno element of $\\calU$ has this cardinality.)\n\\end{proof}\n\n\\begin{corollary}\\label{cor:PCrs-no-term}\nNone of the precoarse categories ($\\CATPCrs$, $\\CATConnPCrs$, \n$\\CATne{\\CATConnPCrs}$, $\\CATUniPCrs$, and $\\CATConnUniPCrs$) has a terminal \nobject.\n\\end{corollary}\n\nThe failure of existence of terminal objects in the precoarse categories is not \njust a failure of uniqueness of maps, but more seriously of existence. Thus we \nwill also get the following on the coarse categories (which are quotients of \nthe precoarse categories).\n\n\\begin{corollary}\\label{cor:Crs-no-term}\nNo quotient of any of the above precoarse categories has a terminal object.\n\\end{corollary}\n\nIt is straightforward to show that the inclusion $\\CATne{\\CATConnPCrs} \\injto \n\\CATConnPCrs$ preserves limits, and moreover that a nonzero limit exists in \n$\\CATne{\\CATConnPCrs}$ if and only if the corresponding set limit is nonempty \n(but the example below shows that $\\CATne{\\CATConnPCrs}$ does \\emph{not} have \nall nonzero limits). On the other hand unitality poses a fatal problem: The \nforgetful functor $\\CATUniPCrs \\to \\CATSet$ still preserves limits, so a \n(nonzero) limit in $\\CATUniPCrs$ can only exist when all the maps in the \ncorresponding limiting set cone are proper (but this is often not the case, \ne.g., in the case of products).\n\n\\begin{example}\nLet $X \\defeq |\\setZplus|_1$ (which is connected and nonempty), $f \\defeq \\id_X \n\\from X \\to X$ be the identity, and define $g \\from X \\to X$ by $g(x) \\defeq \nx+1$. Then the equalizer of $f$ and $g$ in $\\CATPCrs$ is the empty set.\n\\end{example}\n\nTo get ahead of ourselves (see \\S\\ref{sect:Crs}), note that though $f$ and $g$ \nare \\emph{close}, the equalizer of $f$ and itself (which is just $X$ mapping \nidentically to itself) is not \\emph{coarsely equivalent} to the equalizer of \n$f$ and $g$. Indeed, one can obtain other inequivalent equalizers: e.g., the \nequalizer of $h \\from X \\to X$ where $h(x) \\defeq \\min \\set{0,x-1}$ (which also \nclose to $f$) and $f$ is $\\set{0}$ (including into $X$). On the other hand, in \nthe quotient coarse category $\\CATCrs$, $[f] = [g] = [h]$ so the equalizer of \nany pair of these maps is $X$. Since limits in $\\CATPCrs$ are not \n\\emph{coarsely invariant}, they are of limited interest.\n\nWe also see that the quotient functor $\\CATPCrs \\to \\CATCrs$ does not preserve \nequalizers, hence does not preserve limits. However, it \\emph{does} preserve \nproducts. Using this and a method parallel to the one employed in the proof of \nTheorem~\\ref{thm:PCrs-lim}, we will show that $\\CATCrs$ also has all nonzero \nlimits (which will, by definition, be coarsely invariant).\n\nWe will use products extensively. We take this opportunity to mention several \ncanonical coarse maps which arise due to the existence of (nonzero) products \n(all objects are coarse spaces and arrows coarse maps):\n\\begin{enumerate}\n\\item For any $X$ and $Y$, there are \\emph{projection maps} $\\pi_X \\from X \n    \\cross Y \\to X$ and $\\pi_Y \\from X \\cross Y \\to Y$.\n\\item For any $X$, there is a \\emph{diagonal map} $\\Delta_X \\from X \\to X \n    \\cross X$.\n\\item For $f \\from Y \\to X$ and $f' \\from Y' \\to X'$, there is a \\emph{product \n    map} $f \\cross f' \\from Y \\cross Y' \\to X \\cross X'$.\n\\end{enumerate}\nThe above can all be generalized to larger (even infinite) products.\n\n\n\\subsection{Colimits in the precoarse categories}\n\n\\begin{theorem}\\label{thm:PCrs-colim}\nA colimit exists in $\\CATPCrs$ if and only if all the maps from a corresponding \ncolimiting set cone are locally proper. Moreover, the forgetful functor \n$\\Forget \\from \\CATPCrs \\to \\CATSet$ preserves colimits. The same hold with \n$\\CATConnPCrs$ in place of $\\CATPCrs$.\n\\end{theorem}\n\n\\begin{proof}\nThis proof is basically dual to the proof of Theorem~\\ref{thm:PCrs-lim}, only \nwith the added onus of showing the ``only if''. The reason for the local \nproperness requirement is that coarse structures can only be pushed forward \nalong locally proper maps (whereas they can be pulled back along all maps).\n\nRecall that $\\CATSet$ has all colimits. Given $\\calF_Y \\from \\calJ \\to \n\\CATPCrs$, fix a colimiting set cone $\\nu \\from \\Forget \\circ \\calF_Y \\to Y$, \nso that $Y$ is a set and $\\nu_j \\from Y_j \\defeq \\calF_Y(j) \\to Y$, $j \\in \n\\Obj(\\calJ)$, are set maps. Suppose all of the $\\nu_j$ are locally proper. \nTaking the coarse structure on $Y$ to be\n\\[\n    \\calE_Y \\defeq \\langle (\\nu_j)_* \\calE_{Y_j}\n            \\suchthat j \\in \\Obj(\\calJ) \\rangle_Y,\n\\]\nwe clearly get a cone $\\nu \\from \\calF_Y \\to Y$ in $\\CATPCrs$; we must prove \nthat it is universal.\n\nSuppose $\\mu \\from \\calF_Y \\to X$ is another cone in $\\CATPCrs$. Then there is \na canonical set map $t \\from Y \\to X$ such that $\\mu = t \\circ \\nu$ as cones in \n$\\CATSet$. We must show that $t$ is coarse (again uniqueness is clear). \nEntourages $(\\nu_j)^{\\cross 2}(F)$, $F \\in \\calE_{Y_j}$, $j \\in \\Obj(\\calJ)$, \ngenerate $\\calE_Y$. $t$ is locally proper for each such entourage (using $\\mu_j \n= t \\circ \\nu_j$ and \nProposition~\\ref{prop:loc-prop-for-comp}\\enumref{prop:loc-prop-for-comp-III}) \nand clearly preserves each such entourage. Thus $t$ is coarse, as required.\n\nIf the $\\nu_j$ are \\emph{not} all locally proper, we must show that $\\calF_Y$ \ndoes not have a colimit (in $\\CATPCrs$); in fact, we show something stronger, \nthat there is no cone from $\\calF_Y$ in $\\CATPCrs$. We proceed by \ncontradiction, so suppose that $\\nu_{j_0}$ is not locally proper ($j_0 \\in \n\\Obj(\\calJ)$ fixed) and suppose $\\mu \\from \\calF_Y \\to X$ is a cone in \n$\\CATPCrs$. Again there must be a set map $t \\from Y \\to X$ such that $\\mu = t \n\\circ \\nu$ as set cones. But then $\\mu_{j_0} = t \\circ \\nu_{j_0}$ is locally \nproper, which implies that $\\nu_{j_0}$ is locally proper \n(Proposition~\\ref{prop:loc-prop-comp}\\enumref{prop:loc-prop-comp:II}) which is \na contradiction.\n\nTo get the asserted colimits in $\\CATConnPCrs$, simply apply \nCorollary~\\ref{cor:PCrs-ConnPCrs-preserve}. To show that $\\CATConnPCrs$ has no \nmore colimits than $\\CATPCrs$ (i.e., $\\calF_Y \\from \\calJ \\to \\CATConnPCrs$ has \na colimit in $\\CATConnPCrs$ only if $I \\circ \\calF_Y \\from \\calJ \\to \\CATPCrs$ \nhas a colimit in $\\CATPCrs$), it is probably simplest to modify the above \nproof.\n\\end{proof}\n\nThe following are clear.\n\n\\begin{corollary}\n$\\CATPCrs$ and $\\CATConnPCrs$ have all coproducts.\n\\end{corollary}\n\n\\begin{corollary}\nThe empty coarse space is the (unique) initial object in $\\CATPCrs$ and in \n$\\CATConnPCrs$.\n\\end{corollary}\n\n\\begin{corollary}\\label{cor:PCrs-fin-components}\nAny coarse space with only \\emph{finitely} many connected components is \n(isomorphic in $\\CATPCrs$ to) the coproduct in $\\CATPCrs$ of its connected \ncomponents.\n\\end{corollary}\n\nThe above Corollary does not necessarily hold for coarse spaces with infinitely \nmany connected components. One may say, more generally, that any coarse space \nwhose unital subspaces have only finitely many connected components is the \ncoproduct of its connected components.\n\nWe get concrete descriptions of coproducts in $\\CATPCrs$ and in $\\CATConnPCrs$. \nThe coproduct $\\pfx{\\CATPCrs}\\coprod_{j \\in J} Y_j$ in $\\CATPCrs$ is just the \nset coproduct (i.e., disjoint union) $Y \\defeq \\pfx{\\CATSet}\\coprod_{j \\in J} \nY_j$ with entourages finite unions of entourages of the $Y_j$ (included into \n$Y$). The corresponding coproduct in $\\CATConnPCrs$ is the same as a set, but \none may also take an additional union with an arbitrary finite subset of \n$Y^{\\cross 2}$.\n\nThe inclusion $\\CATne{\\CATConnPCrs} \\injto \\CATConnPCrs$ preserves colimits. \n$\\CATne{\\CATConnPCrs}$ does not have a zero colimit (i.e., initial object), but \notherwise has a colimit if the corresponding colimit exists in $\\CATConnPCrs$, \nin which case the two colimits coincide; note that a nonzero colimit of \nnonempty sets is nonempty. Unitality does not pose a problem for colimits: \nTheorem~\\ref{thm:PCrs-colim} also holds with $\\CATUniPCrs$ in place of \n$\\CATPCrs$ (and $\\CATConnUniPCrs$ in place of $\\CATConnPCrs$). In the proof, \none simply takes the unital coarse structure\n\\[\n    \\langle (\\nu_j)_* \\calE_{Y_j} \\suchthat j \\in \\Obj(\\calJ) \\rangle_Y^\\TXTuni\n\\]\ninstead. Of course, in the unital cases, one may substitute ``(globally) \nproper'' for ``locally proper''.\n\nThe ``locally proper'' hypothesis is necessary, as the following shows.\n\n\\begin{example}\nLet $X \\defeq |\\setZplus|_1$, $f \\from X \\to X$ be the identity, and define $g \n\\from X \\to X$ by $g(x) \\defeq \\min \\set{0,x-1}$. The coequalizer of $f$ and \n$g$ in $\\CATSet$ is the one-point set $\\ast$; since $X$ is unital, $f$ and $g$ \ndo not have a coequalizer in $\\CATPCrs$.\n\\end{example}\n\nAgain, to get ahead of ourselves, we see that coequalizers in $\\CATPCrs$ are \nnot coarsely invariant. Even though $f$ is close to $g$ and the coequalizer of \n$f$ and itself is just $X$, $f$ and $g$ do not have a coequalizer in \n$\\CATPCrs$. In the quotient category $\\CATCrs$, there are no problems: the \ncoequalizer of $[f]$ and $[g]$ is $X$, as expected.\n\nThe quotient functor $\\CATPCrs \\to \\CATCrs$ does not preserve coequalizers or \ncolimits in general. However, it \\emph{does} preserve coproducts, and we will \nuse these to show that in fact $\\CATCrs$ has \\emph{all} colimits (which are \nevidently coarsely invariant). In particular, $\\CATCrs$ has all coequalizers, \nwhich contrasts with the situation in $\\CATPCrs$ (recall that having all \ncoproducts and all coequalizers would imply having all colimits).\n\n\n\n\n\\section{The coarse categories}\\label{sect:Crs}\n\n\n\\subsection{Closeness of maps}\n\nIn classical (unital) coarse geometry, two maps $f, f' \\from Y \\to X$ are \n\\emph{close} if $(f \\cross f')(1_Y)$ is an entourage of $X$. Closeness is an \nequivalence relation on maps $Y \\to X$, but note that it does not involve the \ncoarse structure on $Y$ at all! In the nonunital case, we must modify the \ndefinition, lest closeness not even be reflexive (e.g., take $Y \\defeq X$ \nnonunital and $f \\defeq f' \\defeq \\id_X$).\n\n\\begin{definition}\nCoarse maps $f, f' \\from Y \\to X$ are \\emph{close} (write $f \\closeequiv f'$) \nif $(f \\cross f')(F) \\in \\calE_X$ for all $F \\in \\calE_Y$.\n\\end{definition}\n\n\\begin{proposition}\nCloseness of coarse maps $Y \\to X$ is an equivalence relation (on the \n$\\Hom$-set $\\Hom_{\\CATPCrs}(Y,X)$).\n\\end{proposition}\n\n\\begin{proof}\nReflexivity follows since coarse maps preserve entourages. Symmetry follows by \ntaking transposes. Transitivity: Suppose $f, f', f'' \\from Y \\to X$ are coarse \nmaps with $f \\closeequiv f'$ and $f' \\closeequiv f''$. For any $F \\in \\calE_Y$,\n\\[\n    (f \\cross f'')(F)\n        \\subseteq (f \\cross f')(1_{F \\cdot Y}) \\circ (f' \\cross f'')(F)\n\\]\nis an entourage of $X$ since $1_{F \\cdot Y} \\in \\calE_Y$ (since $1_{F \\cdot Y} \n\\subseteq F \\circ F^\\transpose$).\n\\end{proof}\n\nLike local properness, closeness is also determined ``on'' unital subspaces of \nthe domain. Thus for unital coarse spaces, our notion of closeness is just the \nclassical one.\n\n\\begin{proposition}\\label{prop:close-uni}\nCoarse maps $f, f' \\from Y \\to X$ are close if and only if, for every unital \nsubspace $Y' \\subseteq Y$, $f |_{Y'}$ and $f' |_{Y'}$ are close (i.e., $(f \n\\cross f')(1_{Y'}) \\in \\calE_X)$. Thus, for $Y$ unital, $f$ and $f'$ are close \nif and only if $(f \\cross f')(1_Y) \\in \\calE_X$.\n\\end{proposition}\n\n\\begin{proof}\n(\\textimplies): Immediate.\n\n(\\textimpliedby): For $F \\in \\calE_Y$, $Y' \\defeq F \\cdot Y \\union Y \\cdot F$ \nis a unital subspace of $Y$, and $F \\in \\calE_{Y'}$. Then\n\\[\n    (f \\cross f')(F) = (f |_{Y'} \\cross f' |_{Y'})(F) \\in \\calE_X,\n\\]\nas required.\n\\end{proof}\n\nWe have not used local properness at all, so we can actually define closeness \nfor maps which preserve entourages (but are not necessarily locally proper). \nHowever, we will not need this.\n\nThe following observation is rather important.\n\n\\begin{proposition}\\label{prop:term-close}\nSuppose $f, f' \\from Y \\to X$ are coarse maps. If $X = |X|_1$ has the terminal \ncoarse structure, then $f$ and $f'$ are close.\n\\end{proposition}\n\nThus if $X = |X|_1$, then for any coarse space $Y$ there is \\emph{at most} one \n(but possibly no) closeness class of coarse map $Y \\to X$.\n\n\\begin{proof}\nThis follows immediately from Proposition~\\ref{prop:loc-prop-prod}.\n\\end{proof}\n\n\n\\subsection{The coarse categories}\n\nCloseness, an equivalence relation on the $\\Hom$-sets of $\\CATPCrs$, yields a \nquotient category\n\\[\n    \\CATCrs \\defeq \\CATPCrs\/{\\closeequiv}\n\\]\n(see \\S\\ref{subsect:set-cat}), which we call the \\emph{coarse category}, \ntogether with a quotient functor $\\Quotient \\from \\CATPCrs \\to \\CATCrs$. We may \nsimilarly define quotients $\\CATConnCrs$, $\\CATUniCrs$, and $\\CATConnUniCrs$ of \n$\\CATConnPCrs$, $\\CATUniPCrs$, and $\\CATConnUniPCrs$, respectively. These \nlatter categories are full subcategories of $\\CATPCrs$, so their quotients are \nfull subcategories of $\\CATCrs$.\n\nThe following is clear.\n\n\\begin{proposition}\nCloseness is respected by composition: If $f, f' \\from Y \\to X$ and $g, g' \n\\from Z \\to Y$ are coarse maps with $f \\closeequiv f'$ and $g \\closeequiv g'$, \nthen $f \\circ g \\closeequiv f' \\circ g'$.\n\\end{proposition}\n\nThis allows us to describe the arrows of $\\CATCrs$ as closeness equivalence \nclasses of coarse maps. Denote such classes by $[f]_\\TXTclose \\from Y \\to X$ \n(or simply $[f]$ for brevity), where $f$ is usually taken to be a \nrepresentative map $Y \\to X$, i.e., $\\Quotient(f) = [f]$. However, we will use \nthe notation $[f] \\from Y \\to X$ for arrows $Y \\to X$ in $\\CATCrs$ even when we \ndo not have a particular $f$ in mind.\n\nThe notion of isomorphism in $\\CATCrs$ is weaker than in $\\CATPCrs$. A coarse \nmap $f \\from Y \\to X$ is a \\emph{coarse equivalence} if $[f]$ is an isomorphism \nin $\\CATCrs$. In other words, $f$ is a coarse equivalence if and only if there \nis a coarse map $g \\from X \\to Y$ so that the two possible compositions are \nclose to the identities (i.e., $[f \\circ g] = [\\id_X]$ and $[g \\circ f] = \n[\\id_Y]$).\n\nA functor $F \\from \\CATPCrs \\to \\calC$, $\\calC$ any category, is \\emph{coarsely \ninvariant} if $f \\closeequiv f'$ implies $F(f) = F(f')$. Any coarsely invariant \n$F$ induces a functor $[F] \\from \\CATCrs \\to \\calC$ with $F = [F] \\circ \n\\Quotient$. Coarsely invariant functors send coarse equivalences to \nisomorphisms. For functors $F \\from \\CATPCrs \\to \\CATPCrs$ (or with codomain \none of the other precoarse categories), we abuse terminology and also say that \n$F$ is \\emph{coarsely invariant} if $\\Quotient \\circ F \\from \\CATPCrs \\to \n\\CATCrs$ is coarsely invariant in the previous (stronger) sense. Such a \ncoarsely invariant functor $F \\from \\CATPCrs \\to \\CATPCrs$ induces a functor \n$[F] \\from \\CATCrs \\to \\CATCrs$; if $F \\from \\CATPCrs \\to \\CATConnPCrs$, then \n$[F] \\from \\CATCrs \\to \\CATConnCrs$; etc.\n\n\n\\subsection{\\pdfalt{\\maybeboldmath $\\CATCrs$ versus $\\CATConnCrs$}%\n        {Crs versus CCrs}}\n\nThe relation between the quotient categories $\\CATCrs$ and $\\CATConnCrs$ is \nessentially the same as that between $\\CATPCrs$ and $\\CATConnPCrs$ for the \nfollowing reasons, which are easy to check.\n\n\\begin{proposition}\nThe functors $I \\from \\CATConnPCrs \\injto \\CATPCrs$ and $\\Connect \\from \n\\CATPCrs \\to \\CATConnPCrs$ are coarsely invariant, hence induce functors $[I] \n\\from \\CATConnCrs \\to \\CATCrs$ and $[\\Connect] \\from \\CATCrs \\to \\CATConnCrs$, \nrespectively. In fact, $[I]$ is just the inclusion and is fully faithful. \nAgain, $[\\Connect] \\circ [I]$ is the identity functor (now on $\\CATConnCrs$), \nand $[\\Connect]$ is left adjoint to $[I]$.\n\\end{proposition}\n\nConsequently, we get the following (exact) analogues of \nCorollary~\\ref{cor:PCrs-ConnPCrs-preserve}.\n\n\\begin{corollary\n$[I]$ preserves limits and $[\\Connect]$ preserves colimits. If $\\nu$ is a \nlimiting cone to (or colimiting cone from) $[I] \\circ \\calF$, where $\\calF \n\\from \\calJ \\to \\CATConnCrs$, then $[\\Connect] \\circ \\nu$ is a limiting cone to \n(or colimiting cone from, respectively) $\\calF = [\\Connect] \\circ [I] \\circ \n\\calF$.\n\\end{corollary}\n\n\\begin{remark}\nEvidently, $I$ and $\\Connect$ ``commute'' with the quotient functors \n$\\Quotient$ ($\\CATPCrs \\to \\CATCrs$ and its restriction $\\CATConnPCrs \\to \n\\CATConnCrs$) in that\n\\[\n    \\Quotient \\circ I = [I] \\circ \\Quotient\n\\quad\\text{and}\\quad\n    \\Quotient \\circ \\Connect = [\\Connect] \\circ \\Quotient.\n\\]\nThe quotient functors give a map of adjunctions (see, e.g., \n\\cite{MR1712872}*{Ch.~IV \\S{}7}) from $(\\Connect,I)$ to $([\\Connect],[I])$.\n\\end{remark}\n\n\\begin{remark}\n$[I]$ is fully faithful, but $[\\Connect]$ is neither full nor faithful (even \nthough $\\Connect$ is faithful, though also not full): e.g., consider\n\\[\n    \\Hom_{\\CATCrs}(\\ast, \\ast \\copro \\ast)\n\\quad\\text{and}\\quad\n    \\Hom_{\\CATCrs}(\\Connect(|\\setZplus|_1 \\copro |\\setZplus|_1),\n            |\\setZplus|_1 \\copro |\\setZplus|_1),\n\\]\nrespectively.\n\\end{remark}\n\n\n\\subsection{\\pdfalt{\\maybeboldmath $\\CATConnCrs$ versus $\\CATne{\\CATConnCrs}$}%\n        {CCrs versus CCrs\\textcaret{}x}}\n\nAfter passing to the quotients by closeness, the situation with respect to \n\\emph{nonempty} connected coarse spaces is greatly improved. Below, we work in \n$\\CATConnPCrs$ or its quotient $\\CATConnCrs$ (or the nonempty subcategories), \nso all coarse spaces will be connected. Let $I \\from \\CATne{\\CATConnPCrs} \n\\injto \\CATConnPCrs$ denote the inclusion; it is coarsely invariant, hence \ninduces $[I] \\from \\CATne{\\CATConnCrs} \\injto \\CATConnCrs$, which is also the \ninclusion and which is fully faithful. Again, the inclusion functors \n``commute'' with the quotient functors.\n\n\\begin{definition}\nFix a one-point coarse space $\\ast$. Define a functor\n\\[\n    \\AddPt \\from \\CATConnPCrs \\to \\CATne{\\CATConnPCrs}\n\\]\nas follows:\n\\begin{enumerate}\n\\item For a coarse space $X$, $\\AddPt(X) \\defeq X \\copro_{\\CATConnPCrs} \\ast$ \n    (coproduct in $\\CATConnPCrs$).\n\\item For a coarse map $f \\from Y \\to X$, $\\AddPt(f) \\defeq f \n    \\copro_{\\CATConnPCrs} \\id_\\ast$.\n\\end{enumerate}\n\\end{definition}\n\n(It is easy to construct the functor $\\AddPt$ concretely, and all functors \nsatisfying the above are naturally equivalent.) The following are all easy to \nverify.\n\n\\begin{proposition}\n$\\AddPt \\from \\CATConnPCrs \\to \\CATne{\\CATConnPCrs}$ is coarsely invariant and \nhence induces a functor $[\\AddPt] \\from \\CATConnCrs \\to \\CATne{\\CATConnCrs}$.\n\\end{proposition}\n\n$\\AddPt$ is not terribly useful, but $[\\AddPt]$ is.\n\n\\begin{proposition\n$[\\AddPt] \\circ [I]$ is naturally equivalent to the identity on \n$\\CATne{\\CATConnCrs}$. Moreover, $[\\AddPt] \\from \\CATConnCrs \\to \n\\CATne{\\CATConnCrs}$ is left adjoint to $[I]$.\n\\end{proposition}\n\nIt follows that $[\\AddPt]$ is naturally equivalent to a functor $[\\AddPt]'$ \nsuch that $[\\AddPt]' \\circ [I]$ is equal to the identity functor. It is easy to \ngive a natural equivalence $\\Id_{\\CATne{\\CATConnCrs}} \\to [\\AddPt] \\circ [I]$: \nfor each (nonempty, connected) $X$, the canonical inclusion $\\iota_X \\from X \n\\to X \\copro \\ast$ is a coarse equivalence hence an isomorphism $[\\iota_X] \n\\from X \\to \\AddPt(X)$ in $\\CATne{\\CATConnCrs}$.\n\n\\begin{corollary\n$[I]$ preserves limits and $[\\AddPt]$ preserves colimits. If $\\nu$ is a \nlimiting cone to (or, a colimiting cone from) $[I] \\circ \\calF$, where $\\calF \n\\from \\calJ \\to \\CATne{\\CATConnCrs}$, then $[\\AddPt] \\circ \\nu$ is a limiting \ncone to (or, respectively, a colimiting cone from) $[\\AddPt] \\circ [I] \\circ \n\\calF$ (or $\\calF$ after applying a natural equivalence).\n\\end{corollary}\n\n\n\\subsection{Limits in the coarse categories}\n\nWe first prove our assertion that nonzero products in the nonunital coarse \ncategories are just images (under the quotient functor) of products in the \nprecoarse categories. We then show the nonunital coarse categories also have \nall equalizers of pairs of arrows. It then follows by a standard construction \nthat the nonunital coarse categories have all nonzero limits.\n\n\\begin{proposition}\\label{prop:Crs-prod}\nSuppose $\\set{X_j \\suchthat j \\in J}$ ($J$ some index set) is a nonzero \ncollection of coarse spaces. The product of the $X_j$ in $\\CATCrs$ (or in \n$\\CATConnCrs$ or $\\CATne{\\CATConnCrs}$, as appropriate) is just the coarse \nspace\n\\[\n    X \\defeq \\pfx{\\CATPCrs}\\prod_{j \\in J} X_j\n\\]\n(product in $\\CATPCrs$) together with the projections $[\\pi_j] \\from X \\to \nX_j$, $j \\in J$ (closeness classes of the projections). Thus $\\CATCrs$ (and \n$\\CATConnCrs$ and $\\CATne{\\CATConnCrs}$) have all nonzero products.\n\\end{proposition}\n\nRecall, from Corollary~\\ref{cor:Crs-no-term}, that none of the quotient coarse \ncategories has a zero product, i.e., terminal object.\n\n\\begin{proof}\nThe cone $\\pi$ in $\\CATPCrs$ maps (via the quotient functor) to a cone $[\\pi] \n\\defeq \\Quotient \\circ \\pi$ in $\\CATCrs$; we must prove universality. Suppose \n$Y$ is a coarse space and $[\\mu_j] \\from Y \\to X_j$, $j \\in J$, is a collection \nof arrows in $\\CATCrs$. Choosing representative coarse maps $\\mu_j \\from Y \\to \nX_j$, we get (since the cone $\\pi$ is universal) a natural coarse map $t \\from \nY \\to X$ such that $\\mu_j = \\pi_j \\circ t$ for all $j$. Of course, this implies \n$[\\mu_j] = [\\pi_j] \\circ [t]$.\n\nWe must show that this $[t]$ is unique (hence does not depend on our choice of \nrepresentatives $\\mu_j$). Suppose $[t'] \\from Y \\to X$ is a class such that \n$[\\mu_j] = [\\pi_j] \\circ [t']$ for all $j$. Choose a representative $t'$. \nSuppose $F \\in \\calE_Y$, and put $E \\defeq (t \\cross t')(F)$, which is in \n$\\calE_{|X|_1}$ by Proposition~\\ref{prop:loc-prop-prod}. For each $j$, we have \nthat $\\mu'_j \\defeq \\pi_j \\circ t' \\closeequiv \\mu_j = \\pi_j \\circ t$, and \nhence\n\\[\n    (\\pi_j)^{\\cross 2}(E) = ((\\pi_j \\circ t) \\cross (\\pi_j \\circ t'))(F)\n\\]\nis in $\\calE_{X_j}$. Moreover, since $(\\mu_j \\cross \\mu'_j) |_F = \n(\\pi_j)^{\\cross 2} |_E \\circ (t \\cross t') |_E^F$ is proper (by the same \nProposition), $\\pi_j$ is locally proper for $E$. Thus $E \\in (\\pi_j)^* \n\\calE_{X_j}$ for all $j$, so $E \\in \\calE_X$. Hence $t$ is close to $t'$.\n\nFor $\\CATConnCrs$ and $\\CATne{\\CATConnCrs}$, it suffices to recall that nonzero \nproducts of connected coarse spaces are connected, and nonzero products of \nnonempty coarse spaces are nonempty.\n\\end{proof}\n\n\\begin{remark}\nFor obvious reasons, we cannot usually obtain products in the unital coarse \ncategories using the above construction. However, unlike in $\\CATPCrs$, this \ndoes not imply the nonexistence of products. In certain cases (see, e.g., \nRemark~\\ref{rmk:term-unital-prod}), the (nonunital) product above will be \ncoarsely equivalent to a unital coarse space which is a product in \n$\\CATUniCrs$. I do not know, in general, which products exist in $\\CATUniCrs$.\n\\end{remark}\n\nNext, we examine equalizers in the (nonunital) coarse categories. Unlike \nproducts, equalizers in the coarse categories are not usually ``the same'' as \nequalizers in the precoarse categories.\n\n\\begin{definition}\nSuppose $X$ is a coarse space and $f, f' \\from Y \\to X$ are set maps ($Y$ some \nset). $f$ and $f'$ are \\emph{pointwise connected} if $f(y)$ is connected to \n$f'(y)$ for all $y \\in Y$. $f$ and $f'$ are \\emph{close for $F \\in f^* \\calE_X \n\\intersect (f')^* \\calE_X$} if $(f \\cross f')(F) \\in \\calE_X$.\n\\end{definition}\n\nOf course, if $X$ is connected, all set maps into $X$ are pointwise connected. \nIf $f \\from Y \\to X$ is a coarse map, then any coarse map close to $f$ is \npointwise connected to $f$.\n\n\\begin{lemma}\nSuppose $X$ is a coarse space and $f, f' \\from Y \\to X$ are set maps. If $f$ \nand $f'$ are close for both $F, F' \\in \\calE_{|Y|_1}$, then $f$ and $f'$ are \nclose for $F + F'$, $F \\circ F'$, $F^\\transpose$, and all subsets of $F$.\n\\end{lemma}\n\n\\begin{proof}\nAgain, the only (slightly) nontrivial one is $F \\circ F'$:\n\\[\n    (f \\cross f')(F \\circ F') \\subseteq f^{\\cross 2}(F) \\circ (f \\cross f')(F')\n            \\in \\calE_X.\n\\]\n\\end{proof}\n\n\\begin{definition}\nSuppose $X$ is a coarse space and $f, f' \\from Y \\to X$ are pointwise connected \ncoarse maps. The \\emph{equalizing pull-back coarse structure} $(f,f')^* \n\\calE_X$ (on $Y$ along $f$ and $f'$) is\n\\[\n    (f,f')^* \\calE_X \\defeq \\set{F \\in f^* \\calE_X \\intersect (f')^* \\calE_X\n                \\suchthat \\text{$f$ and $f'$ are close for $F$}}.\n\\]\n\\end{definition}\n\nPointwise-connectedness is important: it guarantees that the singletons \n$\\set{1_y}$, $y \\in Y$, are in $(f,f')^* \\calE_X$. It then follows from the \nLemma that $(f,f')^* \\calE_X$ is a coarse structure on $Y$.\n\n\\begin{definition}\nSuppose $f, f' \\from Y \\to X$ are coarse maps. Define the \\emph{equalizer} of \n$[f]$ and $[f']$ is\n\\[\n    \\OBJequalizer_{[f],[f']}\n        \\defeq \\set{y \\in Y \\suchthat \\text{$f(y)$ is connected to $f'(y)$}}\n        \\subseteq Y\n\\]\nwith the coarse structure\n\\[\n    \\calE_{\\OBJequalizer_{[f],[f']}} \\defeq \\calE_Y |_{\\OBJequalizer_{[f],[f']}}\n            \\intersect (f |_{\\OBJequalizer_{[f],[f']}},\n                f' |_{\\OBJequalizer_{[f],[f']}})^* \\calE_X\n\\]\n(where $\\calE_Y |_{\\OBJequalizer_{[f],[f']}}$ is the subspace coarse structure \non $\\OBJequalizer_{[f],[f']} \\subseteq Y$), together with closeness class of \nthe inclusion map\n\\[\n    \\equalizer_{[f,[f']} \\from \\OBJequalizer_{[f],[f']} \\to Y\n\\]\n(which is coarse).\n\\end{definition}\n\nClearly, the restrictions $f |_{\\OBJequalizer_{[f],[f']}}$ and $f' \n|_{\\OBJequalizer_{[f],[f']}}$ are pointwise connected, so \n$\\calE_{\\OBJequalizer_{[f],[f']}}$ really is a coarse structure. Also, the \nabove definition does not depend on order, i.e., $\\OBJequalizer_{[f],[f']} = \n\\OBJequalizer_{[f'],[f]}$.\n\n\\begin{lemma}\nSuppose $f, f' \\from Y \\to X$ are coarse maps. The equalizer of $[f]$ and \n$[f']$ is coarsely invariant in the sense that $\\OBJequalizer_{[f],[f']}$ and \n$[\\equalizer_{[f],[f']}]$ (indeed, $\\equalizer_{[f],[f']}$) only depend on the \ncloseness class of $f$ and $f'$ (hence the notation).\n\\end{lemma}\n\n\\begin{proof}\nSuppose $e, e' \\from Y \\to X$ are close to $f$, $f'$, respectively. Then, for \nall $y \\in Y$, $e(y)$ is connected to $f(y)$ and $e'(y)$ is connected to \n$f'(y)$; for $y \\in \\OBJequalizer_{[f],[f']} \\subseteq Y$, $f(y)$ is connected \nto $f'(y)$ hence $e(y)$ is connected to $e'(y)$. Thus the \\emph{set} \n$\\OBJequalizer_{[f],[f']}$ is coarsely invariant.\n\nIt remains to show that the coarse structure $\\calE_{\\OBJequalizer_{[f],[f']}}$ \nis also coarsely invariant. Observe that\n\\[\n    \\calE_{\\OBJequalizer_{[f],[f']}}\n        = \\set{F \\in \\calE_Y |_{\\OBJequalizer_{[f],[f']}} \\suchthat\n                (f \\cross f')(F) \\in \\calE_X}\n\\]\nand $\\calE_Y |_{\\OBJequalizer_{[f],[f']}} = \\calE_Y \n|_{\\OBJequalizer_{[f],[f']}}$. If $F \\in \\calE_{\\OBJequalizer_{[f],[f']}}$, \nthen\n\\[\n    (e \\cross e')(F) \\subseteq (e \\cross f)(1_{F \\cdot Y})\n        \\circ (f \\cross f')(F) \\circ (f' \\cross e)(1_{Y \\cdot F})\n\\]\nis in $\\calE_X$, and so $F \\in \\calE_{\\OBJequalizer_{[e],[e']}}$; the reverse \ninclusion follows symmetrically.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:Crs-equal}\nThe equalizer of $[f], [f'] \\from Y \\to X$ really is (in the categorical sense) \nthe equalizer of $[f]$ and $[f']$ in $\\CATCrs$ (or in $\\CATConnCrs$ or \n$\\CATne{\\CATConnCrs}$, as appropriate), hence the terminology. Thus $\\CATCrs$ \n(and $\\CATConnCrs$ and $\\CATne{\\CATConnCrs}$) have all equalizers of pairs of \narrows.\n\\end{proposition}\n\n\\begin{proof}\nFix representative coarse maps $f$ and $f'$, and suppose $g \\from Z \\to Y$ is a \ncoarse map such that $f \\circ g \\closeequiv f' \\circ g$. Then clearly the (set) \nimage of $g$ is contained in $\\OBJequalizer_{[f],[f']}$, and indeed\n\\[\n    \\tilde{g} \\defeq g |^{\\OBJequalizer_{[f],[f']}}\n            \\from Z \\to \\OBJequalizer_{[f],[f']}\n\\]\nis clearly coarse with $g = \\equalizer_{[f],[f']} \\circ \\tilde{g}$ (hence $[g] \n= [\\equalizer_{[f],[f']}] \\circ [\\tilde{g}]$).\n\nWe must prove uniqueness of $[\\tilde{g}]$. Suppose $\\tilde{g}' \\from Z \\to \n\\OBJequalizer_{[f],[f']}$ is a coarse map with $g \\closeequiv \n\\equalizer_{[f],[f']} \\circ \\tilde{g}' \\eqdef g'$. Then, for all $G \\in \n\\calE_Z$,\n\\[\n    F \\defeq (\\tilde{g} \\cross \\tilde{g}')(G) = (g \\cross g')(G) \\in \\calE_Y\n\\]\nand, since $f \\circ g' \\closeequiv f \\circ g \\closeequiv f' \\circ g \\closeequiv \nf' \\circ g'$, we have\n\\[\n    (f \\cross f')(F) = ((f \\circ g') \\cross (f' \\circ g'))(F) \\in \\calE_X,\n\\]\nso $F \\in \\calE_{\\OBJequalizer_{[f],[f']}}$. Hence $\\tilde{g}$ is close to \n$\\tilde{g}'$, as required.\n\nIf $X$ and $Y$ are connected, then $\\OBJequalizer_{[f],[f']}$ is clearly \nconnected. Moreover, if $X$ is connected, then $\\OBJequalizer_{[f],[f']} = Y$ \nas a set and hence is nonempty if $Y$ is nonempty.\n\\end{proof}\n\n\\begin{remark}\nThe above construction does not work in the unital coarse categories because \nthe equalizing pull-back coarse structures are not in general unital (and one \ncannot ``unitalize'' them and still have the required properties). Again, this \ndoes not imply the nonexistence of equalizers in $\\CATUniCrs$, and I do not \nknow which equalizers exist in $\\CATUniCrs$.\n\\end{remark}\n\n\\begin{remark}\nWhen $X$ is a coarse space and $f, f' \\from Y \\to X$ are just set maps, one can \ntake\n\\[\n    \\calE_Y \\defeq f^* \\calE_X \\intersect (f')^* \\calE_X\n\\]\nand apply the above Proposition. If $g \\from Z \\to Y$ is another set map, one \ncan then take $\\calE_Z \\defeq g^* \\calE_Y$. Then $f \\circ g$ is close to $f' \n\\circ g$ if and only if $g$ factors through the equalizer of $[f]$ and $[f']$.\n\\end{remark}\n\nWe have now shown that the nonunital coarse categories have all nonzero \nproducts and all equalizers. It follows, using a standard argument, that these \ncategories have all nonzero limits. For completeness, we give this argument.\n\n\\begin{theorem}\\label{thm:Crs-lim}\nThe nonunital coarse categories $\\CATCrs$, $\\CATConnCrs$, and \n$\\CATne{\\CATConnCrs}$ have all nonzero limits.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\calC$ be one of the above categories and suppose $\\calF_X \\from \\calJ \\to \n\\calC$ ($\\calJ$ nonzero, small) is a functor, putting $X_j \\defeq \\calF_X(j)$ \nfor $j \\in \\Obj(\\calJ)$ as usual. If $\\calJ$ has no arrows (i.e., $\\Map(\\calJ) \n= \\emptyset$), then $\\pfx{\\calC}\\OBJlim \\calF_X$ is just a product, and we are \ndone.\n\nOtherwise, let\n\\begin{align*}\n    Y & \\defeq \\pfx{\\calC}\\prod_{\\mathclap{j \\in \\Obj(\\calJ)}} \\; X_j\n\\shortintertext{and}\n    X & \\defeq \\,\\pfx{\\calC}\\prod_{\\mathclap{u \\in \\Map(\\calJ)}}\\;\\,\n            X_{\\target(u)}.\n\\end{align*}\nWe have two collections of arrows $[f_u], [f'_u] \\from Y \\to X_{\\target(u)}$, \n$u \\in \\Map(\\calJ)$:\n\\[\n    [f_u] \\defeq [\\pi_{\\target(u)}]\n\\quad\\text{and}\\quad\n    [f'_u] \\defeq \\calF_X(u) \\circ [\\pi_{\\source(u)}].\n\\]\nBy the universal property of products, these collections of arrows give rise to \ncanonical arrows $[f] \\from Y \\to X$ and $[f'] \\from Y \\to X$, respectively. \nPut\n\\[\n    \\pfx{\\calC}\\OBJlim \\calF_X \\defeq \\OBJequalizer_{[f],[f']},\n\\]\nwith the cone $[nu] \\from \\pfx{\\calC}\\OBJlim \\calF_X \\to \\calF_X$ being defined \nby $[\\nu_j] \\defeq [\\pi_j] \\circ [\\equalizer_{[f],[f']}]$ for $j \\in \n\\Obj(\\calJ)$. It is easy to check that $[\\nu]$ is indeed a limiting cone.\n\\end{proof}\n\n\\begin{remark}\nIt follows from the above proof that, as a set, one can always take the limit \n$\\OBJlim \\calF_X$ to be a subset of the product (set) $\\prod_{j \\in \n\\Obj(\\calJ)} X_j$. When all the coarse spaces $X_j$ are connected (i.e., in \n$\\CATConnCrs$), one can take $\\OBJlim \\calF_X$ to be, as a set, exactly the set \nproduct. Moreover, the proof actually gives a concrete description of limits in \nthe coarse categories. If all the $X_j$ are connected, the coarse structure on\n\\[\n    Y \\defeq \\OBJlim \\calF_X\n        \\defeq \\pfx{\\CATSet}\\prod_{\\mathclap{j \\in \\Obj(\\calJ)}}\\; X_j\n\\]\nconsists of all $F \\in \\calE_{|Y|_1}$ such that, for all arrows $u \\in \n\\Map(\\calJ)$ and (all) representative coarse maps $f_u \\from X_{\\source(u)} \\to \nX_{\\target(u)}$ of $\\calF_X(u)$:\n\\begin{enumerate}\n\\item $((f_u \\circ \\pi_{\\source(u)}) \\cross \\pi_{\\target(u)}) |_F$ is proper; \n    and\n\\item $((f_u \\circ \\pi_{\\source(u)}) \\cross \\pi_{\\target(u)})(F)$ is an \n    entourage of $X_{\\target(u)}$.\n\\end{enumerate}\n(By taking $u$ to be the identity arrow of $j \\in \\Obj(\\calJ)$, one gets that \nthe $\\pi_j$ are coarse for $F$.)\n\\end{remark}\n\n\n\\subsection{Entourages as subspaces of products}\\label{ent-subsp-prod}\n\nIs there a relation between entourages of a coarse space $X$, which are subsets \nof $X^{\\cross 2} \\defeq X \\cross X$, and the product coarse space $X \\cross X$? \nWe first need a coarse space $\\Terminate(X)$ which we will discuss more \nthoroughly in \\S\\ref{subsect:Crs-Term}: For any $X$, $\\Terminate(X) \\defeq X$ \nas a set, with coarse structure\n\\[\n    \\calE_{\\Terminate(X)} \\defeq \\set{E \\in \\calE_{|X|_1}\n            \\suchthat 1_{E \\cdot X}, 1_{X \\cdot E} \\in \\calE_X}.\n\\]\nNote that if $X$ is unital, $\\Terminate(X) = |X|_1$.\n\nThe following will be useful later in conjunction with various universal \nproperties, as well as generalized coarse quotients (which we intend to study \nin \\cite{crscat-quot}).\n\n\\begin{proposition}\\label{prop:ent-prod}\nSuppose $X$ is a coarse space. If $E \\in \\calE_{\\Terminate(X)}$, then $E$ can \nbe considered as a unital subspace $|E|$ of the product coarse space $X \\cross \nX$. If in fact $E \\in \\calE_X$, then the restricted projections $\\pi_1 |_{|E|}, \n\\pi_2 |_{|E|} \\from |E| \\to X$ are close. Conversely, any unital subspace $|E| \n\\subseteq X \\cross X$ determines a subset $E \\in \\calE_{\\Terminate(X)} \n\\subseteq \\calE_{|X|_1}$; if $\\pi_1 |_{|E|}$, $\\pi_2 |_{|E|}$ are close, then \n$E \\in \\calE_X$.\n\\end{proposition}\n\n\\begin{proof}\nIf $E \\in \\calE_{\\Terminate(X)}$, then $1_{|E|}$ is an entourage of $X \\cross \nX$: certainly $1_{|E|} \\in \\calE_{|X \\cross X|_1}$, and $(\\pi_1 \\cross \n\\pi_1)(1_{|E|}) = 1_{E \\cdot X}$ and $(\\pi_2 \\cross \\pi_2)(1_{|E|}) = 1_{X \n\\cdot E}$ are entourages of $X$. If $E \\in \\calE_X$, then $(\\pi_1 \\cross \n\\pi_2)(1_{|E|}) = E \\in \\calE_X$; since $|E|$ is unital, it follows that the \nrestricted projections are close.\n\nConversely, suppose $|E| \\subseteq X \\cross X$ is a unital subspace. Then the \nrestricted projections $\\pi_1 |_E = \\pi_1 |_{|E|}$ and $\\pi_2 |_E = \\pi_2 \n|_{|E|}$ are proper, so $E \\in \\calE_{|X|_1}$. Since $\\pi_1$ maps unital \nsubspaces of $X \\cross X$ to unital subspaces of $X$ and $\\pi_1(|E|) = E \\cdot \nX$, the left support, and symmetrically the right support, of $E$ is a unital \nsubspace of $X$, and so $E \\in \\calE_{\\Terminate(X)}$. If the restricted \nprojections are close, then $E = (\\pi_1 \\cross \\pi_2)(1_{|E|}) \\in \\calE_X$.\n\\end{proof}\n\n\n\\subsection{Colimits in the coarse categories}\n\nWe now do the same for coproducts, coequalizers, and thus colimits in the \ncoarse categories.\n\n\\begin{proposition}\nSuppose that $\\calC$ is one of the coarse categories $\\CATCrs$, $\\CATConnCrs$, \n$\\CATUniCrs$, or $\\CATConnUniCrs$, that $\\calP\\calC$ is the corresponding \nprecoarse category, and that $\\set{Y_j \\suchthat j \\in J}$ ($J$ some index set) \nis a collection of coarse spaces in $\\calC$ (or $\\calP\\calC$). The coproduct of \nthe $Y_j$ in $\\calC$ is just the coarse space\n\\[\n    Y \\defeq \\pfx{\\calP\\calC}\\coprod_{j \\in J} Y_j\n\\]\n(coproduct in $\\calP\\calC$) together with the ``inclusions'' $[\\iota_j] \\from \nY_j \\to Y$, $j \\in J$ (closeness classes of the inclusions). If instead $\\calC \n= \\CATne{\\CATConnCrs}$, then the same holds except when $J = \\emptyset$, in \nwhich case the coproduct is any one-point coarse space. Thus all the coarse \ncategories have all coproducts.\n\\end{proposition}\n\n\\begin{proof}\nWe have shown (or at least mentioned, in the unital cases) the existence of the \ncorresponding coproduct cone $\\iota$ in the corresponding precoarse category, \nleaving aside the special case of $\\calC = \\CATne{\\CATConnCrs}$ and $J = \n\\emptyset$ (which is easily handled). The quotient functor yields a cone \n$[\\iota]$ in the coarse category $\\calC$; we must show that it is universal.\n\nSuppose $X$ is a coarse space and $[\\mu_j] \\from Y_j \\to X$, $j \\in J$, is a \ncollection of arrows in $\\calC$. Choosing representative coarse maps $\\mu_j$, \nwe get a natural coarse map $t \\from Y \\to X$ such that $\\mu_j = t \\circ \n\\iota_j$ (and hence $[\\mu_j] = [\\pi_j \\circ [t]$) for all $j$. We must show \nthis $[t]$ is unique. Suppose $t' \\from Y \\to X$ is such that $\\mu_j \n\\closeequiv t' \\circ \\iota_j$ for all $j$. The coarse structure on the \nprecoarse coproduct $Y$ is generated by $F \\defeq (\\iota_j)^{\\cross 2}(F_j)$, \n$F_j \\in \\calE_{Y_j}$, $j \\in J$, and so to show $t \\closeequiv t'$ it is \nenough to show that $(t \\cross t')(F) \\in \\calE_X$ for such $F$. But\n\\[\n    (t \\cross t')(F) = ((t \\circ \\iota_j) \\cross (t' \\circ \\iota_j))(F_j)\n\\]\nis in $\\calE_X$ since $t \\circ \\iota_j = \\mu_j \\closeequiv t' \\circ \\iota_j$, \nas required.\n\\end{proof}\n\nNext, coequalizers: Unlike coproducts, coequalizers in the coarse categories \ndiffer from coequalizers in the precoarse categories; in particular, they \nalways exist.\n\n\\begin{definition}\nSuppose $Y$ is a coarse space and $f, f' \\from Y \\to X$ ($X$ some set) are \nlocally proper maps. The \\emph{coequalizing push-forward coarse structure} \n$(f,f')_* \\calE_Y$ (on $X$ along $f$ and $f'$) is\n\\[\n    (f,f')_* \\calE_Y \\defeq \\langle f_* \\calE_Y, (f')_* \\calE_Y,\n            \\set{(f \\cross f')(F) \\suchthat F \\in \\calE_Y} \\rangle_Y.\n\\]\n(We may similarly define connected, unital, and connected unital versions.)\n\\end{definition}\n\nBy Proposition~\\ref{prop:loc-prop-prod}, the sets $(f \\cross f')(F)$ satisfy \nthe properness axiom. The coequalizing push-forward coarse structure makes $f$ \nand $f'$ \\emph{close} coarse maps, and is the minimum coarse structure on $X$ \nfor which this is true.\n\n\\begin{definition}\nSuppose $f, f' \\from Y \\to X$ are coarse maps. The \\emph{coequalizer} of $[f]$ \nand $[f']$ is $\\OBJcoequalizer_{[f],[f']} \\defeq X$ equipped the coarse \nstructure\n\\[\n    \\calE_{\\OBJcoequalizer_{[f],[f']}}\n        \\defeq \\langle \\calE_X, (f,f')_* \\calE_Y \\rangle_X,\n\\]\ntogether with the closeness class of ``identity'' map\n\\[\n    \\coequalizer_{[f],[f']} \\from X \\to \\OBJcoequalizer_{[f],[f']}\n\\]\n(which is a coarse map).\n\\end{definition}\n\nObserve that if $X$ is unital so too is the coequalizer, and similarly if $X$ \nis connected.\n\n\\begin{lemma}\nSuppose $f, f' \\from Y \\to X$ are coarse maps. The coequalizer of $[f]$ and \n$[f']$ is coarsely invariant (hence the notation).\n\\end{lemma}\n\n\\begin{proof}\nSuppose $e, e' \\from Y \\to X$ are close to $f$, $f'$, respectively. Observe \nthat, since $f_* \\calE_Y, (f')_* \\calE_Y \\subseteq \\calE_X$,\n\\[\n    \\calE_{\\OBJcoequalizer_{[f],[f']}} = \\langle \\calE_X,\n            \\set{(f \\cross f')(F) \\suchthat F \\in \\calE_Y} \\rangle_X\n\\]\nand similarly for $e$ and $e'$. Thus it suffices to show\n\\[\n    \\set{(e \\cross e')(F) \\suchthat F \\in \\calE_Y}\n        \\subseteq \\calE_{\\OBJcoequalizer_{[f],[f']}}\n\\]\nand similarly symmetrically. But if $F \\in \\calE_Y$, then\n\\[\n    (e \\cross e')(F) \\subseteq (e \\cross f)(1_{F \\cdot Y})\n            \\circ (f \\cross f')(F) \\circ (f' \\cross e')(1_{Y \\cdot F})\n\\]\nis in $\\calE_{\\OBJcoequalizer_{[f],[f']}}$, as required.\n\\end{proof}\n\n\\begin{proposition}\nThe coequalizer of $[f], [f'] \\from Y \\to X$ really is (in the categorical \nsense) the coequalizer of $[f]$ and $[f']$ in $\\CATCrs$ (or in $\\CATConnCrs$, \n$\\CATne{\\CATConnCrs}$, $\\CATUniCrs$, or $\\CATConnUniCrs$, as appropriate), \nhence the terminology. Thus $\\CATCrs$ (and the other coarse categories) have \nall coequalizers of pairs of arrows.\n\\end{proposition}\n\n\\begin{proof}\nFix representative coarse maps $f$ and $f'$, and suppose $g \\from X \\to W$ is a \ncoarse map such that $g \\circ f \\closeequiv g \\circ f'$. Let $\\utilde{g} \\from \n\\OBJcoequalizer_{[f],[f']} \\to W$ be the same, as a set map, as $g$; then \nclearly $g = \\utilde{g} \\circ \\coequalizer_{[f],[f']}$, and hence $[g] = \n[\\utilde{g}] \\circ [\\coequalizer_{[f],[f']}]$, assuming $\\utilde{g}$ is \nactually a coarse map. To show that $\\utilde{g}$ is coarse, it suffices to show \nthat $\\utilde{g}$ coarse for sets $E \\defeq (f \\cross f')(F)$, $F \\in \\calE_Y$. \nSince\n\\[\n    ((g \\circ f) \\cross (g \\circ f')) |_F\n        = g^{\\cross 2} |_E \\circ (f \\cross f') |_F^E\n\\]\nis proper (Proposition~\\ref{prop:loc-prop-prod}), it follows that \n$\\utilde{g}^{\\cross 2} |_E = g^{\\cross 2} |_E$ is proper, hence $\\utilde{g}$ is \nlocally proper for $E$. Since $g \\circ f$ and $g \\circ f'$ are close, it \nfollows that $\\utilde{g}$ preserves $E$.\n\nUniqueness of $[\\utilde{g}]$: Suppose $\\utilde{g}' \\from \n\\OBJcoequalizer_{[f],[f']} \\to W$ is a coarse map such that $g \\closeequiv \n\\utilde{g}' \\circ \\coequalizer_{[f],[f']}$. To show that $\\utilde{g}$ is close \nto $\\utilde{g}'$, we must show that $(\\utilde{g} \\cross \\utilde{g}')(E) \\in \n\\calE_W$ for all $E \\in \\calE_{\\OBJcoequalizer_{[f],[f']}}$. Clearly, this is \nthe case for $E \\in \\calE_X \\subseteq \\calE_{\\OBJcoequalizer_{[f],[f']}}$, so \nit suffices to show this for $E = (f \\cross f')(F)$ for some $F \\in \\calE_Y$. \nThe map $g' \\defeq \\utilde{g}' \\circ \\coequalizer_{[f],[f']}$ is close to $g$, \nhence $g \\circ f \\closeequiv g' \\circ f'$. Therefore,\n\\[\n    (\\utilde{g} \\cross \\utilde{g}')((f \\cross f')(F))\n        = ((g \\circ f) \\cross (g' \\circ f'))(F),\n\\]\nis in $\\calE_W$, as required.\n\nAs previously noted, if $X$ is connected, unital, and\/or nonempty, then \n$\\OBJequalizer_{[f],[f']}$ has the corresponding property or properties, so the \nabove actually proves the result in all the coarse categories.\n\\end{proof}\n\nSince the coarse categories have all coproducts and coequalizers, we \nimmediately get the following.\n\n\\begin{theorem}\\label{thm:Crs-colim}\nThe coarse categories $\\CATCrs$, $\\CATConnCrs$, $\\CATne{\\CATConnCrs}$, \n$\\CATUniCrs$, and $\\CATConnUniCrs$ have all colimits.\n\\end{theorem}\n\n\n\\subsection{The termination functor}\\label{subsect:Crs-Term}\n\nFor essentially set theoretic reasons, $\\CATCrs$ does not have a terminal \nobject (Corollary~\\ref{cor:Crs-no-term}). However, for many purposes, one can \nfind a suitable substitute. We begin with some general definitions which are \napplicable in any category $\\calC$.\n\n\\begin{definition}\nIn $\\calC$, an object $\\tilde{X}$ \\emph{terminates} an object $X$ if:\n\\begin{enumerate}\n\\item there is a (unique) arrow $\\tau_X \\from X \\to \\tilde{X}$; and\n\\item for all $Y \\in \\Obj(\\calC)$, there is at most one arrow $Y \\to \\tilde{X}$.\n\\end{enumerate}\nI.e., $\\tilde{X}$ is terminal in the full subcategory of $\\calC$ consisting of \n$X$ and all objects mapping to $\\tilde{X}$. $\\tilde{X}$ \\emph{universally \nterminates} $X$ if it is the smallest object terminating $X$ (i.e., for all \n$\\tilde{X}'$ terminating, $X$ there is an arrow $\\tilde{X} \\to \\tilde{X}'$).\n\\end{definition}\n\nIf $\\tilde{X}$ terminates $X$, then for all $Y$ and pairs of arrows $f, g \\from \nY \\to X$, $\\tau_X \\circ f = \\tau_X \\circ g$. Two objects universally \nterminating $X$ are canonically and uniquely isomorphic. If $\\tilde{X}$ \nterminates any object, then it universally terminates itself.\n\nIn a category with a terminal object $1$, the product of any object $Y$ and $1$ \nis just $Y$. The following generalizes this.\n\n\\begin{proposition}\\label{prop:term-id}\nIf there is some arrow $f \\from Y \\to X$ in $\\calC$ and $\\tilde{X}$ \nterminates $X$ in $\\calC$, then $Y$ is the (categorical) product of \n$\\tilde{X}$ and $Y$ (in $\\calC$).\n\\end{proposition}\n\n\\begin{proof}\nThe two ``projections'' from $Y$ are $\\pi_{\\tilde{X}} \\defeq \\tau_X \\circ f \n\\from Y \\to \\tilde{X}$ and $\\pi_Y \\defeq \\id_Y \\from Y \\to Y$. Suppose $Z \\in \n\\Obj(\\calC)$ is equipped with arrows $p_{\\tilde{X}} \\from Z \\to \\tilde{X}$ and \n$p_Y \\from Z \\to Y$. Both these arrows factor through $p_Y$: evidently $p_Y = \n\\pi_Y \\circ p_Y$, but also $p_{\\tilde{X}} = \\pi_{\\tilde{X}} \\circ p_Y$ since \nthere is only one arrow $Z \\to \\tilde{X}$.\n\\end{proof}\n\nIf $\\calC$ is known to have products (of pairs of objects), we can restate the \nabove Proposition in the following way: Whenever there is an arrow $f \\from Y \n\\to X$ and $\\tilde{X}$ terminates $X$, the projection $\\pi_Y \\from \n\\tilde{X} \\cross Y \\to Y$ is an isomorphism. Moreover, one the inverse \nisomorphism is given by the composition\n\\[\n    Y \\nameto{\\smash{\\Delta_Y}} Y \\cross Y\n        \\nameto{\\smash{(\\tau_X \\circ f) \\cross \\id_Y}} \\tilde{X} \\cross Y.\n\\]\n\n\\begin{definition}\nA \\emph{termination functor} on $\\calC$ is a functor $\\calC \\to \\calC$ \n(temporarily denoted $X \\mapsto \\tilde{X}$) which sends each $X$ to an object \n$\\tilde{X}$ terminating $X$; such a functor is \\emph{universal} if $\\tilde{X}$ \nalways universally terminates $X$.\n\\end{definition}\n\nThe following is implied: Whenever there is an arrow $f \\from Y \\to X$, there \nis a unique arrow $\\tilde{Y} \\to \\tilde{X}$ (namely $\\tilde{f}$). Note that \nuniversality is meant in the ``pointwise'' sense, and we do not assert \nuniversality as a termination functor. Universal termination functors are \nunique up to natural equivalence. Also observe that universal termination \nfunctors are idempotent up to natural equivalence.\n\n\\begin{example}\nIf $\\calC$ has a terminal object $1$, then $1$ terminates all objects, and $X \n\\mapsto 1$ is a termination functor (not necessarily universal). In \n$\\CATpt{\\CATSet}$ or $\\CATpt{\\CATTop}$ (pointed sets or topological spaces, \nrespectively), the functor $X \\mapsto \\ast$, where $\\ast$ is any one-point set \nor space, is a universal termination functor. More generally, in any category \n$\\calC$ with a zero object $0$ (i.e., $0$ is initial and terminal), $X \\mapsto \n0$ is a universal termination functor.\n\\end{example}\n\n\\begin{example}\\label{ex:Set-Top-univ-term}\nIn $\\CATSet$ or $\\CATTop$, the functor given by\n\\[\n    X \\mapsto \\begin{cases}\n            \\emptyset & \\text{if $X = \\emptyset$, or} \\\\\n            \\ast & \\text{if $X \\neq \\emptyset$,}\n        \\end{cases}\n\\]\nis a universal termination functor.\n\\end{example}\n\n\\begin{example}\nIn $\\CATCrs$ (and our various full subcategories), $|X|_1$ terminates any \ncoarse space $X$ (Proposition~\\ref{prop:term-close}). However, $X \\mapsto \n|X|_1$ does not define a functor on $\\CATPCrs$ (or $\\CATCrs$). E.g., for any \nset $X$, there is always a (unique) coarse map from $|X|_0$ to a one-point \ncoarse space $\\ast$, but no coarse map $|X|_1 = |\\,|X|_0\\,|_1 \\to \\ast$ when \n$X$ is infinite. The problem is that coarse maps from $|X|_1$ must be globally \nproper; in the unital categories this is not a problem, so $X \\mapsto |X|_1$ \ndoes define a coarsely invariant functor $\\CATUniPCrs \\to \\CATUniPCrs$ (for \nexample). The induced functor on unital coarse category $\\CATUniCrs$ is a \nuniversal termination functor. We wish to generalize this to all of $\\CATCrs$.\n\\end{example}\n\nWe recall the definition of the coarse space $\\Terminate(X)$ (for $X$ a coarse \nspace) from \\S\\ref{ent-subsp-prod}, and extend $\\Terminate$ to a functor in the \nobvious way.\n\n\\begin{definition}\nFor any coarse space $X$, $\\Terminate(X)$ is the coarse space which is just $X$ \nas a set with coarse structure\n\\[\n    \\calE_{\\Terminate(X)} \\defeq \\set{E \\in \\calE_{|X|_1}\n            \\suchthat 1_{E \\cdot X}, 1_{X \\cdot E} \\in \\calE_X};\n\\]\n$\\tau_X \\from X \\to \\Terminate(X)$ is the ``identity'' map. If $f \\from Y \\to \nX$, $\\Terminate(f) \\from \\Terminate(Y) \\to \\Terminate(X)$ is the same as $f$ as \na set map.\n\\end{definition}\n\nObserve the following:\n\\begin{enumerate}\n\\item $E \\subseteq X^{\\cross 2}$ is an entourage of $\\Terminate(X)$ if and only \n    if $E$ satisfies the properness axiom (i.e., $E \\in \\calE_{|X|_1}$) and the \n    left and right supports of $E$ are unital subspaces of $X$.\n\\item $\\Terminate(X)$ has the same unital subspaces as $X$ and is the maximum \n    coarse structure on $X$ with this property. (Consequently, if $X$ is \n    unital, $\\Terminate(X) = |X|_1$. It also follows that $\\Terminate$ is \n    idempotent, and hence so too is the induced functor $[\\Terminate]$; see \n    below.)\n\\end{enumerate}\n\n\\begin{proposition}\n$\\Terminate$ is a coarsely invariant functor $\\CATPCrs \\to \\CATPCrs$. The \ninduced functor $[\\Terminate] \\from \\CATCrs \\to \\CATCrs$ is a universal \ntermination functor.\n\\end{proposition}\n\n\\begin{proof}\nThat $\\Terminate(f)$ is a coarse map follows from the above observations, and \nhence $\\Terminate(f)$ is a functor. Moreover, using the above observations, we \nsee that, for all $X$, all coarse maps to $\\Terminate(X)$ are close. In \nparticular, this implies first that $\\Terminate$ is coarsely invariant and \nsecond that $[\\Terminate]$ is a termination functor on $\\CATCrs$.\n\nIt only remains to show universality. Suppose $\\tilde{X}$ terminates $X$, so \nthere is a unique $[t] \\from X \\to \\tilde{X}$, represented by a coarse map $t$, \nsay. It suffices to show that there is a coarse map $t' \\from \\Terminate(X) \\to \n\\tilde{X}$; since $\\tilde{X}$ terminates $X$ in $\\CATCrs$, uniqueness of $[t']$ \nfollows, as does the equality $[t] = [t'] \\circ [\\tau_X]$.\n\nTake $t' \\defeq t \\from \\Terminate(X) = X \\to \\tilde{X}$ as a set map. Local \nproperness of $t'$ follows from the above observations and \nProposition~\\ref{prop:loc-prop}\\enumref{prop:loc-prop:II}. To see that $t'$ \npreserves entourages, we use Proposition~\\ref{prop:ent-prod}: If $E \\in \n\\calE_{\\Terminate(X)}$, consider the unital subspace $|E|$ of the product \ncoarse space $X \\cross X$. Since $\\tilde{X}$ terminates $X$, $t \\circ \\pi_1 \n|_{|E|} \\closeequiv t \\circ \\pi_2 |_{|E|}$, and hence\n\\[\n    ((t \\circ \\pi_1 |_{|E|}) \\cross (t \\circ \\pi_2 |_{|E|}))(1_{|E|})\n        = (t')^{\\cross 2}(E)\n\\]\nis an entourage of $\\tilde{X}$, as required.\n\\end{proof}\n\nIn the above proof, one could instead consider the map $\\Terminate(t) \\from \n\\Terminate(X) \\to \\Terminate(\\tilde{X})$, and show that $\\Terminate(\\tilde{X}) =\n\\tilde{X}$.\n\n\\begin{remark}\n$\\Terminate$ restricts to (coarsely invariant) endofunctors on the other \nprecoarse categories, and hence $[\\Terminate]$ restricts to universal \ntermination functors on the other coarse categories. (The proof of the above \nProposition requires only unital coarse spaces $|E|$ and not actually the \nnonunital products $X \\cross X$, and hence works even in the unital cases.) Of \ncourse, in the unital cases, $\\Terminate$ is just the functor $X \\mapsto \n|X|_1$.\n\\end{remark}\n\nBy applying Proposition~\\ref{prop:term-id}, we immediately get the following, \nwhich will play a crucial role in the development of exponential objects in the \ncoarse categories \\cite{crscat-II}.\n\n\\begin{corollary}\\label{cor:Crs-term-id}\nIf there is a coarse map $Y \\to \\Terminate(X)$, where $X$ and $Y$ are coarse \nspaces, then\n\\[\n    \\pi_Y \\from \\Terminate(X) \\cross Y \\to Y\n\\]\nis a coarse equivalence. The maps\n\\[\n    D_\\tau \\defeq (\\tau \\cross \\id_Y) \\circ \\Delta_Y\n            \\from Y \\to \\Terminate(X) \\cross Y,\n\\]\nwhere $\\tau \\from Y \\to \\Terminate(X)$ is any coarse map (they are all close), \nare coarsely inverse to $\\pi_Y$. Hence, if there is a coarse map $Y \\to \n\\Terminate(X)$, then $Y \\cong \\Terminate(X) \\cross Y$ canonically in $\\CATCrs$ \n(or in $\\CATConnCrs$ or $\\CATne{\\CATConnCrs}$). In the case $Y \\defeq X$, we \nget that $\\pi_X \\from \\Terminate(X) \\cross X \\to X$ and $D_X \\defeq D_{\\tau_X} \n\\from X \\to \\Terminate(X) \\cross X$ are coarsely inverse coarse equivalences, \nso $X \\cong \\Terminate(X) \\cross X$ canonically in $\\CATCrs$ (or in \n$\\CATConnCrs$ or $\\CATne{\\CATConnCrs}$).\n\\end{corollary}\n\n\\begin{remark}\\label{rmk:term-unital-prod}\nFor any set $X$, $\\Terminate(|X|_1) = |X|_1$, so $|X|_1 \\cross |X|_1$ is \n(canonically) coarsely equivalent to $|X|_1$. While $|X|_1$ is always unital, \n$|X|_1 \\cross |X|_1$ is unital only when $X$ is finite. In particular, \nunitality is \\emph{not} coarsely invariant. It also follows easily that $|X|_1$ \nis actually the product of $|X|_1$ with itself in the unital coarse category \n$\\CATUniCrs$. More generally, for any coarse space $X$, the product of $X$ and \n$|X|_1$ in $\\CATUniCrs$ is just $X$. (As previously mentioned, $\\CATUniCrs$ has \nsome products of infinite spaces, even though the natural construction of the \ncorresponding products in $\\CATCrs$ are nonunital.)\n\\end{remark}\n\n\n\\subsection{Monics and images}\n\n\\begin{example}\nPull-back coarse structures are not coarsely invariant. That is, suppose $f, f' \n\\from Y \\to X$ are coarse maps. Even if $f \\closeequiv f'$, it may not be the \ncase that $f^* \\calE_X = (f')^* \\calE_X$. To see this, take $Y \\defeq \n|\\setN|_0^\\TXTconn$, $X \\defeq |\\setN|_1$, $f$ to be the ``identity'' map (as a \nset map), and $f'$ to be a constant map. Then $f^* \\calE_X = \\calE_{|Y|_1}$ \nwhereas $(f')^* \\calE_X = \\calE_Y$.\n\\end{example}\n\n\\begin{proposition}\nIf $f, f' \\from Y \\to X$ are coarse maps with $f \\closeequiv f'$, then\n\\[\n    \\calE_{\\Terminate(Y)} \\intersect f^* \\calE_X\n        = \\calE_{\\Terminate(Y)} \\intersect (f')^* \\calE_X.\n\\]\n\\end{proposition}\n\n\\begin{proof}\nWe prove inclusion $\\subseteq$; containment $\\supseteq$ follows symmetrically. \nSuppose $F \\in \\calE_{\\Terminate(Y)} \\intersect f^* \\calE_X$. Since $F \\in \n\\Terminate(Y)$, $f'$ is locally proper for $F$ \n(Proposition~\\ref{prop:loc-prop}\\enumref{prop:loc-prop:II}). It only remains to \nshow that $(f')^{\\cross 2}(F) \\in \\calE_X$. But\n\\[\\begin{split}\n    (f')^{\\cross 2}(F) \\subseteq (f' \\cross f)(1_{F \\cdot Y})\n            \\circ f^{\\cross 2}(F) \\circ (f \\cross f')(1_{Y \\cdot F}) \\in \\calE_X\n\\end{split}\\]\nsince $f \\closeequiv f'$ (and the left and right supports of $F$ are unital \nsubspaces of $Y$) and $f^{\\cross 2}(F) \\in \\calE_X$.\n\\end{proof}\n\n\\begin{definition}\nSuppose $[f] \\from Y \\to X$. The \\emph{coarsely invariant pull-back coarse \nstructure} $[f]^* \\calE_X$ on $Y$ (along $[f]$) is given by\n\\[\n    [f]^* \\calE_X \\defeq \\calE_{\\Terminate(Y)} \\intersect f^* \\calE_X\n\\]\n(where $f \\from Y \\to X$ is any representative coarse map).\n\\end{definition}\n\n\\begin{proposition}\\label{prop:Crs-factor-I}\nIf $[f] \\from Y \\to X$ is represented by a coarse map $f$, then $[f]$ factors \nas\n\\[\n    Y \\nameto{\\smash{[\\beta]}} |Y|_{[f]^* \\calE_X}\n        \\nameto{\\smash{[\\utilde{f}]}} X,\n\\]\nwhere $\\beta = \\id_Y$ and $\\utilde{f} = f$ as set maps (i.e., $\\calE_Y \n\\subseteq [f]^* \\calE_X$). Moreover, $[\\utilde{f}]$ depends only on $[f]$ (and \nnot on the particular $f$) and is unique in the above factorization.\n\\end{proposition}\n\n\\begin{proof}\nThe factorization follows immediately from Corollary~\\ref{cor:crs-factor-I}. We \nnow show that $f \\closeequiv f'$ implies $\\utilde{f} \\closeequiv \\utilde{f}'$ \n(noting that $[f]^* \\calE_X = [f']^* \\calE_X$). If $F \\in [f]^* \\calE_X$, then\n\\[\\begin{split}\n    (f \\cross f')(F) & = (f \\cross f')(F \\circ 1_{Y \\cdot F}) \\\\\n        & \\subseteq f^{\\cross 2}(F) \\circ (f \\cross f')(1_{Y \\cdot F})\n\\end{split}\\]\nis in $\\calE_X$ since $f^{\\cross 2}(F) \\in \\calE_X$ and $1_{Y \\cdot F} \\in \n\\calE_Y$ so $(f \\cross f')(1_{Y \\cdot F}) \\in \\calE_X$ as $f \\closeequiv f'$. \nUniqueness: If $[f] = [g] \\circ [\\beta]$, where $[g] \\from |Y|_{[f]^* \\calE_X} \n\\to X$ and $g$ is any representative, then $f \\closeequiv g \\circ \\beta$, so \n$\\utilde{f} \\closeequiv (g \\circ \\beta)\\utilde{\\mathstrut} = g$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:Crs-monic}\n$[f] \\from Y \\to X$ is monic in $\\CATCrs$ if and only if $\\calE_Y = [f]^* \n\\calE_X$ (i.e., if and only if $Y = |Y|_{[f]^* \\calE_X}$).\n\\end{proposition}\n\n\\begin{proof}\nFix a representative coarse map $f \\from Y \\to X$ and let $Y \n\\nameto{\\smash{\\beta}} |Y|_{[f]^* \\calE_X} \\nameto{\\smash{\\utilde{f}}} X$ be \nthe canonical factorization.\n\n(\\textimplies): Suppose there exists some $F \\in [f]^* \\calE_X \\setminus \n\\calE_Y$. Consider $|F|$ as a unital subspace of the product $Y \\cross Y$, with \nprojections $\\pi_1 |_{|F|}, \\pi_2 |_{|F|} \\from |F| \\to Y$. Then\n\\[\n    (\\pi_1 |_{|F|} \\cross \\pi_2 |_{|F|})(1_{|F|}) = F,\n\\]\nso $\\pi_1 |_{|F|}$ is not close to $\\pi_2 |_{|F|}$, but $\\beta \\circ \\pi_1 \n|_{|F|}$ is close to $\\beta \\circ \\pi_2 |_{|F|}$. Hence $[\\pi_1 |_{|F|}] \\neq \n[\\pi_2 |_{|F|}]$ but\n\\[\n    [f] \\circ [\\pi_1 |_{|F|}] = [\\utilde{f}] \\circ [\\beta] \\circ [\\pi_1 |_{|F|}]\n        = [\\utilde{f}] \\circ [\\beta] \\circ [\\pi_2 |_{|F|}]\n        = [f] \\circ [\\pi_2 |_{|F|}],\n\\]\nso $[f]$ is not monic.\n\n(\\textimpliedby): Suppose $g, g' \\from Z \\to Y$ are coarse maps such that $[f] \n\\circ [g] = [f] \\circ [g']$. Then, for each $G \\in \\calE_Z$,\n\\[\n    ((f \\circ g) \\cross (f \\circ g'))(G)\n        = f^{\\cross 2}((g \\cross g')(G)) \\in \\calE_X.\n\\]\nBut then $(g \\cross g')(G) \\in [f]^* \\calE_X = \\calE_Y$, so $[g] = [g']$, as \nrequired.\n\\end{proof}\n\n\\begin{corollary}\nFor any $[f] \\from Y \\to X$, the canonical arrow\n\\[\n    [\\utilde{f}] \\from |Y|_{[f]^* \\calE_X} \\to X\n\\]\nis monic in $\\CATCrs$.\n\\end{corollary}\n\n\\begin{definition}\\label{def:Crs-image}\nSuppose $[f] \\from Y \\to X$. Denote $\\OBJim [f] \\defeq |Y|_{[f]^* \\calE_X}$ and \n$\\im [f] \\defeq [\\utilde{f}] \\from \\OBJim [f] \\to X$, where $[\\utilde{f}]$ is \ndefined as above. We will also sometimes write $[f](Y) \\defeq \\OBJim [f]$.\n\\end{definition}\n\nDespite the notation, $[f](Y)$ should not be considered as a subspace of $X$ \n(however, see Proposition~\\ref{prop:Crs-subsp-images} and the discussion which \nprecedes it).\n\n\\begin{theorem}\nFor any $[f] \\from Y \\to X$, the subobject of $X$ represented by the arrow $\\im \n[f] \\from \\OBJim [f] \\monto X$ is the (categorical) image of $[f]$ in \n$\\CATCrs$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $[f]$ also factors as $Y \\nameto{\\smash{[h]}} Z \\nameto{\\smash{[g]}} X$ \nwhere $[g]$ is monic, so that $\\calE_Z = [g]^* \\calE_X$. We must show that \nthere is a unique $[\\underline{h}] \\from \\OBJim [f] \\to Z$ such that $\\im [f] = \n[g] \\circ [\\underline{h}]$.\n\nPick a representative coarse map $h \\from Y \\to Z$, and put $\\underline{h} \n\\defeq h$ as a set map $\\OBJim [f] = Y \\to Z$. First, $\\underline{h}$ is a \ncoarse map: Local properness is equivalent to properness when restricted to \nunital subspaces (Corollary~\\ref{cor:loc-prop-uni}); since $\\OBJim [f]$ and $Y$ \nhave the same unital subspaces (and $\\underline{h} = h$ as set maps), \n$\\underline{h}$ is locally proper. Reasoning similarly, for any $F \\in \n\\calE_{\\OBJim [f]} = [f]^* \\calE_X$, $\\underline{h}^{\\cross 2}(F)$ is in \n$\\calE_{\\Terminate(Z)}$. Then, since $\\calE_Z = [g]^* \\calE_X$, it follows that \n$\\underline{h}$ is coarse. From the uniqueness assertion of \nProposition~\\ref{prop:Crs-factor-I}, we get that $[g] \\circ [\\underline{h}] = \n\\im [f]$. Uniqueness of $[\\underline{h}]$: If $[h'] \\from \\OBJim [f] \\to Z$ and \n$[g] \\circ [h'] = \\im [f] = [g] \\circ [h]$, then $[h] = [h']$ since $[g]$ is \nmonic.\n\\end{proof}\n\n\n\\subsection{Epis and coimages}\n\nFor rather trivial reasons, push-forward coarse structures are not coarsely \ninvariant. Recall that coarse structures are semirings, which gives rise to an \nobvious notion of ideals.\n\n\\begin{definition}\\label{def:ideal}\nSuppose $\\calE_X$ is a coarse structure on a set $X$. A subset $\\calE \\subseteq \n\\calE_X$ is an \\emph{ideal} of $\\calE_X$ if it is a coarse structure on $X$ \nsuch that $E \\circ E', E' \\circ E \\in \\calE$ for all $E \\in \\calE$, $E' \\in \n\\calE_X$. Note that any intersection of ideals is again an ideal. The \n\\emph{ideal} $\\lAngle \\calE \\rAngle_X$ (of $\\calE_X$ generated by $\\calE$) is \nthe smallest ideal of $\\calE_X$ which contains $\\calE$.\n\\end{definition}\n\n\\begin{proposition}\nSuppose $f, f' \\from Y \\to X$ are coarse maps with $f \\closeequiv f'$. Then\n\\[\n    \\lAngle f_* \\calE_Y \\rAngle_X = \\lAngle (f')_* \\calE_Y \\rAngle_X.\n\\]\n\\end{proposition}\n\n\\begin{proof}\nElements $E \\in \\lAngle f_* \\calE_Y \\rAngle_X$ are exactly subsets\n\\[\n    E \\subseteq E' \\circ f^{\\cross 2}(F) \\circ E'' \\union E'''\n\\]\nfor some $F \\in \\calE_Y$ and some $E', E'', E''' \\in \\calE_X$ with $E'''$ \nfinite. But then\n\\[\n    E \\subseteq (E' \\circ (f \\cross f')(1_{F \\cdot Y})) \\circ (f')^{\\cross 2}(F)\n            \\circ ((f' \\cross f)(1_{Y \\cdot F}) \\circ E'') \\union E'''\n\\]\nis in $\\lAngle (f')_* \\calE_Y \\rAngle_X$ (and symmetrically) as required.\n\\end{proof}\n\n\\begin{definition}\nSuppose $[f] \\from Y \\to X$. The \\emph{coarsely invariant push-forward coarse \nstructure} $[f]_* \\calE_Y$ on $X$ (along $[f]$) is given by\n\\[\n    [f]_* \\calE_Y \\defeq \\lAngle f_* \\calE_Y \\rAngle_X\n\\]\n(where $f \\from Y \\to X$ is any representative coarse map).\n\\end{definition}\n\nDespite the obvious parallels with coarsely invariant pull-backs, the coarsely \ninvariant push-forward $[f]_* \\calE_Y$ depends very little on $\\calE_Y$. In \nfact, it depends only on the set of unital subspaces of $Y$ (recall from \nProposition~\\ref{prop:close-uni} that closeness is entirely determined on the \nunital subspaces). Thus we have the following.\n\n\\begin{proposition}\nFor any $[f] \\from Y \\to X$,\n\\[\n    [f]_* \\calE_Y = (\\im [f])_* \\calE_{\\OBJim [f]}.\n\\]\n\\end{proposition}\n\n\\begin{proof}\nRecall that $\\OBJim [f] \\defeq |Y|_{[f]^* \\calE_X}$, where $[f]^* \\calE_X \n\\defeq \\calE_{\\Terminate(Y)} \\intersect f^* \\calE_X$ (for any representative \nmap $f$) and $\\im [f] \\defeq f$ as a set map. Since $\\calE_Y \\subseteq [f]^* \n\\calE_X$, $[f]_* \\calE_Y \\subseteq (\\im [f])_* \\calE_{\\OBJim [f]}$. For the \nopposite inclusion, it suffices to show that, for $F \\in [f]^* \\calE_X$,\n\\[\n    E \\defeq f^{\\cross 2}(F) \\in \\lAngle f_* \\calE_Y \\rAngle_X;\n\\]\nbut $F \\cdot Y$ is a unital subspace of $Y$ (hence $1_{F \\cdot Y} \\in \\calE_Y$) \nand $f^{\\cross 2}(F) \\in \\calE_X$, so\n\\[\n    E = f^{\\cross 2}(1_{F \\cdot Y}) \\circ E \\in \\lAngle f_* \\calE_Y \\rAngle_X,\n\\]\nas required.\n\\end{proof}\n\nSuppose $[f] \\from Y \\to X$, represented by a coarse map $f$. Denote\n\\[\n    X_{[f]} \\defeq \\set{x \\in X\n        \\suchthat \\text{$x$ is connected to some $x' \\in f(Y)$}} \\subseteq X,\n\\]\na subspace of $X$. It is easy to see that $X_{[f]}$ really only depends on the \ncloseness class $[f]$, as the notation indicates. (If $X$ is connected, then of \ncourse $X_{[f]} = X$.)\n\nThe subspace $X_{[f]} \\subseteq X$ contains the set image of $f$ (and indeed of \nany coarse map close to $f$), and hence we may take the range restriction $f \n|^{X_{[f]}}$ which is evidently a coarse map $Y \\to X_{[f]}$. It is easy to see \nthat the closeness class $[f |^{X_{[f]}}]$ only depends on the closeness class \n$[f]$, and hence we also temporarily denote\n\\[\n    [f] |^{X_{[f]}} \\defeq [f |^{X_{[f]}}] \\from Y \\to X_{[f]}.\n\\]\nNow, we may coarsely invariantly push $\\calE_Y$ forward along $[f] |^{X_{[f]}}$ \nto get a coarse space $|X_{[f]}|_{([f] |^{X_{[f]}})_* \\calE_Y}$. We get the \nfollowing.\n\n\\begin{proposition}\\label{prop:Crs-factor-II}\nIf $[f] \\from Y \\to X$ is represented by a coarse map $f$, then $[f]$ factors \nas\n\\[\n    Y \\nameto{\\smash{[\\tilde{f}]}} |X_{[f]}|_{([f] |^{X_{[f]}})_* \\calE_Y}\n        \\nameto{\\smash{[\\alpha]}} X,\n\\]\nwhere $\\tilde{f} = f |^{X_{[f]}}$ and $\\alpha$ is the inclusion as set maps \n(thus $([f] |^{X_{[f]}})_* \\calE_Y \\subseteq \\calE_X$). Moreover, $[\\tilde{f}]$ \ndepends only on $[f]$ (and not $f$) and is unique in the above factorization.\n\\end{proposition}\n\n\\begin{proof}\nNearly all the assertions are clear from the definitions, \nCorollary~\\ref{cor:crs-factor-II}, and the previous remarks. We show that $f \n\\closeequiv f'$ implies $\\tilde{f} \\closeequiv \\tilde{f}'$: If $F \\in \\calE_Y$, \nthen\n\\[\n    (\\tilde{f} \\cross \\tilde{f}')(F) = (f \\cross f')(F)\n        \\subseteq f^{\\cross 2}(F) \\circ (f \\cross f')(1_{Y \\cdot F})\n\\]\nis in $([f] |^{X_{[f]}})_* \\calE_Y$ since $f^{\\cross 2}(F) \\in (f \n|^{X_{[f]}})_* \\calE_Y$ and $(f \\cross f')(1_{Y \\cdot F}) \\in \\calE_X \n|_{X_{[f]}}$. Uniqueness: If $[f] = [\\alpha] \\circ [g]$, where $[g] \\from Y \\to \n|X_{[f]}|_{([f] |^{X_{[f]}})_* \\calE_Y}$ and $g$ is any representative, then $f \n\\closeequiv \\alpha \\circ g$, so $\\tilde{f} \\closeequiv (\\alpha \\circ \ng)\\tilde{\\mathstrut} = g$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:Crs-epi}\n$[f] \\from Y \\to X$ is epi in $\\CATCrs$ if and only if $X_{[f]} = X$ and $[f]_* \n\\calE_Y = \\calE_X$ (i.e., if and only if $|X_{[f]}|_{([f] |^{X_{[f]}})_* \n\\calE_Y} = X$).\n\\end{proposition}\n\n\\begin{proof}\n(\\textimplies): Consider the push-out square\n\\[\\begin{CD}\n    Y @>{[f]}>> X \\\\\n    @V{[f]}VV @V{[e_1]}VV \\\\\n    X @>{[e_2]}>> X \\copro_Y X\n\\end{CD}\\]\n(in $\\CATCrs$). Fix a representative coarse map $f \\from Y \\to X$. As a set, \none may take $X \\copro_Y X \\defeq X_1 \\disjtunion X_2$ (disjoint union of sets) \nwhere $X_1 \\defeq X_2 \\defeq X$, with coarse structure\n\\[\n    \\langle \\calE_{X_1}, \\calE_{X_2},\n        \\set{(f_1 \\cross f_2)(F) \\suchthat F \\in \\calE_Y}\n            \\rangle_{X_1 \\disjtunion X_2},\n\\]\nwhere $\\calE_{X_j} \\defeq \\calE_X \\subseteq \\powerset((X_j)^{\\cross 2})$ and \n$f_j \\defeq f \\from Y \\to X = X_j$, for $j = 1, 2$. As set maps, one may take \n$e_1$, $e_2$ to be the two inclusions.\n\nIf $X_{[f]} \\neq X$, then there exists $x_0 \\in X$ not connected to any $f(y)$, \n$y \\in Y$. The entourage $\\set{1_{x_0}} \\in \\calE_X$ then shows that $e_1$ is \nnot close to $e_2$, hence $[e_1] \\neq [e_2]$ while $[e_1] \\circ [f] = [e_2] \n\\circ [f]$ so $[f]$ is not epi. Similarly, if $E \\in \\calE_X \\setminus [f]_* \n\\calE_Y$, then one can show that $(e_1 \\cross e_2)(E)$ is not an entourage of \n$X \\copro_Y X$, hence again $[f]$ is not epi.\n\n(\\textimpliedby): It suffices to show that $|X_{[f]}|_{([f] |^{X_{[f]}})_* \n\\calE_Y} = X$ implies that $[e_1] = [e_2]$ in the push-out square considered \nabove. If $|X_{[f]}|_{([f] |^{X_{[f]}})_* \\calE_Y} = X$, then every entourage \nof $[f]_* \\calE_Y$ is a subset of an entourage of the form $E_1 \\circ f^{\\cross \n2}(F) \\circ E_2$ for $F \\in \\calE_Y$ and $E_1, E_2 \\in \\calE_X$. Thus if $[f]_* \n\\calE_Y = \\calE_X$, given $E \\in \\calE_X$ choose $F$, $E_1$, and $E_2$ so that \n$E \\subseteq E_1 \\circ f^{\\cross 2}(F) \\circ E_2$, and then\n\\[\n    (e_1 \\cross e_2)(E) \\subseteq E_1 \\circ (f_1 \\cross f_2)(F) \\circ E_2\n\\]\n(where we now consider $E_j \\in \\calE_{X_j} = \\calE_X$ for $j = 1, 2$) is an \nentourage of $X \\copro_Y X$. Thus $e_1$ is close to $e_2$ as required.\n\\end{proof}\n\n\\begin{corollary}\nFor any $[f] \\from Y \\to X$, the canonical arrow\n\\[\n    [\\tilde{f}] \\from Y \\to |X_{[f]}|_{([f] |^{X_{[f]}})_* \\calE_Y}\n\\]\nis epi in $\\CATCrs$.\n\\end{corollary}\n\n\\begin{corollary}\\label{cor:Crs-epi-crs-structs}\nSuppose $\\calE$, $\\calE'$ are coarse structures on a set $X$ with $\\calE' \n\\subseteq \\calE$. If every unital subspace of $|X|_{\\calE}$ is a unital \nsubspace of $|X|_{\\calE'}$, then the class $[q]$ of the ``identity'' map\n\\[\n    q \\from |X|_\\calE' \\to |X|_\\calE\n\\]\nis epi in $\\CATCrs$.\n\\end{corollary}\n\n\\begin{proof}\nTrivially, $(|X|_\\calE)_{[q]} = |X|_\\calE$. We have that\n\\[\n    [q]_* \\calE' = \\lAngle \\calE' \\rAngle_{|X|_{\\calE}}\n\\]\nis an ideal of $\\calE$; we must prove equality, so suppose $E \\in \\calE$. Then \n$1_{E \\cdot X}$ is in $\\calE$ hence also in $\\calE'$, so\n\\[\n    E = 1_{E \\cdot X} \\circ E\n\\]\nis in $[q]_* \\calE'$, as required.\n\\end{proof}\n\n\\begin{definition}\\label{def:Crs-coimage}\nSuppose $[f] \\from Y \\to X$. Denote $\\OBJcoim [f] \\defeq |X_{[f]}|_{([f] \n|^{X_{[f]}})_* \\calE_Y}$ and $\\coim [f] \\defeq [\\tilde{f}] \\from Y \\to \\OBJcoim \n[f]$, where $[\\tilde{f}]$ is defined as above.\n\\end{definition}\n\n\\begin{theorem}\nFor any $[f] \\from Y \\to X$, the quotient object of $Y$ represented by the \narrow $\\coim [f] \\from Y \\surto \\OBJcoim [f]$ is the (categorical) coimage of \n$[f]$ in $\\CATCrs$.\n\\end{theorem}\n\n\\begin{proof}\nSuppose $[f]$ also factors as $Y \\nameto{\\smash{[h]}} Z \\nameto{\\smash{[g]}} X$ \nwhere $[h]$ is epi, so that $Z_{[h]} = Z$ and $\\calE_Z = [h]_* \\calE_Y$. We \nmust show that there is a unique $[\\bar{g}] \\from Z \\to \\OBJcoim [f]$ such that \n$\\coim [f] = [\\bar{g}] \\circ [h]$.\n\nPick representative coarse maps $g \\from Z \\to X$ and $h \\from Y \\to Z$. We may \nthen take $f \\defeq g \\circ h$ as a representative for $[f]$. Since $Z_{[f]} = \nZ$ and $[g] \\circ [h] = [f]$, it follows that $g$ has set image contained in \n$X_{[f]}$. Thus we may put $\\bar{g} \\defeq g |^{X_{[f]}}$ as a set map $Z \\to \nX_{[f]} = \\OBJcoim [f]$. $\\bar{g}$ is a coarse map: It is locally proper since \n$g = \\alpha \\circ \\bar{g}$ is locally proper. Since $\\calE_Z = [h]_* \\calE_Y$, \nevery entourage of $Z$ is contained in one of the form $G_1 \\circ h^{\\cross \n2}(F) \\circ G_2$, for $F \\in \\calE_Y$, $G_1, G_2 \\in \\calE_Z$. For such an \nentourage,\n\\[\n    \\bar{g}^{\\cross 2}(G_1 \\circ h^{\\cross 2}(F) \\circ G_2)\n        \\subseteq g^{\\cross 2}(G_1) \\circ (g \\circ h)^{\\cross 2}(F)\n                \\circ g^{\\cross 2}(G_2)\n\\]\nis in $([f] |^{X_{[f]}})_* \\calE_Y$ since $g^{\\cross 2}(G_1), g^{\\cross 2}(G_2) \n\\in \\calE_X$ (and $g$ has set image in $X_{[f]}$) and $(g \\circ h)^{\\cross \n2}(F) = f^{\\cross 2}(F)$. Thus $\\bar{g}$ is coarse. From the uniqueness \nassertion of Proposition~\\ref{prop:Crs-factor-II} (or, since $\\tilde{f} = \n\\bar{g} \\circ h$), we get that $\\coim [f] = [\\bar{g}] \\circ [h]$. Uniqueness of \n$[\\bar{g}]$ follows immediately from the hypothesis that $[h]$ is epi.\n\\end{proof}\n\n\n\\subsection{Monic and epi arrows}\n\nI do not know if $\\CATCrs$ is a \\emph{balanced} category, i.e., whether every \narrow in $\\CATCrs$ which is both monic and epi is an isomorphism (the converse \nis always true, of course). To show that a monic and epi $[f] \\from Y \\to X$ is \nan isomorphism one must show that there is an inverse $[f]^{-1} \\from X \\to Y$. \nWhen $X$ is unital, this is fairly straightforward (see below), but I do not \nknow how to prove it when $X$ is not.\n\n\\begin{theorem}\\label{thm:Crs-unital-balanced}\nIf $[f] \\from Y \\to X$ is monic and epi in $\\CATCrs$ and $X$ is a unital coarse \nspace, then $[f]$ is an isomorphism in $\\CATCrs$.\n\\end{theorem}\n\n\\begin{proof}\nFix a representative coarse map $f \\from Y \\to X$. Since $[f]$ is epi, by \nProposition~\\ref{prop:Crs-epi}, $X_{[f]} = X$ and\n\\[\n    [f]_* \\calE_Y \\defeq \\lAngle f_* \\calE_Y \\rAngle_X = \\calE_X.\n\\]\nThen every entourage $E_0 \\in \\calE_X$ is contained in one of the form $E_1 \n\\circ E_2 \\circ E_3$, where $E_1, E_3 \\in \\calE_X$ and $E_2 \\in f_* \\calE_Y$. \nEvery $E_2 \\in f_* \\calE_Y$ is contained in an entourage of the form\n\\[\n    \\bigl(f^{\\cross 2}(F_2^1) \\circ \\dotsb \\circ f^{\\cross 2}(F_2^N)\\bigr)\n        \\union \\bigunion_{j \\in J} (K_j \\cross K'_j),\n\\]\nwhere $F_2^1, \\dotsc, F_2^N \\in \\calE_Y$ (some $N \\geq 0$), $J$ is the set of \nconnected components of $X$, and $K_j$, $K'_j$ are finite subsets of $j$ for \neach $j \\in J$. Since $X_{[f]} = X$ (and $f^{\\cross 2}(F_2^k) \\in \\calE_X$ for \n$k = 2, \\dotsc, N$), it follows that every $E \\in \\calE_X$ is contained in a \nsome entourage\n\\[\n    E_0 \\circ f^{\\cross 2}(F_0) \\circ E'_0,\n\\]\nwhere $E_0, E'_0 \\in \\calE_X$ and $F_0 \\in \\calE_Y$.\n\nWe specialize the above discussion to the case $E = 1_X$ which is in $\\calE_X$ \nby unitality. Fix $E_0, E'_0 \\in \\calE_X$ and $F_0 \\in \\calE_Y$, so that $1_X \n\\subseteq E_0 \\circ f^{\\cross 2}(F_0) \\circ E'_0$. Define a set map $e \\from X \n\\to Y$ as follows. For each $x \\in X$, there are $x', x'' \\in X$ and $y', y'' \n\\in Y$ such that $(x,x') \\in E_0$, $(x'',x) \\in E'_0$, $f(y') = x'$, $f(y'') = \nx''$, and $(y',y'') \\in F_0$; choosing such a $y'' \\in Y$ in particular, put \n$e(x) \\defeq y''$.\n\nWe must verify that (any) $e \\from X \\to Y$ as constructed above is a coarse \nmap. Local properness: $X$ is unital, so $e$ is locally proper if and only if \nit is proper. For any $y \\in Y$, $e^{-1}(\\set{y}) \\subseteq (E_0 \\circ \nf^{\\cross 2}(F_0)) \\cdot \\set{f(y)}$ is finite, since $E_0 \\circ f^{\\cross \n2}(F_0) \\in \\calE_X \\subseteq \\calE_{|X|_1}$ satisfies the properness axiom. \n$e$ preserves entourages: Fix $E \\in \\calE_X$ and put $F \\defeq e^{\\cross \n2}(E)$. Since $[f]$ is monic, by Proposition~\\ref{prop:Crs-monic},\n\\[\n    \\calE_Y = [f]^* \\calE_X \\defeq \\calE_{\\Terminate(Y)} \\intersect f^* \\calE_X.\n\\]\nSince $e$ is (locally) proper, $F$ satisfies the properness axiom; since the \nimage of $e$ is contained in the unital subspace $Y \\cdot F_0$ of $Y$, it then \nfollows that $F \\in \\calE_{\\Terminate(Y)}$ and hence also that $f$ is locally \nproper for $F$. To show that $F \\in f^* \\calE_X$, it only remains to show that \n$f^{\\cross 2}(F) \\in \\calE_X$: Since\n\\[\n    G_0 \\defeq (\\id_X \\cross (f \\circ e))(1_X)\n        \\subseteq E_0 \\circ f^{\\cross 2}(F_0)\n\\]\nis in $\\calE_X$,\n\\[\n    f^{\\cross 2}(F)\n        = (f \\circ e)^{\\cross 2}(E)\n        \\subseteq (G_0)^\\transpose \\circ E \\circ G_0\n\\]\nis also in $\\calE_X$.\n\nSince $G_0 \\in \\calE_X$, we also get that $f \\circ e$ is close to $\\id_X$, \ni.e., $[f \\circ e] = [f] \\circ [e]$ is the identity arrow $[\\id_X]$ of $X$ in \n$\\CATCrs$. Since $[e]$ is monic (and $[f] \\circ [e] \\circ [f] = [f] = [f] \\circ \n[\\id_Y]$), it also follows that $[e] \\circ [f] = [\\id_Y]$. Thus $[e] = \n[f]^{-1}$, as required.\n\\end{proof}\n\n\\begin{corollary}\nIf $[f] \\from Y \\to X$ is monic and epi in $\\CATCrs$ and $X$ is coarsely \nequivalent to a unital coarse space, then $[f]$ is an isomorphism in $\\CATCrs$.\n\\end{corollary}\n\nThe problem with the above Corollary, of course, is that I do not know when a \ncoarse space is coarsely equivalent to a unital one. If $\\iota \\from X' \\injto \nX$ is the inclusion of a subspace of $X$ into $X$, then $[\\iota]$ is monic (and \n$\\OBJim [\\iota] = X'$), so $\\coim [\\iota] \\from X' \\to \\OBJcoim [\\iota]$ is \nboth monic and epi. (If $X$ is connected and $X'$ nonempty, $\\OBJcoim [\\iota]$ \nis just the set $X$ equipped with the coarse structure of entourages in \n$\\calE_X$ ``supported near $X'$''.) However, I do not know when $\\coim [\\iota]$ \nis a coarse equivalence.\n\nMore generally, for any $[f] \\from Y \\to X$, the natural arrow $Y \\to \\OBJim \n[f]$ is epi (either use Proposition~\\ref{prop:Crs-epi}, or the fact that \n$\\CATCrs$ has equalizers and, e.g., \\cite{MR0202787}*{Ch.~I Prop.~10.1}) and \nhence there is a natural epi arrow $[\\gamma] \\from \\OBJim [f] \\to \\OBJcoim [f]$ \nthrough which $\\im [f] \\from \\OBJim [f] \\to X$ factors; as $\\im [f]$ is monic, \n$[\\gamma]$ must also be monic. (One may dually show that the natural arrow \n$\\OBJcoim [f] \\to X$ is monic, but this yields the same arrow $[\\mu]$.) Of \ncourse, I do not know when $[\\mu]$ is an isomorphism. But when it is an \nisomorphism, one can, in a coarsely invariant way, describe the image of $[f]$ \nas a subset of $X$ with a certain coarse structure. This would be an appealing \n``generalization'' of the following, which is not coarsely invariant in the \ndesired sense.\n\n\\begin{proposition}\\label{prop:Crs-subsp-images}\nIf $f \\from Y \\to X$ is a coarse map and $Y$ is unital, then $\\OBJim [f] = \nf(Y)$ (where $f(Y)$ is the subspace of $X$ determined by the set image of $f$) \nas subobjects of $X$ in $\\CATCrs$.\n\\end{proposition}\n\n\\begin{proof}\nIf $Y$ is unital, $X' \\defeq f(Y)$ is also unital. The range restriction $f \n|^{X'} \\from Y \\to X$ is a coarse map, and $[f]^* \\calE_X = [f |^{X'}]^* \n\\calE_X$ hence $\\OBJim [f] = \\OBJim [f |^{X'}]$. Using this equality, we get \n$\\im [f] = [\\iota] \\circ \\im [f |^{X'}]$, where $\\iota \\from X' \\injto X$ is \nthe inclusion. But it is easy to check that $\\calE_{X'} \\defeq \\calE_X |_{X'} = \n[f |^{X'}]_* \\calE_Y$, so $[f |^{X'}]$ is epi. Hence $\\im [f |^{X'}] \\from \n\\OBJim [f] = \\OBJim [f |^{X'}] \\to X'$ is both monic and epi, hence an \nisomorphism by Theorem~\\ref{thm:Crs-unital-balanced}.\n\\end{proof}\n\n\n\\subsection{Quotients of coarse spaces}\\label{subsect:Crs-quot}\n\nWe now discuss a notion of quotient coarse spaces in $\\CATCrs$. The quotient \nspaces below are not the most general possible; rather, they appear to be a \nspecial case of a more general notion (of quotients by \\emph{coarse equivalence \nrelations}). However, I have not fully explored the more general notion, and so \nI leave it to a future paper.\n\nSuppose $\\calC$ is a category with zero object $0$ (e.g., an abelian category), \ni.e., $0$ is both initial and terminal. Given an arrow $f \\from Y \\to X$ (often \ntaken to be monic) in $\\calC$, a standard way of defining the quotient, denoted \n$X\/f(Y)$, is as the push-out $X \\copro_Y 0$ (assuming it exists); i.e., \n$X\/f(Y)$ fits into a push-out square\n\\[\\begin{CD}\n    Y @>f>> X \\\\\n    @VVV @VVV \\\\\n    0 @>>> X\/f(Y)\n\\end{CD}\\quad.\\]\nThe quotient $X\/f(Y)$ comes equipped with an arrow $X \\to X\/f(Y)$ and, in the \nabove case, also an arrow $0 \\to X\/f(Y)$.\n\nIn an abelian category, $X\/f(Y)$ is by definition just the cokernel of $f$. If \n$\\calC = \\CATpt{\\CATSet}$ or $\\CATpt{\\CATTop}$ (pointed sets or spaces), then \none has $0 = \\ast$ (a one-point set\/space) and $X\/f(Y)$ is (isomorphic to) just \n$X$ with the image of $f$ collapsed to the base point. The situation in \n$\\CATSet$ or $\\CATTop$ is slightly more complicated: If $Y \\neq \\emptyset$, one \ncan again take the push-out $X\/f(Y) \\defeq X \\copro_Y \\ast$. However, if $Y = \n\\emptyset$, then $X\/f(Y) \\cong X$; one should instead take $X\/f(Y) \\defeq X \n\\copro_Y \\emptyset$. In other words, one takes $X\/f(Y) \\defeq X \\copro_Y \n\\tilde{Y}$, where $\\tilde{Y}$ universally terminates $Y$ (see \nExample~\\ref{ex:Set-Top-univ-term}). This is exactly what we do in the coarse \ncategories.\n\n\\begin{definition}\nSuppose $[f] \\from Y \\to X$ (in $\\CATCrs$). The \\emph{quotient coarse space} \n$X\/[f](Y)$ is the push-out $X \\copro_Y \\Terminate(Y)$ in $\\CATCrs$, i.e., \n$X\/[f](Y)$ fits into a push-out square\n\\[\\begin{CD}\n    Y @>{[f]}>> X \\\\\n    @V{[\\tau_Y]}VV @V{[q]}VV \\\\\n    \\Terminate(Y) @>{[f]\/[f]}>> X\/[f](Y)\n\\end{CD}\\quad.\\]\nIf $Y \\subseteq X$ is a subspace, we will write $X\/[Y] \\defeq X\/[\\iota(Y)]$, \nwhere $\\iota \\from Y \\injto X$ is the inclusion.\n\\end{definition}\n\nThe justification for our notation is the following.\n\n\\begin{proposition}\nFor any $[f] \\from Y \\to X$, the quotient coarse space $X\/[f](Y)$ and the \nnatural map $X \\to X\/[f](Y)$ only depend on the image of $[f]$.\n\\end{proposition}\n\n\\begin{proof}\n$[f]$ factorizes canonically as\n\\[\n    Y \\nameto{\\smash{[\\beta]}} [f](Y) \\nameto{\\smash{\\im [f]}} X\n\\]\n(Proposition~\\ref{prop:Crs-factor-I} and Definition~\\ref{def:Crs-image}). Thus \n$X\/[f](Y)$ is also the colimit of the diagram\n\\[\\begin{CD}\n    Y @>{[\\beta]}>> [f](Y) @>{\\im [f]}>> X \\\\\n    @V{[\\tau_Y]}VV @V{[\\tau_{[f](Y)}]}VV \\\\\n    \\Terminate(Y) @>{[\\Terminate(\\beta)]}>> \\Terminate([f](Y)),\n\\end{CD}\\]\nand hence also of the cofinal subdiagram obtained by deleting $Y$ and \n$\\Terminate(Y)$.\n\\end{proof}\n\nThe coarse categories have all push-outs and we have seen how to describe them \nconcretely; the standard construction would take $X\/[f](Y)$ to be, as a set, \nthe disjoint union of $X$, $Y$, and $\\Terminate(Y)$. Taking a representative \ncoarse map $f \\from Y \\to X$, we have two ``smaller'' descriptions of the \nquotient:\n\\begin{enumerate}\n\\item Take $X\/[f](Y) \\defeq X \\disjtunion \\Terminate(Y)$ as a set with the \n    coarse structure generated by $\\calE_X$, $\\calE_{\\Terminate(Y)}$, and \n    $\\set{(f \\cross \\tau_Y)(F) \\suchthat F \\in \\calE_Y}$, where we consider \n    $\\Terminate(Y)$ and $X$ as subsets of $X \\disjtunion \\Terminate(Y)$. (This \n    is a particular instance of a ``smaller'' construction of push-outs in \n    $\\CATCrs$.)\n\\item Take $X\/[f](Y) \\defeq X$ as a set with the coarse structure generated by \n    $\\calE_X$ and $f_* \\calE_{\\Terminate(Y)}$, where we treat $f$ as a set map \n    $\\Terminate(Y) = Y \\to X$.\n\\end{enumerate}\n\nUsing the second description above and applying \nCorollary~\\ref{cor:Crs-epi-crs-structs} (the left and right supports of \nentourages in $f_* \\calE_{\\Terminate(Y)}$ are already unital subspaces of $X$), \nwe immediately get the following.\n\n\\begin{proposition}\nFor any $[f] \\from Y \\to X$, $X\/[f](Y)$ is a quotient of $X$ in the categorical \nsense (i.e., the natural map $[q] \\from X \\to X\/[f](Y)$ is epi).\n\\end{proposition}\n\n\n\\subsection{Restricted coarse categories}\\label{subsect:rest-Crs}\n\nThe lack of restriction on the size of coarse spaces (other than that imposed \nby the choice of universe) may be somewhat bothersome, and moreover prevent \n$\\CATCrs$ from having a terminal object. It is tempting to restrict the \ncardinality of coarse spaces, i.e., consider the full subcategory of $\\CATCrs$ \nof the coarse spaces of cardinality at most $\\kappa$, for some fixed, small \n(probably infinite) cardinal $\\kappa$. This is not the correct thing to do: \nFirst, one would no longer have all small limits and colimits (though as long \nas $\\kappa$ is infinite one have all finite limits and colimits). Second, and \nmore importantly, it would bar constructions involving the set of (set) \nfunctions $Y \\to X$ ($\\#X, \\#Y \\leq \\kappa$) which will be important in \n\\cite{crscat-II}.\n\nA better way to proceed is to consider the full subcategory of $\\CATCrs$ of \ncoarse spaces $X$ for which there exists a coarse map $X \\to R$, where $R \n\\defeq \\Terminate(R_0)$ for some fixed $R_0$. (Of particular interest is the \ncase when $R_0$ is a unital coarse space of some infinite cardinality $\\kappa$, \nin which case $R = |R_0|_1$ only depends on $\\kappa$ up to coarse equivalence.)\n\nWe will first discuss this in full generality, using terminology from the \nbeginning of \\S\\ref{subsect:Crs-Term}. In the following, suppose $\\calC$ is \nsome category and that is some object which $\\tilde{X}$ terminates any object \n(e.g., itself) in $\\calC$.\n\n\\begin{definition}\nThe \\emph{$\\tilde{X}$-restriction} $\\calC_{\\preceq \\tilde{X}}$ of $\\calC$ is \nthe full subcategory of $\\calC$ consisting of all the objects $Y$ in $\\calC$ \nsuch that there exists some (unique) arrow $Y \\to \\tilde{X}$.\n\\end{definition}\n\nIn other words, $\\calC_{\\preceq \\tilde{X}}$ consists of all objects which are \nterminated by $\\tilde{X}$. Equivalently, one may consider the comma category \n$(\\calC \\CATover \\tilde{X})$. It is easy to check that the range restricted \nprojection functor $(\\calC \\CATover \\tilde{X}) \\to \\calC_{\\preceq \\tilde{X}}$ \nis an isomorphism of categories.\n\nLet $I \\from \\calC_{\\preceq \\tilde{X}} \\to \\calC$ denote the inclusion functor. \nWhen a \\emph{nonzero} limit $\\calC_{\\preceq \\tilde{X}}$ already exists in \n$\\calC$, the limits are the same. More precisely, we have the following.\n\n\\begin{proposition}\\label{prop:restricted-lim}\nSuppose $\\calF \\from \\calJ \\to \\calC_{\\preceq \\tilde{X}}$, where $\\calJ$ is \nnonempty. If the limit $\\pfx{\\calC}\\OBJlim (I \\circ \\calF)$ exists, then\n\\[\n    \\pfx{\\calC_{\\preceq \\tilde{X}}}\\OBJlim \\calF\n        = \\pfx{\\calC}\\OBJlim (I \\circ \\calF);\n\\]\ni.e., the limit of $\\calF$ in $\\calC_{\\preceq \\tilde{X}}$ exists and any \nlimiting cone in $\\calC$ gives a limiting cone in $\\calC_{\\preceq \\tilde{X}}$.\n\\end{proposition}\n\n\\begin{proof}\nThe nonemptiness of $\\calJ$ ensures that the object $\\pfx{\\calC}\\OBJlim (I \n\\circ \\calF)$ is in $\\calC_{\\preceq \\tilde{X}}$ (since it must map to some \nobject of $\\calC_{\\preceq \\tilde{X}}$, hence to $\\tilde{X}$). The rest follows \neasily, since the inclusion functor $I$ is fully faithful.\n\\end{proof}\n\nThe following is trivial.\n\n\\begin{proposition}\n$\\tilde{X}$ is a terminal object (i.e., zero limit) in $\\calC_{\\preceq \n\\tilde{X}}$.\n\\end{proposition}\n\nThus $\\calC_{\\preceq \\tilde{X}}$ has all the limits that $\\calC$ does (to the \nextent that this makes sense), but also has a terminal object, which $\\calC$ \nmay not have. However, $\\calC$ may have a terminal object which is not \nisomorphic to $\\tilde{X}$ (in which case $\\calC_{\\preceq \\tilde{X}}$ is a \nproper subcategory of $\\calC$), so the inclusion functor $I$ may not preserve \nlimits.\n\nThe result dual to Proposition~\\ref{prop:restricted-colim} is true without the \nnonemptiness criterion.\n\n\\begin{proposition}\\label{prop:restricted-colim}\nSuppose $\\calF \\from \\calJ \\to \\calC_{\\preceq \\tilde{X}}$. If the colimit \n$\\pfx{\\calC}\\OBJcolim (I \\circ \\calF)$ exists, then\n\\[\n    \\pfx{\\calC_{\\preceq \\tilde{X}}}\\OBJcolim \\calF\n        = \\pfx{\\calC}\\OBJcolim (I \\circ \\calF).\n\\]\n\\end{proposition}\n\n\\begin{proof}\nIf $\\pfx{\\calC}\\OBJcolim (I \\circ \\calF)$ exists, then it maps to $\\tilde{X}$ \nsince there is a (unique) cone $\\calJ \\to \\tilde{X}$; thus the colimiting cone \nis actually in $\\calC_{\\preceq \\tilde{X}}$ and is universal since $I$ is fully \nfaithful.\n\\end{proof}\n\nNow, we return to our coarse context. Suppose $R \\defeq \\Terminate(R_0)$ for \nsome coarse space $R_0$. The \\emph{$R$-restricted coarse category} \n$\\CATCrs_{\\preceq R}$ is, as the notation indicates, the $R$-restriction of \n$\\CATCrs$. We similarly get $R$-restricted connected and connected, nonempty \ncoarse categories. We refer to the above categories collectively (i.e., for all \n$R$ and the various cases) as the \\emph{restricted coarse categories}.\n\n\\begin{theorem}\nThe restricted coarse categories have all (small) limits and colimits.\n\\end{theorem}\n\n\\begin{proof}\nThis follows immediately from Theorems \\ref{thm:Crs-lim} \nand~\\ref{thm:Crs-colim}, and Propositions \\ref{prop:restricted-lim} \nand~\\ref{prop:restricted-colim}.\n\\end{proof}\n\nOne can also check that all the earlier facts on monics and images, epis and \ncoimages, quotients, etc. hold in the restricted coarse categories.\n\n\n\n\n\\section{Topology and coarse spaces}\\label{sect:top-crs}\n\nOur coarse spaces are discrete, as opposed to the more standard definition of \n\\emph{proper coarse spaces} which allows coarse spaces to carry topologies and \nthus has different properness requirements (see the works of Roe, e.g., \n\\cite{MR2007488}*{Def.~2.22}). Our aim here is not to provide a general \ndiscussion of topological coarse spaces but to provide a means from going from \nRoe's \\emph{proper coarse spaces} to our (discrete) coarse spaces.\n\nWe will use the terms \\emph{compact} and \\emph{locally compact} in the sense of \nBourbaki \\cite{MR1712872}*{Ch.~I \\S{}9}, including the Hausdorff condition; in \nfact, all spaces will be Hausdorff unless otherwise stated. Throughout, $X$ and \n$Y$ will denote paracompact, locally compact topological spaces. Recall that a \nsubset $K$ of a space $X$ is \\emph{relatively compact} if it is contained in \nsome compact subspace of $X$. (If $X$ is Hausdorff, $K$ is relatively compact \nif and only if $\\overline{K}$ is compact.)\n\n\n\\subsection{Roe coarse spaces}\\label{subsect:Roe-crs-sp}\n\nWe will diverge from the standard terminology to avoid confusion with our \npreviously defined terms. \\emph{Roe coarse spaces} will be what are usually \ncalled proper coarse spaces. Let us recall these definitions (compare \nDefinitions \\ref{def:prop-ax} and~\\ref{def:crs-sp}).\n\n\\begin{definition}[see, e.g., \\cite{MR2007488}*{Def.~2.1}]%\n        \\label{def:Roe-prop-ax}\nA subset $E \\subseteq X^{\\cross 2}$ satisfies the \\emph{Roe properness axiom} \nif $E \\cdot K$ and $K \\cdot E$ are relatively compact subsets of $X$ for all \n(relatively) compact $K \\subseteq X$.\n\\end{definition}\n\n\\begin{definition}[see, e.g., \\cite{MR2007488}*{Def.~2.22}]%\n        \\label{def:Roe-crs-sp}\nA \\emph{Roe coarse structure} on $X$ is a subset $\\calR_X \\subseteq\n\\powerset(X^{\\cross 2})$ such that:\n\\begin{enumerate}\n\\item each $E \\in \\calR_X$ satisfies the Roe properness axiom;\n\\item $\\calR_X$ is closed under the operations of addition, multiplication, \n    transpose, and the taking of subsets;\n\\item\\label{def:Roe-crs-sp:III} if $K \\subseteq X$ is \\emph{bounded} in the \n    sense that $K^{\\cross 2} \\in \\calR_X$, then $K$ is relatively compact; and\n\\item\\label{def:Roe-crs-sp:IV} there is a neighbourhood (with respect to the \n    product topology on $X^{\\cross 2}$ of the unit (i.e., diagonal) $1_X$ which \n    is in $\\calR_X$.\n\\end{enumerate}\nA \\emph{Roe coarse space} is a paracompact, locally compact space $X$ equipped \nwith a Roe coarse structure $\\calR_X$ on $X$.\n\\end{definition}\n\n\\enumref{def:Roe-crs-sp:IV} implies Roe coarse spaces are always unital (in the \nobvious sense; see Definition~\\ref{def:uni-conn}) and that any Roe coarse space \n$X$ has an open cover $\\calU \\subseteq \\powerset(X)$ which is \\emph{uniformly \nbounded} in the sense that $\\bigunion_{U \\in \\calU} U^{\\cross 2}$ is in \n$\\calR_X$. Paracompactness implies that this cover can be taken to be locally \nfinite. The local compactness requirement is redundant, since it is implied by \n\\enumref{def:Roe-crs-sp:III} and \\enumref{def:Roe-crs-sp:IV}.\n\n\\begin{definition}\\label{def:top-prop}\nA continuous map $f \\from Y \\to X$ between locally compact spaces is \n\\emph{topologically proper} if $f^{-1}(K)$ is compact for every compact $K \n\\subseteq X$. More generally, also say that a (not necessarily continuous) map \n$f \\from Y \\to X$ between locally compact spaces is \\emph{topologically proper} \nif $f^{-1}(K)$ is relatively compact for every relatively compact $K \\subseteq \nX$.\n\\end{definition}\n\n\\begin{definition}[see, e.g., \\cite{MR2007488}*{Def. 2.21 and~2.14}]\nA (not necessarily continuous) map $f \\from Y \\to X$ between Roe coarse spaces \nis a \\emph{Roe coarse map} if it is topologically proper and \\emph{preserves \nentourages} in the sense that $f^{\\cross 2}(F) \\in \\calR_X$ for all $F \\in \n\\calR_Y$. (Roe coarse maps are usually called \\emph{proper coarse maps}.) Roe \ncoarse maps $f, f' \\from Y \\to X$ are \\emph{close} if $(f \\cross f')(1_Y) \\in \n\\calR_X$ (or equivalently if $(f \\cross f')(F) \\in \\calR_X$ for all $F \\in \n\\calR_Y$).\n\\end{definition}\n\nWe get an obvious \\emph{Roe precoarse category} $\\CATRoePCrs$ with objects all \n(small) Roe coarse spaces and arrows Roe coarse maps, and a quotient \\emph{Roe \ncoarse category} $\\CATRoeCrs$ with the same objects but whose arrows are \ncloseness classes of Roe coarse maps. \\emph{Roe coarse equivalences} are Roe \ncoarse maps which represent isomorphisms in $\\CATRoeCrs$.\n\n\n\\subsection{Discretization of Roe coarse spaces}\\label{subsect:Disc}\n\nWe now provide a way of passing from Roe coarse spaces to our (discrete) coarse \nspaces.\n\n\\begin{definition}\nA set $E \\in \\powerset(X^{\\cross 2})$ satisfies the \\emph{topological \nproperness axiom} (with respect to the topology of $X$) if, for all compact \nsubspaces $K \\subseteq X$, $(\\pi_1 |_E)^{-1}(K)$ and $(\\pi_2 |_E)^{-1}(K)$ are \nfinite.\n\\end{definition}\n\nSince all our spaces are Hausdorff hence $\\text{T}_{\\text{1}}$, the topological \nproperness axiom implies the (discrete) properness axiom \n(Definition~\\ref{def:prop-ax}).\n\nThe following is easy to check.\n\n\\begin{proposition}\nA set $E \\in \\powerset(X^{\\cross 2})$ satisfies the topological properness \naxiom if and only if $E$ is a (closed) discrete subset of $X^{\\cross 2}$ and \nthe restricted projections $\\pi_1 |_E, \\pi_2 |_E \\from E \\to X$ are \ntopologically proper maps.\n\\end{proposition}\n\n\\begin{remark}\\label{rmk:top-crs-sp}\nWe provide only a means from passing from Roe coarse spaces to our coarse \nspaces and not a complete discussion of ``topological coarse spaces'' since the \ntopological properness axiom does not encompass the axioms of \nDefinition~\\ref{def:Roe-crs-sp} (\\enumref{def:Roe-crs-sp:IV} in particular). We \nwould like not just a direct translation of Roe's definition to our setting, \nbut a proper generalization: First, we would like to allow nonunital \ntopological coarse spaces. Second, we do not want to impose local compactness \nfor two (possibly related) reasons:\n\\begin{inparaenum}\n\\item The ``topological coarse category'' should have all colimits (including \n    infinite ones). In particular, we are interested in ``large'' simplicial \n    complexes which may not be locally finite.\n\\item We wish to be able to analyze Hilbert space and other Banach spaces \n    directly as coarse spaces. This seems especially relevant as methods \n    involving uniform (i.e., coarse) embeddings into such spaces have gained \n    prominence in recent years (e.g., in \\cite{MR1728880}, Yu shows that the \n    Coarse Baum--Connes Conjecture is true for metric spaces of bounded \n    geometry which uniformly embed in Hilbert space).\n\\end{inparaenum}\n\nInstead of requiring that spaces be paracompact and locally compact, we \nshould probably require that spaces be \\emph{compactly generated} (i.e., be \nweak Hausdorff $k$-spaces). The topological properness axiom makes sense for \nsuch spaces (weak Hausdorffness still implies the $\\text{T}_{\\text{1}}$ \ncondition), but the problem of translating axioms \\enumref{def:Roe-crs-sp:III} \nand \\enumref{def:Roe-crs-sp:IV} becomes more complicated. Moreover, in the \ncompactly generated case, there are different, inequivalent definitions for \n``topological properness'' (whereas they all agree in the locally compact case; \nsee, e.g., \\cite{MR1712872}*{Ch.~I \\S{}10}), though perhaps one could still use \nDefinition~\\ref{def:top-prop} verbatim. In that case, the above Proposition \nremains true so long as $X^{\\cross 2}$ is given the categorically appropriate \ntopology, namely the $k$-ification of the standard product topology. We leave \nthese problems to a future paper \\cite{crscat-III}.\n\\end{remark}\n\nCompare the following, which is easy, with Proposition~\\ref{prop:prop-ax-alg}.\n\n\\begin{proposition}\nIf $E, E' \\in \\powerset(X^{\\cross 2})$ satisfy the topological properness \naxiom, then $E + E'$, $E \\circ E'$, $E^\\transpose$, and all subsets of $E$ \nsatisfy the topological properness axiom. Also, all singletons $\\set{e}$, $e \n\\in X^{\\cross 2}$, and hence all finite subsets of $X^{\\cross 2}$ satisfy the \nproperness axiom. Consequently,\n\\[\n    \\calE_{|X|_\\tau}\n        \\defeq \\set{E \\in \\calE_{|X|_1} \\subseteq \\powerset(X^{\\cross 2})\n            \\suchthat \\text{$E$ satisfies the topological properness axiom}}\n\\]\nis a coarse structure on the set $X$ (in the sense of \nDefinition~\\ref{def:crs-sp}).\n\\end{proposition}\n\n\\begin{definition}\\label{def:Disc-Roe}\nThe \\emph{discretization} of a Roe coarse space $X$ is the coarse space \n$\\Disc(X) \\defeq X$ as a set with the coarse structure\n\\[\n    \\calE_{\\Disc(X)} \\defeq \\calR_X \\intersect \\calE_{|X|_\\tau}\n\\]\n(consisting of all elements of $\\calR_X$ which satisfy the topological \nproperness axiom).\n\\end{definition}\n\nIt is easy to check that $\\calE_{\\Disc(X)}$ is in fact a coarse structure on \nthe set $X$. Unless $X$ is discrete, the coarse space $\\Disc(X)$ is not unital, \neven though the Roe coarse space $X$ is.\n\n\\begin{proposition}\nIf $f \\from Y \\to X$ is a Roe coarse map, then the set map $\\Disc(f) \\defeq f$ \nis coarse as a map $\\Disc(Y) \\to \\Disc(X)$.\n\\end{proposition}\n\n\\begin{proof}\nThe only thing to check is that if $f$ (not necessarily continuous) is \ntopologically proper and $F \\subseteq Y^{\\cross 2}$ satisfies the topological \nproperness axiom, then $E \\defeq f^{\\cross 2}(F) \\subseteq X^{\\cross 2}$ also \nsatisfies the topological properness axiom. This follows since\n\\[\n    E \\cdot K \\subseteq f(F \\cdot f^{-1}(K))\n\\]\nand $f$ is topologically proper (and similarly symmetrically).\n\\end{proof}\n\nSince, trivially, $\\Disc(f \\circ g) = \\Disc(f) \\circ \\Disc(g)$, we get the \nfollowing.\n\n\\begin{corollary}\n$\\Disc$ is a functor from the Roe precoarse category $\\CATRoePCrs$ to the \nprecoarse category $\\CATPCrs$.\n\\end{corollary}\n\n$\\Disc$ is coarsely invariant in the following way, which yields a canonical \nfunctor $[\\Disc] \\from \\CATRoeCrs \\to \\CATCrs$ between the closeness quotients. \n(We continue to write $\\Disc(X)$ instead of $[\\Disc](X)$ for Roe coarse \nspaces.)\n\n\\begin{proposition}\nIf Roe coarse maps $f, f' \\from Y \\to X$ are close, then\n\\[\n    \\Disc(f), \\Disc(f') \\from \\Disc(Y) \\to \\Disc(X)\n\\]\nare close coarse maps.\n\\end{proposition}\n\n\\begin{proof}\nThe result follows easily from the following fact (which is also easy): If $f, \nf' \\from Y \\to X$ are topologically proper and $F \\subseteq Y^{\\cross 2}$ \nsatisfies the topological properness axiom, then $(f \\cross f')(F) \\subseteq \nX^{\\cross 2}$ also satisfies the topological properness axiom.\n\\end{proof}\n\n\\begin{corollary}\nIf $f \\from Y \\to X$ is a Roe coarse equivalence, then $\\Disc(f) \\from \\Disc(Y) \n\\to \\Disc(X)$ is a coarse equivalence.\n\\end{corollary}\n\n\n\\subsection{Properties of the discretization functors}\n\nLet $\\CATDiscRoePCrs \\subseteq \\CATRoePCrs$ and $\\CATDiscRoeCrs \\subseteq \n\\CATRoeCrs$ be the full subcategories of \\emph{discrete} Roe coarse spaces \n(call them the \\emph{discrete Roe precoarse} and \\emph{coarse categories}, \nrespectively). On the discrete subcategories, $\\Disc$ and $[\\Disc]$ are fully \nfaithful.\n\n\\begin{proposition}\\label{prop:DiscRoePCrs-Disc-fullfaith}\nIf $X$, $Y$ are Roe coarse spaces with $Y$ discrete, then the map\n\\[\n    \\Disc_{Y,X} \\from \\Hom_{\\CATRoePCrs}(Y,X)\n        \\to \\Hom_{\\CATPCrs}(\\Disc(Y),\\Disc(X))\n\\]\nis a bijection. Hence, in particular, the restriction of $\\Disc$ to \n$\\CATDiscRoePCrs$ (which actually maps into $\\CATUniPCrs$) is a fully faithful \nfunctor.\n\\end{proposition}\n\n\\begin{proof}\n$\\Disc_{Y,X}$ is trivially injective, so it only remains to show surjectivity. \nSuppose $f \\from \\Disc(Y) \\to \\Disc(X)$ is a coarse map. If $K \\subseteq X$ is \nrelatively compact, then\n\\[\n    f^{-1}(K) = f^{-1}(f^{\\cross 2}(1_Y) \\cdot K)\n\\]\nis finite: since $Y$ is discrete, $\\Disc(Y)$ is unital so $f^{\\cross 2}(1_Y) \n\\in \\calE_{\\Disc(X)}$ satisfies the topological properness axiom (so $f^{\\cross \n2}(1_Y) \\cdot K$ is finite) and $f$ is (discretely) globally proper. Thus $f$ \nis topologically proper. Since $Y$ is discrete, $\\calE_{\\Disc(Y)} = \\calR_Y$, \nso $f$ preserves entourages of $\\calR_Y$ (of course, $\\calE_{\\Disc(X)} \n\\subseteq \\calR_X$). Thus $f$ is Roe coarse as a map $Y \\to X$.\n\\end{proof}\n\nThe unrestricted functor $\\Disc \\from \\CATRoePCrs \\to \\CATPCrs$ is \\emph{not} \nfull.\n\n\\begin{example}\\label{ex:RoePCrs-Disc-notfull}\nLet $X \\defeq \\setRplus$ equipped with the Euclidean metric Roe coarse \nstructure (see \\S\\ref{subsect:prop-met}), and $Y \\defeq \\setRplus \\union \n\\set{\\infty}$ be the one-point compactification of $\\setRplus$ equipped with \nthe unique Roe coarse structure $\\calR_Y \\defeq \\powerset(Y^{\\cross 2})$ (which \nis also the metric Roe coarse structure for any metric which metrizes $Y$ \ntopologically). Define $f \\from Y \\to X$ by\n\\[\n    f(t) \\defeq \\begin{cases}\n            t & \\text{if $t \\in \\setRplus$, and} \\\\\n            0 & \\text{if $t = \\infty$.}\n        \\end{cases}\n\\]\nThen $f$ is actually coarse as a map $\\Disc(Y) \\to \\Disc(X)$. However, clearly \n$f$ does not preserve entourages of $\\calR_Y$, hence does \\emph{not} define a \nRoe coarse map $Y \\to X$. As a map $\\Disc(Y) \\to \\Disc(X)$, $f$ is close to any \nconstant map $\\Disc(Y) \\to \\Disc(X)$ (sending all of $Y$ to some fixed element \nof $X$); every such constant map \\emph{does} define a Roe coarse map $Y \\to X$.\n\\end{example}\n\n\\begin{proposition}\\label{prop:DiscRoeCrs-Disc-fullfaith}\nIf $X$, $Y$ are Roe coarse spaces with $Y$ discrete, then the map\n\\[\n    [\\Disc]_{Y,X} \\from \\Hom_{\\CATRoeCrs}(Y,X)\n        \\to \\Hom_{\\CATCrs}(\\Disc(Y),\\Disc(X))\n\\]\nis a bijection. Hence the restriction of $[\\Disc]$ to $\\CATDiscRoeCrs$ (which \nactually maps into $\\CATUniCrs$) is fully faithful.\n\\end{proposition}\n\n\\begin{proof}\nBy the previous Proposition, $[\\Disc]_{Y,X}$ is surjective, so it only remains \nto show injectivity. Suppose $f, f' \\from Y \\to X$ are Roe coarse maps. If \n$\\Disc(f)$ is close to $\\Disc(f')$, then since $\\Disc(Y)$ is unital,\n\\[\n    (f \\cross f')(1_Y) = (\\Disc(f) \\cross \\Disc(f'))(1_Y)\n            \\in \\calE_{\\Disc(X)} \\subseteq \\calR_X,\n\\]\nso $f$ is close to $f'$, as required.\n\\end{proof}\n\nIf $X' \\subseteq X$ is a \\emph{closed} subspace of a Roe coarse space, then the \nobvious \\emph{Roe subspace coarse structure} $\\calR_{X'} \\defeq \\calR_X |_{X'} \n\\defeq \\calR_X \\intersect \\powerset((X')^{\\cross 2})$ is actually Roe coarse \nstructure on $X'$ (this is not the case if $X'$ is not closed), which makes \n$X'$ into a \\emph{Roe coarse subspace} of $X$. The inclusion of any Roe coarse \nsubspace into the ambient space is a Roe coarse map. The following result is \nwell known.\n\n\\begin{proposition}\\label{prop:Roe-disc-subsp}\nFor any Roe coarse space $X$, there is a (closed) discrete Roe coarse subspace \n$X' \\subseteq X$ such that the inclusion $\\iota \\from X' \\to X$ is a Roe coarse \nequivalence.\n\\end{proposition}\n\n\\begin{proof}\nFix a locally finite, uniformly bounded open cover $\\calU$ of $X$ by nonempty \nsets. For each $U \\in \\calU$, pick a point $x'_U \\in U$ and put $X' \\defeq \n\\set{x'_U \\suchthat U \\in \\calU}$. Since $\\calU$ is locally finite, it is easy \nto check that $X'$ is closed and discrete.\n\nInvoking the Axiom of Choice, fix a map $\\kappa \\from X \\to X'$ such that, for  \nall $x \\in X$, $\\kappa(x) \\in U$ for some $U \\in \\calU$ such that $x \\in U$. We \nmay also ensure that $\\kappa(x') = x'$ for all $x' \\in X'$. $\\kappa$ is \ntopologically proper: For any $x' \\in X'$,\n\\[\n    \\kappa^{-1}(\\set{x'})\n        \\subseteq \\bigunion_{\\substack{U \\in \\calU \\suchthat \\\\ x' \\in U}} U\n\\]\nwhich is a finite union of relatively compact sets, hence \n$\\kappa^{-1}(\\set{x'})$ is relatively compact (this suffices to show \ntopological properness since $X'$ is discrete). $\\kappa$ preserves entourages \nof $X$: Put\n\\[\n    E_\\calU \\defeq \\bigunion_{U \\in \\calU} U^{\\cross 2} \\in \\calR_X;\n\\]\nfor any $E \\in \\calR_X$,\n\\[\n    \\kappa^{\\cross 2}(E) \\subseteq E_\\calU \\circ E \\circ E_\\calU \\in \\calR_X,\n\\]\nhence $\\kappa^{\\cross 2}(E) \\in \\calR_X |_{X'}$, as required. Thus $\\kappa$ is \na Roe coarse map.\n\nTrivially, $\\kappa \\circ \\iota = \\id_{X'}$. Finally, $\\iota \\circ \\kappa$ is \nclose to $\\id_X$: Letting $E_\\calU$ be as above, we have\n\\[\n    (\\kappa \\cross \\id_X)(1_X) \\subseteq E_\\calU \\in \\calR_X,\n\\]\nas required.\n\\end{proof}\n\n\\begin{remark}\nThough we do not so insist, Roe coarse maps are sometimes required to be Borel \n(see, e.g., \\cite{MR1451755}*{Def.~2.2}). In that case, the map $\\kappa$ used \nin the above proof may not suffice. However, if one insists that all Roe coarse \nspaces be, e.g., second countable, then one can construct $\\kappa$ to be Borel. \nThus, as long as one so constrains the allowable Roe coarse spaces, the above \nProposition remains true.\n\\end{remark}\n\n\\begin{corollary}\nThe inclusion functor $\\CATDiscRoeCrs \\injto \\CATRoeCrs$ is fully faithful and \nin fact an equivalence of categories.\n\\end{corollary}\n\n\\begin{theorem}\\label{thm:RoeCrs-Disc-fullfaith}\nThe functor $[\\Disc] \\from \\CATRoeCrs \\to \\CATCrs$ is fully faithful.\n\\end{theorem}\n\n\\begin{proof}\nThis is immediate upon combining Propositions \n\\ref{prop:DiscRoeCrs-Disc-fullfaith} and~\\ref{prop:Roe-disc-subsp}.\n\\end{proof}\n\nEvery unital coarse space (in our sense) becomes a Roe coarse space when it is \ngiven the discrete topology, with coarse maps between unital coarse spaces \nbecoming Roe coarse maps. Thus $\\CATUniPCrs$ and $\\CATDiscRoePCrs$ are \nisomorphic as categories, and hence so too are $\\CATUniCrs$ and \n$\\CATDiscRoeCrs$.\n\n\\begin{corollary}\\label{cor:UniCrs-RoeCrs-equiv}\nOur unital coarse category $\\CATUniCrs$ is equivalent to the Roe coarse \ncategory $\\CATRoeCrs$, with the functor which sends a unital coarse space to \nthe ``identical'' discrete Roe coarse space an equivalence of categories.\n\\end{corollary}\n\n\n\n\n\\section{Examples and applications}\\label{sect:ex-appl}\n\nAs stated in the Introduction, we will not discuss even the standard \napplications of coarse geometry. We will first discuss a couple of basic \nexamples which we will need later, namely proper metric spaces and continuous \ncontrol, and then briefly examine a few things which arise from the categorical \npoint of view (some of which are not obviously possible in standard, unital \ncoarse geometry).\n\n\n\\subsection{Proper metric spaces}\\label{subsect:prop-met}\n\nSuppose that $(X,d) \\defeq (X,d_X)$ is a proper metric space (i.e., its closed \nballs are compact). We wish to produce a coarse space from $X$; we have already \ndiscussed the discrete case in Example~\\ref{ex:disc-met}, and what follows is a \ngeneralization of that.\n\nThere is a well known way to produce a Roe coarse space $|X|_d^\\TXTRoe$ from \n$(X,d)$ (noting that properness implies local compactness, and metrizability \nimplies paracompactness), taking the Roe coarse structure to be consist of the \n$E \\subseteq X^{\\cross 2}$ satisfying inequality \\eqref{ex:disc-met:eq} of \nExample~\\ref{ex:disc-met} (see, e.g., \\cite{MR2007488}*{Ex.~2.5}). One can then \napply the discretization functor $\\Disc$ to this Roe coarse space to obtain the \n\\emph{($d$-)metric coarse space} $|X| \\defeq |X|_d$. More directly and entirely \nequivalently, $|X|_d$ has as entourages the $E \\in \\calE_{|X|_\\tau}$ (i.e., the \n$E$ satisfying the topological properness axiom) which also satisfy the same \ninequality \\eqref{ex:disc-met:eq}. As in the discrete case, we may also allow \n$d(x,x') = \\infty$ (for $x \\neq x'$), and $|X|_d$ is connected if and only if \n$d(x,x') < \\infty$ for all $x$ and $x'$. If $X' \\subseteq X$ is a closed \n(topological) subspace, then the restriction of $d$ to $X'$ makes $X'$ into a \nproper metric space; the subspace coarse structure on $X'$ is the same as the \ncoarse structure coming from the restricted metric.\n\nSuppose $(Y,d_Y)$ is another proper metric space. A (not necessarily \ncontinuous) map $f \\from Y \\to X$ is Roe coarse as a map $|Y|_{d_Y}^\\TXTRoe \\to \n|X|_{d_X}^\\TXTRoe$ if and only if it is topologically proper and\n\\begin{equation}\\label{subsect:prop-met:Roe-crs:eq}\n    \\sup \\set{d_X(f(y),f(y'))\n            \\suchthat \\text{$y, y' \\in Y$ and $d_Y(y,y') \\leq r$}} < \\infty\n\\end{equation}\nfor every $r \\geq 0$. Since $X$, $Y$ are proper metric spaces, $f$ is \ntopologically proper if and only if it is \\emph{metrically proper} in the sense \nthat inverse images of metrically bounded subsets of $X$ are metrically bounded \nin $Y$. Roe coarse maps $f, f' \\from |Y|_{d_Y}^\\TXTRoe \\to |X|_{d_X}^\\TXTRoe$ \nare close if and only if\n\\begin{equation}\\label{subsect:prop-met:Roe-close:eq}\n    \\sup \\set{d_X(f(y),f'(y)) \\suchthat y \\in Y} < \\infty.\n\\end{equation}\n\nWe must warn that there may be a map $f \\from Y \\to X$ which is coarse (in our \nsense) as a map $|Y|_{d_Y} \\to |X|_{d_X}$, yet does not satisfy \n\\eqref{subsect:prop-met:Roe-crs:eq}. Similarly, there may be coarse maps $f, f' \n\\from |Y|_{d_Y} \\to |X|_{d_X}$ which are close but do not satisfy \n\\eqref{subsect:prop-met:Roe-close:eq}. Example~\\ref{ex:RoePCrs-Disc-notfull}, \nwhich shows that $\\Disc$ is not full, exhibits both phenomena. In the former \ncase, Theorem~\\ref{thm:RoeCrs-Disc-fullfaith} shows that every coarse map $f' \n\\from |Y|_{d_Y} \\to |X|_{d_X}$ is close to some coarse map $f \\from |Y|_{d_Y} \n\\to |X|_{d_X}$ which satisfies \\ref{subsect:prop-met:Roe-crs:eq}. (The \ncorresponding statement in the latter case is trivial.) Alternatively, one may \navoid both ``problems'' by considering only discrete, proper metric spaces \n(Proposition~\\ref{prop:DiscRoePCrs-Disc-fullfaith}); every proper metric space \nis Roe coarsely equivalent to a discrete one \n(Proposition~\\ref{prop:Roe-disc-subsp}).\n\n\\begin{remark}[see, e.g., \\cite{MR2007488}*{\\S{}1.3}]\\label{rmk:lsLip-qisom}\nIf $X$ and $Y$ are proper \\emph{length spaces}, then one can characterize the \nRoe coarse maps, and indeed Roe coarse equivalences, $Y \\to X$ a bit more \nstrictly: A map $f \\from Y \\to X$ (not necessarily continuous) is Roe coarse if \nand only if it is (metrically\/topologically) proper and \\emph{large-scale \nLipschitz} in the sense that there exist constants $C > 0$ and $R \\geq 0$ such \nthat\n\\[\n    d_X(f(y),f(y')) \\leq C d_Y(y,y') + R\n\\]\nfor all $y, y' \\in Y$. $f$ is a Roe coarse equivalence if and only if it is a \n\\emph{quasi-isometry} in that there are constants $c, C > 0$ and $r, R \\geq 0$ \nsuch that\n\\[\n    c d_Y(y,y') - r \\leq d_X(f(y),f(y')) \\leq C d_Y(y,y') + R\n\\]\nfor all $y, y' \\in Y$ (evidently, one can always take $c = 1\/C$ and $r = R$, as \nis conventional) and there is a constant $D \\geq 0$ such that every point of \n$X$ is within distance $D$ of a point in the image of $f$.\n\nOne can replace the length space hypothesis with a weaker condition, but some \nhypothesis is necessary; for general metric spaces there are Roe coarse maps, \nand indeed Roe coarse equivalences, which are not large-scale Lipschitz. \nHowever, every proper large-scale Lipschitz map is evidently also Roe coarse, \nand every quasi-isometry is a coarse equivalence.\n\\end{remark}\n\n\n\\subsection{Continuous control}\\label{subsect:cts-ctl}\n\nMost of the following originates from \\cites{MR1277522, MR1451755}, but see \nalso, e.g., \\cite{MR2007488}*{\\S{}2.2}. In the following, all topological \nspaces will be assumed to be second countable and locally compact (and \nHausdorff), whence paracompact. $X$ and $Y$ will always denote such spaces.\n\n\\begin{definition}\nA \\emph{compactified space} is a (second countable, locally compact) \ntopological space $X$ equipped with a (second countable) compactification \n$\\overline{X}$; its \\emph{boundary} is the space $\\die X \\defeq \\overline{X} \n\\setminus X$.\n\\end{definition}\n\nThe \\emph{continuously controlled Roe coarse structure} $\\calR_{|X|_{\\die \nX}^\\TXTRoe}$ on $X$ (for the compactification $\\overline{X}$, or for the \nboundary $\\die X$) consists of the $E \\subseteq X^{\\cross 2}$ such that\n\\begin{equation}\\label{subsect:cts-ctl:eq}\n    \\overline{E} \\subseteq X^{\\cross 2} \\union 1_{\\die X}\n        \\subseteq \\overline{X}^{\\cross 2},\n\\end{equation}\nwhere $1_{\\die X}$ is the diagonal subset of $(\\die X)^{\\cross 2}$ and the \nclosure is taken in $\\overline{X}$ (for the proof that this a Roe coarse \nstructure, see, e.g., \\cite{MR2007488}*{Thm.~2.27}). The associated coarse \nspace (resulting from applying the discretization functor $\\Disc$ to the above \nRoe coarse space $|X|_{\\die X}^\\TXTRoe$) is the \\emph{continuously controlled \ncoarse space} $|X|_{\\die X}$ (for the compactification $\\overline{X}$, or for \nthe boundary $\\die X$) whose entourages are the $E \\in \\calE_{|X|_\\tau}$ (i.e., \n$E$ satisfying the topological properness axiom) which also satisfy \n\\eqref{subsect:cts-ctl:eq}.\n\n\\begin{remark}\nIf $X$ is compact (so $\\overline{X} = X$ and $\\die X = \\emptyset$), then \n$|X|_{\\die X} = |X|_0^\\TXTconn$ (i.e., $X$ equipped with the initial connected \ncoarse structure).\n\\end{remark}\n\nThe following is standard.\n\n\\begin{proposition}\nSuppose $X$, $Y$ are compactified spaces. Any Roe coarse map $f \\from |Y|_{\\die \nY}^\\TXTRoe \\to |X|_{\\die X}^\\TXTRoe$ determines a canonical \\emph{continuous} \nmap $\\die Y \\to \\die X$ which we denote by $\\die [f]$. Moreover, Roe coarse \nmaps $f, f' \\from |Y|_{\\die Y}^\\TXTRoe \\to |X|_{\\die X}^\\TXTRoe$ are close if \nand only if $\\die [f] = \\die [f']$ (which justifies our notation).\n\\end{proposition}\n\nThe ``converse'' is also true: Any set map $Y \\to X$ (not necessarily \ncontinuous, but necessarily topologically proper) which ``extends \ncontinuously'' to a continuous map $\\die Y \\to \\die X$ is Roe coarse as a map \n$|Y|_{\\die Y}^\\TXTRoe \\to |X|_{\\die X}^\\TXTRoe$. This is essentially \ntautological, since the definition of ``extends continuously'' is exactly the \ndefinition of ``is continuously controlled''.\n\n\\begin{proof}\nFix a Roe coarse map $f$. Given $y_\\infty \\in \\die Y$, define $(\\die \n[f])(y_\\infty)$ as follows: By second countability, there is a sequence \n$\\seq{y_n}_{n=1}^\\infty$ in $Y$ which converges to $y_\\infty$. Then the \ndiagonal set $1_{\\set{y_n \\suchthat n \\in \\setN}}$ is in $\\calR_{|Y|_{\\die \nY}^\\TXTRoe}$, so $1_{\\set{f(y_n) \\suchthat n \\in \\setN}}$ must be in \n$\\calR_{|X|_{\\die X}^\\TXTRoe}$. By topological properness, the limit points of \n$\\seq{f(y_n)}_{n=1}^\\infty$ in $\\overline{X}$ (which exist by compactness) are \nall in $\\die X \\subseteq \\overline{X}$; in fact there is only one limit point \nwhich we call $(\\die [f])(y_\\infty)$. Well-definedness follows from the \nobservation that if $\\seq{y'_n}_{n=1}^\\infty \\subseteq Y$ (possibly a \nsubsequence of $\\seq{y_n}_{n=1}^\\infty$) also converges to $y_\\infty$, then \n$1_{\\set{(y_n,y'_n) \\suchthat n \\in \\setN}} \\in \\calR_{|Y|_{\\die Y}^\\TXTRoe}$ \nhence $1_{\\set{(f(y_n),f(y'_n)) \\suchthat n \\in \\setN}} \\in \\calR_{|X|_{\\die \nX}^\\TXTRoe}$, so $\\seq{f(y'_n)}_{n=1}^\\infty$ and $\\seq{f(y_n)}_{n=1}^\\infty$ \nhave the same limit points. To see that $\\die [f]$ is continuous, one proves \nsequential continuity (which suffices) using the obvious diagonal argument.\n\n$f$ (and similarly $f'$) ``extend continuously'' to maps $\\overline{Y} \\to \n\\overline{X}$: e.g.,\n\\[\n    \\bar{f}(y) \\defeq \\begin{cases}\n            f(y) & \\text{if $y \\in Y$, and} \\\\\n            (\\die [f])(y) & \\text{if $y \\in \\die Y$.}\n        \\end{cases}\n\\]\nThe second assertion then follows using the observation that, for any $F \\in \n\\calR_{|Y|_{\\die Y}^\\TXTRoe}$,\n\\[\n    \\overline{(f \\cross f')(F)}\n        = (\\bar{f} \\cross \\bar{f}')(\\overline{F})\n\\]\n(closures $\\overline{X}^{\\cross 2}$ and $\\overline{Y}^{\\cross 2}$).\n\\end{proof}\n\nTemporarily let $\\calC$ be the category of second countable, compact spaces \n(and continuous maps). If $M \\in \\Obj(\\calC)$, $\\setRplus \\cross M$ \ncompactified with boundary $M$ (so $\\overline{\\setRplus \\cross M}$ is \nhomeomorphic to $\\ccitvl{0,1} \\cross M$) is a compactified space. Then $M \n\\mapsto |\\setRplus \\cross M|_M^\\TXTRoe$ (on objects; $g \\mapsto \\id_{\\setRplus} \n\\cross g$ on functions) defines a (Roe) coarsely invariant functor \n$\\calO_\\TXTtop^\\TXTRoe \\from \\calC \\to \\CATRoePCrs$. By the above Proposition,\n\\[\n    [\\calO_\\TXTtop^\\TXTRoe] \\defeq \\Quotient \\circ \\calO_\\TXTtop^\\TXTRoe\n            \\from \\calC \\to \\CATRoeCrs\n\\]\nis fully faithful. As $[\\Disc] \\from \\CATRoeCrs \\to \\CATCrs$ is also fully \nfaithful (Proposition~\\ref{thm:RoeCrs-Disc-fullfaith}), the resulting \ncomposition\n\\[\n    [\\calO_\\TXTtop] \\defeq [\\Disc] \\circ [\\calO_\\TXTtop^\\TXTRoe]\n            \\from \\calC \\to \\CATCrs\n\\]\nis again fully faithful. Note that\n\\[\n    [\\calO_\\TXTtop] = \\Quotient \\circ \\calO_\\TXTtop,\n\\]\nwhere $\\calO_\\TXTtop \\defeq \\calO_\\TXTtop^\\TXTRoe \\from \\calC \\to \\CATRoePCrs$ \n(a coarsely invariant functor).\n\n\\begin{definition}[see, e.g., \\cite{MR1817560}*{\\S{}6.2}]%\n        \\label{def:ctsctl-cone}\nFor any second countable, compact space $M$, the \\emph{continuously controlled \nopen cone} on $M$ is the coarse space\n\\[\n    \\calO_\\TXTtop M \\defeq |\\setRplus \\cross M|_M.\n\\]\n\\end{definition}\n\nWe saw above that $M \\mapsto \\calO_\\TXTtop M$ is a coarsely invariant functor \nfrom the category of second countable, compact topological spaces to the \nprecoarse category $\\CATPCrs$.\n\n\\begin{remark}[compare \\cite{MR1341817}*{Thm.~1.23 and Cor.~1.24}]%\n        \\label{rmk:ctsctl-cones}\nAll continuously controlled coarse spaces can be described as cones in a \nnatural way. That is, for any compactified space $X$, there is a natural coarse \nequivalence\n\\[\n    \\calO_\\TXTtop(\\die X) \\isoto |X|_{\\die X}\n\\]\n(indeed, there is a natural Roe coarse equivalence $|\\setRplus \\cross \\die \nX|_{\\die X}^\\TXTRoe \\isoto |X|_{\\die X}^\\TXTRoe$). Thus, up to coarse \nequivalence, $|X|_{\\die X}$ only depends on the topology of the boundary $\\die \nX$, and not of $X$ itself. We leave this to the reader.\n\\end{remark}\n\n\\begin{remark}\\label{rmk:ctsctl-quot}\nSuppose $M$ is a second countable, compact space, and $N \\subseteq M$ is a \nclosed subspace. There is a natural (coarse) inclusion $\\iota \\from \n\\calO_\\TXTtop N \\injto \\calO_\\TXTtop M$ of continuously controlled open cones, \nhence a quotient coarse space\n\\[\n    (\\calO_\\TXTtop M)\/[\\calO_\\TXTtop N]\n        \\defeq (\\calO_\\TXTtop M)\/[\\iota](\\calO_\\TXTtop N)\n\\]\n(see \\S\\ref{subsect:Crs-quot}). One can check that the quotient $(\\calO_\\TXTtop \nM)\/[\\calO_\\TXTtop N]$ is naturally coarsely equivalent to the continuously \ncontrolled open cone $\\calO_\\TXTtop (M\/N)$ on the topological quotient $M\/N$.\n\\end{remark}\n\n\\begin{remark}\\label{rmk:ctsctl-ray}\nThe continuously controlled ray\n\\[\n    |\\coitvl{0,1}|_{\\set{1}} \\cong |\\setRplus|_{\\ast} \\cong |\\setZplus|_{\\ast} \n        \\cong \\calO_\\TXTtop \\ast\n\\]\n(where $\\ast$ is a one-point space) is coarsely equivalent to $|\\setZplus|_1$, \ni.e., a countable set with the terminal coarse structure.\n\\end{remark}\n\n\n\\subsection{Metric coarse simplices}\\label{subsect:met-simpl}\n\nWe index our simplices in the same way as Mac~Lane \\cite{MR1712872}*{Ch.~VII \n\\S{}5}, shifted by $1$ from most topologists' indexing. That is, our \n$n$-simplices are topologists' $(n-1)$-simplices (which have geometric \ndimension $n-1$) and we include the ``true'' $0$-simplex.\n\n\\begin{definition}\nAs sets, put $\\Delta_0 \\defeq \\set{0}$, $\\Delta_1 \\defeq \\setRplus \\defeq \n\\coitvl{0,\\infty}$, \\ldots, $\\Delta_n \\defeq (\\setRplus)^n$, \\ldots. For each \n$n = 0, 1, 2, \\dotsc$, let $d \\defeq d_n$ be the $l^1$ metric on $\\Delta_n$, \ni.e.,\n\\[\n    d_n((x_0,\\dotsc,x_{n-1}),(x'_0,\\dotsc,x'_{n-1}))\n        \\defeq |x_0 - x'_0| + \\dotsb + |x_{n-1} - x'_{n-1}|,\n\\]\nand denote the resulting coarse space, called the \\emph{metric coarse \n$n$-simplex}, by\n\\[\n    |\\Delta_n| \\defeq |\\Delta_n|_\\TXTmet \\defeq |\\Delta_n|_{d_n}\n\\]\n(the metric coarse space defined in \\S\\ref{subsect:prop-met}). We may also \nsubstitute the coarsely equivalent unital subspaces $(\\setZplus)^n \\subseteq \n(\\setRplus)^n$ for the $\\Delta_n$ when convenient.\n\\end{definition}\n\nNote that we may replace the $l^1$ metric with any $l^p$-metric ($1 \\leq p \\leq \n\\infty$), since\n\\[\n    \\norm{x}_\\infty \\leq \\norm{x}_p \\leq \\norm{x}_1 \\leq n \\norm{x}_\\infty\n\\]\n(for all $1 \\leq p \\leq \\infty$, $x \\in \\Delta_n \\subseteq \\setR^n$); all these \nmetrics yield the same Roe coarse structure and hence the same coarse structure \non $\\Delta_n$. See Proposition~\\ref{prop:met-simp-univ} below for a bit more \nabout the ``universality'' of metric coarse simplices.\n\nFor each $n = 0, 1, 2, \\dotsc$, $j = 0, \\dotsc, n$, define a coarse map \n$\\delta_j \\defeq \\delta_j^n \\from |\\Delta_n| \\to |\\Delta_{n+1}|$ by\n\\begin{equation}\\label{subsect:met-simpl:eq:delta}\n    \\delta_j(x_0, \\dotsc, x_{n-1})\n        \\defeq (x_0, \\dotsc, x_{j-1}, 0, x_j, \\dotsc, x_{n-1})\n\\end{equation}\n(for $n = 0$, let $\\delta_0^0$ be the inclusion). For each $n = 1, 2, 3, \n\\dotsc$, $j = 0, \\dotsc, n-1$, define a coarse map $\\sigma_j \\defeq \\sigma_j^n \n\\from |\\Delta_{n+1}| \\to |\\Delta_n|$ by\n\\begin{equation}\\label{subsect:met-simpl:eq:sigma}\n    \\sigma_j(x_0, \\dotsc, x_n)\n        \\defeq (x_0, \\dotsc, x_{j-1}, x_j+x_{j+1}, x_{j+2}, \\dotsc, x_n).\n\\end{equation}\nIt is easy to verify that the above maps are coarse and satisfy the \n\\emph{cosimplicial identities} (see, e.g., \\cite{MR1711612}*{I.1} or equations \n(11)--(13) in \\cite{MR1712872}*{Ch.~VII \\S{}5}). Consequently, we get a functor \nfrom the \\emph{simplicial category} $\\CATSimp$ to $\\CATPCrs$. Composing with \nthe quotient functor yields the \\emph{metric coarse simplex functor}\n\\[\n    |\\Delta|_\\TXTmet \\from \\CATSimp \\to \\CATCrs;\n\\]\nfor $n \\in \\Obj(\\CATSimp) = \\set{0, 1, 2, \\dotsc}$, $|\\Delta|_\\TXTmet(n) = \n|\\Delta_n|_\\TXTmet$.\n\nProceeding as standard (see, e.g., \\cite{MR1711612}), we may obtain \n\\emph{metric coarse realizations} of any simplicial set (since $\\CATCrs$ has \nall colimits), get a corresponding notion of (metric coarse) ``weak \nequivalence'', define \\emph{metric coarse singular sets} and a resulting \n\\emph{metric coarse singular homology}, and so on. We leave all of this to a \nfuture paper (or to the reader).\n\n\\begin{remark}\nMitchener has defined a related notion of \\emph{coarse $n$-cells} and \n\\emph{coarse $(n-1)$-spheres} (and resulting \\emph{coarse $CW$-complexes}) \n\\cites{MR1834777, MR2012966}. We will also defer the comparison of these with \nour coarse simplices (and resulting coarse simplicial complexes) to a future \npaper.\n\\end{remark}\n\nThe $l^1$ (or any $l^p$, $1 \\leq p \\leq \\infty$) metric coarse structure on a \n$\\Delta_n$ is the minimal ``good'' one, in the following sense. Fix $n \\geq 0$, \nand consider the maps\n\\[\n    \\delta_{j_1}^m \\circ \\dotsb \\circ \\delta_{j_{n-m}}^{n-1}\n        \\from \\Delta_m \\to \\Delta_n\n\\]\nfor all $0 \\leq m < n$. (The $\\delta_j$ all topologically embed their domains \nas closed subspaces of their codomains, and hence the same is true of \ncompositions of the $\\delta_j$.) Let us call the (set or topological) images of \nthe each of the above maps a \\emph{boundary simplex} of the topological space \n$\\Delta_n$. We will not prove the following in full detail.\n\n\\begin{proposition}\\label{prop:met-simp-univ}\nSuppose $|\\Delta_n|_\\calR$ is a Roe coarse space with underlying topological \nspace $\\Delta_n \\defeq (\\setRplus)^n$ and Roe coarse structure $\\calR$. Then \nthere is a Roe coarse map $i \\from |\\Delta_n|_\\TXTmet \\to |\\Delta_n|_\\calR$ \nsuch that (as a set map) $i$ maps each boundary simplex of $\\Delta_n$ to \nitself.\n\\end{proposition}\n\nIn fact, with a bit more trouble, one can even take $i$ to be a homeomorphism. \nThe obvious discrete version of the above, with $(\\setZplus)^n$ in place of \n$\\Delta_n \\defeq (\\setRplus)^n$, is rather trivial. To get a nontrivial \nversion, one should replace $|\\Delta_n|_\\calR$ with a ``sector'' which grows \narbitrarily quickly away from the origin.\n\n\\begin{proof}[Sketch of proof]\nIt is trivial for $n = 0$, so suppose that $n \\geq 1$. Fix an open \nneighbourhood $E_0 \\in \\calR$ of the diagonal $1_{\\Delta_n}$. We will say that \n$B \\subseteq \\Delta_n$ is \\emph{$E_0$-bounded} if $B^{\\cross 2} \\subseteq E_0$. \nIn the following, \\emph{disc} will mean ``closed $l^1$ metric disc in \n$\\Delta_n$''; the \\emph{diameter} of a disc will always be measured in the \n$l^1$ metric.\n\nTesselate $\\Delta_n$ by discs diameter $1$ as in \nFigure~\\ref{prop:met-simp-univ:fig-I} (we illustrate the case $n = 2$), and let\n\\[\n    L_{2j} \\defeq \\set{x \\in \\Delta_n \\suchthat j \\leq \\norm{x}_1 \\leq j+1}\n\\]\nfor $j = 0, 1, 2, \\dotsc$ be the ``layers'' of the tesselation. Then there is a \nrefinement of this tesselation by discs as in \nFigure~\\ref{prop:met-simp-univ:fig-II} such that each ``small'' disc of the \nrefinement is $E_0$-bounded; label the layers of this tesselation $L'_{2j_0}, \nL'_{2j_1}, \\dotsc$ as indicated in the Figure.\n\n\\begin{figure}\n\\resizebox{0.95\\linewidth}{!}{\\input{crscat-I-fig-subdiv-1.pdf_t}}\n\\caption{\\label{prop:met-simp-univ:fig-I}%\nThe tesselation of $\\Delta_2$ by discs of $l^1$-diameter $1$.}\n\\end{figure}\n\n\\begin{figure}\n\\resizebox{0.95\\linewidth}{!}{\\input{crscat-I-fig-subdiv-2.pdf_t}}\n\\caption{\\label{prop:met-simp-univ:fig-II}%\nA refinement of the tesselation by $E_0$-controlled discs.}\n\\end{figure}\n\nDefine a continuous, ``tesselation preserving'' map $i \\from \\Delta_n \\to \n\\Delta_n$ which sends $L_{2j_0}$ to $L'_{2j_0}$, $L_{2j_1}$ to $L'_{2j_1}$, \netc., collapsing the $L_{2j}$ which do not occur in the sequence $L_{2j_0}, \nL_{2j_1}, \\dotsc$; in the example illustrated in the Figures, $L_2$ is \ncollapsed to the ``level set'' $\\set{x \\in \\Delta_2 \\suchthat \\norm{x}_1 = 1}$, \n$L_{12}$ through $L_{22}$ are collapsed to $\\set{x \\in \\Delta_2 \\suchthat \n\\norm{x}_1 = 3}$, etc.\n\nThe map $i$ is (Roe) coarse. This map is proper and ``preserves'' the boundary \nsimplices. Consider the cover $\\set{B_1, B_2, \\dotsc}$ of $\\Delta_n$ by \n(overlapping) discs $B_k$ of diameter $2$, each a union of $2^n$ adjacent discs \nin the tesselation of Figure~\\ref{prop:met-simp-univ:fig-I}. We have that\n\\[\n    \\bigunion_{k=1}^\\infty (B_k)^{\\cross 2}\n\\]\ngenerates the Roe coarse structure of $|\\Delta_n|_\\TXTmet$. The collection \n$\\set{i(B_1), i(B_2), \\dotsc}$ is uniformly $(E_0 \\circ E_0)$-bounded: the \ncollection of unions of $2^n$, adjacent, \\emph{equal-sized} discs in the \ntesselation of Figure~\\ref{prop:met-simp-univ:fig-II} is uniformly $(E_0 \\circ \nE_0)$-bounded, and each $i(B_k)$ is contained in such a disc (there are four \ncases to check in the latter assertion: (1) $i$ does not collapse $B_k$ at all, \n(2) $i$ completely collapses $B_k$, (3) $i$ collapses the ``top half'' of \n$B_k$, or (4) $i$ collapses the ``bottom half'' of $B_k$). This suffices to \nshow that $i$ preserves all (Roe) entourages of $|\\Delta_n|_\\TXTmet$.\n\\end{proof}\n\n\\begin{remark}\nThe above Proposition is not entirely satisfactory. $|\\Delta_n|_\\TXTmet$ should \nsatisfy a stronger universal property (which I have not yet proven): \n$|\\Delta_n|_\\TXTmet$ should be \\emph{coarse-homotopy}-universal with the above \nproperty. That is, if $\\calS$ is any Roe coarse structure on $\\Delta_n$ such \nthat the above is true with $|\\Delta_n|_\\calS$ in place of \n$|\\Delta_n|_\\TXTmet$, then $|\\Delta_n|_\\calS$ should be coarse homotopy \nequivalent to $|\\Delta_n|_\\TXTmet$ in such a way that its boundary simplices \nare preserved (compare \\cite{MR1243611}*{Thm.~7.3}).\n\\end{remark}\n\n\n\\subsection{Continuously controlled coarse simplices}\n\nIf the previously defined metric coarse structure on a simplex $\\Delta_n$ is \nthe minimal ``good'' one, the continuously controlled coarse structure on \n$\\Delta_n$ defined below is the maximal ``good'' one (again, we will not make \nthis precise in this paper).\n\nFor $n = 1, 2, 3, \\dotsc$, let $\\overline{\\Delta}_n$ be the obvious \ncompactification of the topological space $\\Delta_n \\defeq (\\setRplus)^n$ by \nthe standard topological simplex of geometric dimension $n-1$. Alternatively \n(and equivalently, for our purposes), put\n\\begin{align*}\n    \\Delta_n & \\defeq \\Bigset{(x_0, \\dotsc, x_{n-1}) \\in (\\setRplus)^n\n            \\suchthat \\sum_{j=0}^{n-1} x_j < 1}\n\\quad\\text{and} \\\\\n    \\overline{\\Delta}_n & \\defeq \\Bigset{(x_0, \\dotsc, x_{n-1})\n            \\in (\\setRplus)^n \\suchthat \\sum_{j=0}^{n-1} x_j \\leq 1},\n\\end{align*}\nso that $\\die \\Delta_n \\defeq \\overline{\\Delta}_n \\setminus \\Delta_n$ really is \nthe standard topological $(n-1)$-simplex. Put $\\Delta_0 \\defeq \\set{0}$ which \nis compact, so $\\overline{\\Delta}_0 = \\Delta_0$ and $\\die \\Delta_0 = \n\\emptyset$.\n\n\\begin{definition}\nFor $n = 0, 1, 2, \\dotsc$, the \\emph{continuously controlled coarse \n$n$-simplex} is the continuously controlled coarse space\n\\[\n    |\\Delta_n| \\defeq |\\Delta_n|_\\TXTtop \\defeq |\\Delta_n|_{\\die \\Delta_n}.\n\\]\n\\end{definition}\n\nEquivalently (see Remark~\\ref{rmk:ctsctl-cones}), we can define \n$|\\Delta_n|_\\TXTtop$ to be the continuously controlled open cone \n$\\calO_\\TXTtop(\\die \\Delta_n)$ (with underlying set $\\setRplus \\cross (\\die \n\\Delta_n)$).\n\nAgain, as in \\S\\ref{subsect:met-simpl}, we can define various coarse maps \n$\\delta_j$ and $\\sigma_j$ between the continuously controlled coarse simplices. \nIndeed (using either of the above descriptions of the $\\Delta_n$), we may \ndefine them using the same formul\\ae\\ \\eqref{subsect:met-simpl:eq:delta} and \n\\eqref{subsect:met-simpl:eq:sigma}, and hence they also satisfy the \ncosimplicial identities. Consequently, we get a \\emph{continuously controlled \ncoarse simplex functor}\n\\[\n    |\\Delta|_\\TXTtop \\from \\CATSimp \\to \\CATCrs,\n\\]\nand everything that comes along with it: \\emph{continuously controlled coarse \nrealizations} of simplicial sets, a notion of (continuously controlled coarse) \n``weak equivalence'', \\emph{continuously controlled coarse singular sets} and \n\\emph{homology}, etc.\n\n\\begin{remark\nIf $X = \\calO_\\TXTtop M$ for a second countable compact topological space $M$ \n(where $\\calO_\\TXTtop M$ is the continuously controlled open cone on $M$ from \nDef.~\\ref{def:ctsctl-cone}), then it is easy to see that the continuously \ncontrolled coarse singular homology of $\\calO_\\TXTtop M$ is exactly the \nsingular homology of $M$ (in this case, we would want to discard our \n$0$-simplices and shift our indexing to match the topologists').\nContinuously controlled coarse simplices have another nice feature: \n$|\\Delta_1|_\\TXTtop$ is the continuously controlled ray, which is coarsely \nequivalent to $|\\setZplus|_1$, so $|\\Delta_1|_\\TXTtop$ is a product identity \nfor most coarse spaces which arise in practice (those in $\\CATCrs_{\\preceq \n|\\Delta_1|_\\TXTtop}$, which includes all those which are coarsely equivalent to \ncountable coarse spaces). However, continuously controlled simplices have a \nfundamental problem: they are too coarse, and so many coarse spaces $X$ of \ninterest (e.g., metric coarse spaces) do not even admit a coarse map \n$|\\Delta_1|_\\TXTtop \\to X$.\n\\end{remark}\n\n\n\\begin{comment}\n\\subsection{Coarse homotopy}\n\nThroughout this section, $X$ and $Y$ will be second countable Roe coarse \nspaces.\n\n\\begin{definition}[\\cite{MR1243611}*{Def.~1.2}]\nContinuous Roe coarse maps $f_0, f_1 \\from Y \\to X$ are \\emph{directly coarsely \nhomotopic} if there is a continuous, topologically proper map $(h_t) \\defeq H \n\\from Y \\cross \\ccitvl{0,1} \\to X$ such that $f_j = h_j$ for $j = 0, 1$ (where \nwe write $h_t \\defeq H(\\cdot,t)$ for $t \\in \\ccitvl{0,1}$) and, for every $F \n\\in \\calR_Y$,\n\\[\n    \\bigunion_{t \\in \\ccitvl{0,1}} (h_t)^{\\cross 2}(F) \\in \\calR_X.\n\\]\nA \\emph{direct coarse homotopy} is a map $H$ of the above form. (Generalized) \n\\emph{coarse homotopy} of Roe coarse maps is the equivalence relation generated \nby direct coarse homotopy and closeness.\n\\end{definition}\n\nDirect coarse homotopy is an equivalence relation, but is evidently not (Roe) \ncoarsely invariant; (generalized) coarse homotopy is the invariant version.\n\nDenote $P \\defeq P_\\TXTtop \\defeq |\\Delta_1|_\\TXTtop$ (recall our slightly \nnonstandard indexing of simplices); of course, $P$ is coarsely equivalent (and \nuniquely isomorphic in $\\CATCrs$) to $|\\setZplus|_1$, and $P = \\Terminate(P)$. \nAlso denote $I \\defeq I_\\TXTtop \\defeq |\\Delta_2|_\\TXTtop$. Recall that we have \ntwo coarse maps $\\delta_0, \\delta_1 \\from P \\to I$ and $\\sigma_I \\defeq \n\\sigma_0 \\from I \\to P$, with the latter unique modulo closeness.\n\nRecall that we may describe $|\\Delta_1|_\\TXTtop$ and $|\\Delta_2|_\\TXTtop$ as \ncontinuously controlled open cones: $|\\Delta_1|_\\TXTtop \\defeq \\calO_\\TXTtop \n\\ast$ (where $\\ast$ is a one-point space) and $|\\Delta_2|_\\TXTtop \\defeq \n\\calO_\\TXTtop \\ccitvl{0,1}$, which are just $\\setRplus$ and $\\setRplus \\cross \n\\ccitvl{0,1}$ as sets, respectively. Using these descriptions, we may take\n\\begin{align*}\n    \\delta_0(r) & \\defeq (r,0), \\\\\n    \\delta_1(r) & \\defeq (r,1), \\quad\\text{and} \\\\\n    \\sigma_I(r,t) & \\defeq r.\n\\end{align*}\n\nSince $Y$ is second countable, it is Roe coarsely equivalent to a countable \n(discrete) space. Hence there is a canonical arrow $[\\sigma_Y] \\from \\Disc(Y) \n\\to P$ in $\\CATCrs$ (choosing a representative coarse map $\\sigma_Y$ whenever \nnecessary), and $\\Disc(Y) \\cross P$ is canonically isomorphic to $\\Disc(Y)$ in \n$\\CATCrs$. From the coarse maps $\\delta_0$ and $\\delta_1$, we get two canonical \narrows $\\Disc(Y) \\to \\Disc(Y) \\cross I$.\n\n\\begin{proposition}\nSuppose $(h_t) \\defeq H \\from Y \\cross \\ccitvl{0,1} \\to X$ is a direct coarse \nhomotopy, and define a set map\n\\[\n    \\tilde{H} \\from Y \\cross I = Y \\cross \\setRplus \\cross \\ccitvl{0,1} \\to X\n\\]\nby $\\tilde{H}(y,r,t) \\defeq H(y,t)$. Then $\\tilde{H}$ is actually coarse as a \nmap from the product coarse space $\\Disc(Y) \\cross I$ to $\\Disc(X)$.\n\\end{proposition}\n\n(Recall that the set underlying the product of connected coarse spaces can be \ntaken, in a natural way, to be the set product of the underlying sets.)\n\n\\begin{proof}\nLet us write $|X| \\defeq \\Disc(X)$ and $|Y| \\defeq \\Disc(Y)$. Define \n\\[\n    \\tilde{H}' \\from Y \\cross I \\to X \\cross P\n\\]\nby $\\tilde{H}'(y,r,t) \\defeq (H(y,t),r)$. It suffices to show that $\\tilde{H}'$ \nis coarse as a map $|Y| \\cross I \\to |X| \\cross P$ since $\\tilde{H} = \\pi_{|X|} \n\\circ \\tilde{H}'$ and $\\pi_{|X|} \\from |X| \\cross P \\to |X|$ is coarse.\n\nWe have a commutative square\n\\[\\begin{CD}\n    |Y| \\cross I @>{\\tilde{H}'}>> |X| \\cross P \\\\\n    @V{\\pi_I}VV @V{\\pi_P}VV \\\\\n    I @>{\\sigma_I}>> P\n\\end{CD}\\quad;\\]\nsince $\\sigma_I \\circ \\pi_I$ is coarse hence locally proper, $\\tilde{H}'$ must \nbe locally proper by Proposition~\\ref{prop:loc-prop-comp}.\n\nIt remains to show that $\\tilde{H}'$ preserves entourages, so fix\n$G \\in \\calE_{\\Disc(Y) \\cross I}$. Since $\\tilde{H}'$ is locally proper, \n$(\\tilde{H}')^{\\cross 2}(G)$ satisfies the properness axiom. Using the above \ncommutative square, we also get that $(\\pi_P)^{\\cross 2}((\\tilde{H}')^{\\cross \n2}(G))$ is in $\\calE_P$. Thus it only remains to show that\n\\[\n    E \\defeq (\\pi_{|X|})^{\\cross 2}((\\tilde{H}')^{\\cross 2}(G))\n        = \\tilde{H}^{\\cross 2}(G)\n\\]\nis in $\\calE_{|X|}$. Let us also put $F \\defeq (\\pi_{|Y|})^{\\cross 2}(G) \\in \n\\calE_{|Y|}$.\n\n$E$ satisfies the topological properness axiom: Fix a compact $K \\subseteq X$; \nwe must show that $E \\cdot K$, $K \\cdot E$ are finite sets (we omit the latter, \nsymmetric case). Since $H^{-1}(K)$ is relatively compact (recall that $H$ is \ntopologically proper), $L \\defeq \\pi_Y(H^{-1}(K)) \\subseteq Y$ is also \nrelatively compact. As $F$ satisfies the topological properness axiom, $F \\cdot \nL$ is a finite set, so $G \\cdot (\\pi_{|Y|})^{-1}(L)$ is finite since $\\pi_{|Y|} \n\\from |Y| \\cross I \\to |Y|$ is locally proper. If $x \\in E \\cdot K$, there is \nan $x' \\in K$ such that $(x,x') \\in E$, and then a $((y,r,t),(y',r',t')) \\in G$ \nsuch that $H(y,t) = x$ and $H(y',t') = x'$; but  then $(y',r',t') \\in \n(\\pi_{|Y|})^{-1}(L)$, so $(y,r,t) \\in G \\cdot (\\pi_{|Y|})^{-1}(L)$. Hence $E \n\\cdot K$ is the image of a finite set, hence finite.\n\n$E$ is in $\\calR_X$:\n\n\n\\ldots\n\\end{proof}\n\n\n\n\n\n\n\n\n\\ldots\n\\end{comment}\n\n\n\\subsection{\\pdfalt{\\maybeboldmath $\\sigma$-coarse spaces and $\\sigma$-unital\n        coarse spaces}{sigma-coarse spaces and sigma-unital coarse spaces}}\n\nIn \\cite{MR2225040}*{\\S{}2}, Emerson--Meyer consider increasing sequences of \ncoarse spaces. Their coarse spaces are equipped with topologies and are \nconnected and unital (i.e., are \\emph{Roe coarse spaces} in the terminology of \n\\S{}\\ref{subsect:Roe-crs-sp}). We will simply handle the discrete case. (This \nis perhaps at significant loss of generality, since in a sense Emerson--Meyer \nare largely interested in ``non-locally-compact coarse spaces'' which we do not \nreally examine in this paper; see Remark~\\ref{rmk:top-crs-sp}.) For our \npurposes, we may safely discard the connectedness assumption, though we still \nneed unitality.\n\n\\begin{definition}[\\cite{MR2225040}*{\\S{}2}]\nA (discrete) \\emph{$\\sigma$-coarse space} $(X_m)$ is a nondecreasing sequence\n\\[\n    X_0 \\subseteq X_1 \\subseteq X_2 \\subseteq \\dotsb\n\\]\nof unital coarse spaces such that, for all $m \\geq 0$, $X_m$ is a coarse \nsubspace of $X_{m+1}$ (i.e., is a subset and has the subspace coarse \nstructure).\n\\end{definition}\n\n\\begin{remark}\nGiven a sequence $(X_m)$ which is a $\\sigma$-coarse space in the sense of \nEmerson--Meyer (i.e., each $X_m$ is a Roe coarse space thus may have nontrivial \ntopology), one can obtain a nondecreasing sequence of coarse spaces by applying \nour discretization functor $\\Disc$ to each $X_m$. However, $\\Disc(X_m)$ is \ntypically not unital. It may be interesting to remove the unitality assumption \nfrom the above Definition, and thus be able to consider $(\\Disc(X_m))$ as a \n``nonunital $\\sigma$-coarse space''.\n\\end{remark}\n\nUntil otherwise stated (near the end of this section), $(X_m)$ and $(Y_n)$ will \nalways denote $\\sigma$-coarse spaces.\n\n\\begin{definition}[\\cite{MR2225040}*{\\S{}4}]\nA \\emph{coarse map} $(f_n) \\from (Y_n) \\to (X_m)$ of $\\sigma$-coarse spaces is \na map of directed systems in $\\CATPCrs$ (taken modulo cofinality).\n\\end{definition}\n\nThat is, a coarse map $(f_n) \\from (Y_n) \\to (X_m)$ is represented by a \nsequence of coarse maps $f_n \\from Y_n \\to X_{m(n)}$, $n = 0, 1, \\dotsc$, \n(where $0 \\leq m(0) \\leq m(1) \\leq \\dotsb$ is a nondecreasing sequence) such \nthat the obvious diagram commutes (in $\\CATPCrs$, not modulo closeness); two \nrepresentative sequences $(f_n)$, $(f'_n)$ are considered to be equivalent if, \nfor all $n$, the compositions\n\\begin{equation}\\label{sect:sigma-crs:eq:maps}\n    Y_n \\nameto{\\smash{f_n}} X_{m(n)} \\injto X_{\\max\\set{m(n),m'(n)}}\n\\quad\\text{and}\\quad\n    Y_n \\nameto{\\smash{f'_n}} X_{m(n)} \\injto X_{\\max\\set{m(n),m'(n)}}\n\\end{equation}\nare equal.\n\nActually, Emerson--Meyer consider maps $\\bigunion_n Y_n \\to \\bigunion_m X_m$, \ni.e., maps between set colimits which restrict to give sequences of coarse \nmaps. This is equivalent to our definition (which avoids set colimits).\n\n\\begin{definition}[\\cite{MR2225040}*{\\S{}4}]\\label{def:sigma-crs}\nCoarse maps $(f_n), (f'_n) \\from (Y_n) \\to (X_m)$ are \\emph{close} if, for all \n$n$ (and any, hence all, representative sequences $(f_n)$, $(f'_n)$, \nrespectively), the compositions \\eqref{sect:sigma-crs:eq:maps} are close. We \ndenote the \\emph{closeness} (equivalence) \\emph{class} of $(f_n)$ by $[f_n]$.\n\\end{definition}\n\nEquivalently, coarse maps $(f_n)$, $(f'_n)$ are close if they yield maps of \ndirected systems in $\\CATCrs$ which are equivalent modulo cofinality.\n\nSince the system $X_0 \\to X_1 \\to \\dotsb$ consists of inclusion maps, the \nprecoarse colimit $\\pfx{\\CATPCrs}\\OBJcolim X_m$ exists; one may take it to be\n\\[\n    X \\defeq \\pfx{\\CATPCrs}\\OBJcolim X_m \\defeq \\bigunion_m X_m\n\\]\nas a set, with coarse structure\n\\[\n    \\calE_X \\defeq \\langle \\calE_{X_m} \\suchthat m = 0, 1, \\dotsc \\rangle_X\n\\]\ngenerated by the coarse structures of all the $X_m$. In fact, since $X_m$ is a \ncoarse subspace of $X_{m+1}$ for all $m$,\n\\[\n    \\calE_X = \\bigunion_m \\calE_{X_m}\n\\]\n(and $X_m$ is a subspace of $X$); conversely, we get, for each $m$, that\n$\\calE_{X_m} = \\calE_X |_{X_m}$.\n\nUntil otherwise stated, let $X$ be as above and similarly $Y \\defeq \n\\pfx{\\CATPCrs}\\OBJcolim Y_m \\defeq \\bigunion_n Y_n$.\n\nThe coarse colimit $\\pfx{\\CATCrs}\\OBJcolim X_m$ also exists (since all colimits \nin $\\CATCrs$ exist), and maps canonically to $X$ in $\\CATCrs$. The following is \neasy to show.\n\n\\begin{proposition}\n$\\pfx{\\CATCrs}\\OBJcolim X_m = \\pfx{\\CATPCrs}\\OBJcolim X_m \\eqdef X$. More \nprecisely, the canonical arrow\n\\[\n    \\pfx{\\CATCrs}\\OBJcolim X_m \\to \\pfx{\\CATPCrs}\\OBJcolim X_m \\eqdef X\n\\]\nis an isomorphism (in $\\CATCrs$).\n\\end{proposition}\n\nBy definition, any coarse map $(f_n) \\from (Y_n) \\to (X_m)$ of $\\sigma$-coarse \nspaces yields a well-defined coarse map $f \\from Y \\to X$. (Of course, $f$ is \njust, as a set map, given by $f(y_n) \\defeq f_n(y_n)$ for all $n$ and $y_n \\in \nY_n$.) Likewise, its closeness class $[f_n]$ yields a well-defined closeness \nclass $[f] \\from Y \\to X$.\n\nLet $\\calP\\calS$ be the category of $\\sigma$-coarse spaces and coarse maps, and \n$\\calS$ be the category of $\\sigma$-coarse spaces and closeness classes of \ncoarse maps. We have defined functors\n\\[\n    \\calL \\defeq \\pfx{\\CATPCrs}\\OBJcolim \\from \\calP\\calS \\to \\CATPCrs\n\\quad\\text{and}\\quad\n    [\\calL] \\defeq \\pfx{\\CATCrs}\\OBJcolim \\from \\calS \\to \\CATCrs.\n\\]\n\n\\begin{proposition}\\label{prop:sigma-crs:L-full-faith}\nThe functor $\\calL \\from \\calP\\calS \\to \\CATPCrs$ is fully faithful.\n\\end{proposition}\n\n(Recall that ``faithful'' does not require injectivity on object sets!)\n\n\\begin{proof}\nFaithfulness: Clear, since representative sequences $(f_n), (f'_n) \\from (Y_n) \n\\to (X_m)$ are cofinally equivalent if and only if they are equal on colimits \n(i.e., $f = f'$).\n\nFullness: To show that $\\calL$ maps $\\Hom_{\\calP\\calS}((Y_n),(X_m))$ to \n$\\Hom_{\\CATPCrs}(Y,X)$ surjectively, we must use the unitality of the $Y_n$. \nSuppose $f \\from Y \\to X$ is a coarse map (not a priori in the image of \n$\\calL$). For each $n$, $Y_n$ is a unital subspace of $Y$, and hence $f(Y_n)$ \nis a unital subspace of $X$. Then $1_{f(Y_n)}$ must be an entourage of some \n$X_m$; let $m(n)$ be the least such $m$. Since $\\calE_{X_{m(n)}} = \\calE_X \n|_{X_{m(n)}}$, $f_n \\defeq f |_{Y_n}^{X_{m(n)}} \\from Y_n \\to X_{m(n)}$ is a \ncoarse map. It follows that $(f_n)$ is a coarse map of $\\sigma$-coarse spaces, \nand that $\\calL((f_n)) = f$.\n\\end{proof}\n\nThe following shows that unitality of the $Y_n$ really is needed for fullness.\n\n\\begin{example}\\label{ex:sigma-crs:L-nonunital-not-full}\nPut, for each $m$, $X_m \\defeq |\\set{0, \\dotsc, m-1}|_1$, so that $X \\defeq \n\\calL((X_m))$ is just $\\setZplus$ as a set, with entourages the finite subsets \nof $(\\setZplus)^{\\cross 2}$. Put, for all $n$, $Y_n \\defeq X$, so that colimit \n$Y \\defeq X$ is nonunital ($(Y_n)$ is not a $\\sigma$-coarse space in our \nterminology). The identity map $Y \\to X$ is coarse, but its image is not \ncontained in any single $X_m$ so is no ``coarse map'' $(f_n) \\from (Y_n) \\to \n(X_m)$ which yields $f$.\n\\end{example}\n\nA $\\sigma$-coarse space $(X_m)$ includes as a part of its structure the \n``filtration'' $X_0 \\subseteq X_1 \\subseteq \\dotsb$. However, the particular \nchoice of ``filtration'' is not important, since maps of $\\sigma$-coarse spaces \nare taken modulo cofinality.\n\n\\begin{corollary}\nIf $X \\defeq \\calL((X_m))$ is isomorphic in $\\CATPCrs$ to $Y \\defeq \n\\calL((Y_n))$ (i.e., there is a \\emph{bijection} of sets $f \\from Y \\to X$ such \nthat $f$ and $f^{-1}$ are both coarse maps), then $(X_m)$ is isomorphic to \n$(Y_n)$ in $\\calP\\calS$ (in particular, this is the case when $X = Y$ as coarse \nspaces).\n\\end{corollary}\n\nThe situation modulo closeness parallels the above.\n\n\\begin{proposition}\\label{prop:sigma-crs:QL-full-faith}\nThe functor $[\\calL] \\from \\calS \\to \\CATCrs$ is fully faithful.\n\\end{proposition}\n\n\\begin{proof}\nFaithfulness: Since each $Y_n$ is a subspace of $Y$ and each $X_m$ a subspace \nof $X$, closeness of $f = \\calL((f_n))$ to $f' = \\calL((f'_n))$ implies \ncloseness of the compositions \\eqref{sect:sigma-crs:eq:maps} (noting that $f_n \n= f |_{Y_n}^{X_{m(n)}}$ and similarly for $f'_n$).\n\nFullness: Here, we implicitly use the unitality condition. We have a \ncommutative diagram\n\\[\\begin{CD}\n    \\calP\\calS @>{\\calL}>> \\CATPCrs \\\\\n    @V{\\Quotient}VV @V{\\Quotient}VV \\\\\n    \\calS @>{[\\calL]}>> \\CATCrs\n\\end{CD}\\,.\\]\nSince $\\calL$ is full and evidently the quotient functors are also full and map \nsurjectively onto object sets, $[\\calL]$ is full.\n\\end{proof}\n\nIt is not clear to me whether fullness of $[\\calL]$ fails if the unitality \ncondition is removed from Definition~\\ref{def:sigma-crs}; the counterexample of \nExample~\\ref{ex:sigma-crs:L-nonunital-not-full} fails.\n\n\\begin{corollary}\nIf $X \\defeq \\calL((X_m))$ is coarsely equivalent (i.e., isomorphic in \n$\\CATCrs$) to $Y \\defeq \\calL((Y_n))$, then $(X_m)$ is isomorphic to $(Y_n)$ in \n$\\calS$.\n\\end{corollary}\n\nIt follows from Propositions \\ref{prop:sigma-crs:QL-full-faith} \nand~\\ref{prop:sigma-crs:L-full-faith} that $\\calL$ and $[\\calL]$ are \nequivalences (of categories) onto their images. We now consider what the images \nof these functors are (and how one constructs ``inverse'' functors).\n\nLet us ``reset'' our notation: $X$, $Y$ are just coarse spaces, not necessarily \ncoming from $\\sigma$-coarse spaces, and $(X_m)$, $(Y_n)$ are not assumed to \nhave any meaning.\n\n\\begin{definition}\nA coarse space $X$ is \\emph{$\\sigma$-unital} if there is a nondecreasing \nsequence\n\\[\n    X_0 \\subseteq X_1 \\subseteq \\dotsb \\subseteq X\n\\]\nof unital subspaces of $X$ such that each unital subspace $X' \\subseteq X$ is \ncontained in some $X_m$ ($m$ depending on $X'$).\n\\end{definition}\n\nIt is implied that $X = \\bigunion_m X_m$, though this equality certainly does \nnot imply that each unital subspace of $X$ is contained in some $X_m$.\n\nLet $\\CATPCrs_{\\bfsigma} \\subseteq \\CATPCrs$ and $\\CATCrs_{\\bfsigma} \\subseteq \n\\CATCrs$ denote the full subcategories of $\\sigma$-unital coarse spaces. \nClearly, $\\calL$ and $[\\calL]$ map both into and onto $\\CATPCrs_{\\bfsigma}$ and \n$\\CATCrs_{\\bfsigma}$, respectively. We get the following.\n\n\\begin{theorem}\nThe functors $\\calL \\from \\calP\\calS \\to \\CATPCrs_{\\bfsigma}$ and $[\\calL] \n\\from \\calS \\to \\CATCrs_{\\bfsigma}$ are equivalences of categories.\n\\end{theorem}\n\nIt is also easy to construct ``inverse'' functors. Choose, for each \n$\\sigma$-unital $X$, a ``filtration'' $(X_m)$. Then $X \\mapsto (X_m)$ (and, for \n$f \\from Y \\to X$, $f \\mapsto (f_n)$, where $f_n$ is an appropriate range \nrestriction of $f |_{Y_n}$) gives a functor ``inverse'' to $\\calL \\from \n\\calP\\calS \\to \\CATPCrs_{\\bfsigma}$. Choosing representative coarse maps, one \ndoes the same to obtain an ``inverse'' to $[\\calL] \\from \\calS \\to \n\\CATCrs_{\\bfsigma}$.\n\n\n\\subsection{Quotients and Roe algebras}\\label{subsect:quot-Cstar}\n\nWe shall assume that the reader is familiar with the definition and \nconstruction of the Roe algebras $C^*(X)$ for $X$ a (Roe) coarse space (see, \ne.g., \\cite{MR1451755}); the generalization to our nonunital situation is \nstraightforward. We will follow the standard, abusive practice of pretending \nthat $X \\mapsto C^*(X)$ is a functor. (The situation is slightly complicated by \nour nonunital situation. However, there are a number of ways of obtaining an \nactual functor, just not to the category of $C^*$-algebras. One could, for \nexample, construct a coarsely invariant functor from $\\CATCrs$ to the category \nof $C^*$-categories \\cite{MR1881396}.) The important fact is that, applying \n$K$-theory, one gets a coarsely invariant functor $X \\mapsto \nK_\\grstar(C^*(X))$. The following should be regarded as a sketch, with more \ndetails to follow in a future paper.\n\nFix a coarse space $X$ and a subspace $Y \\subseteq X$, and denote the inclusion \n$Y \\injto X$ by $\\iota$. We note that the following does not depend on our \ngeneralizations, and even works in the ``classical'' unital context; if $X$ is \na Roe coarse space in the sense of \\S\\ref{sect:top-crs}, $Y$ should be closed \nin $X$. Recall that we simply denote the quotient $X\/[\\iota](Y)$ (defined in \n\\S\\ref{subsect:Crs-quot}) by $X\/[Y]$. The coarse space $X\/[Y]$ is easy to \ndescribe: It is just $X$ as a set, with coarse structure generated by the \nentourages of $X$ and those of $\\Terminate(Y)$ (if $X$ is unital, the latter \nare just those of the terminal coarse structure on $Y$).\n\nThe quotient $Y\/[Y] \\Terminate(Y)$ is a subspace of $X\/[Y]$, with\n$\\utilde{\\iota} \\from Y\/[Y] \\injto X\/[Y]$ an inclusion. We get a commutative \nsquare\n\\[\\begin{CD}\n    Y @>{\\iota}>> X \\\\\n    @V{\\tilde{q}}VV @V{q}VV \\\\\n    Y\/[Y] @>{\\utilde{\\iota}}>> X\/[Y]\n\\end{CD}\\quad,\\]\nwhere $\\tilde{q}$ and $q$ represent the quotient maps (which one can take to be \nidentity set maps). This square gives rise to a commutative diagram\n\\[\\begin{CD}\n    0 @>>> C^*_X(Y) @>{\\iota_*}>> C^*(X) @>>> Q_{X,Y} @>>> 0 \\\\\n    @. @V{\\tilde{q}_*}VV @V{q_*}VV @V{\\utilde{q}_*}VV \\\\\n    0 @>>> C^*_{X\/[Y]}(Y\/[Y]) @>{\\utilde{\\iota}_*}>> C^*(X\/[Y])\n        @>>> Q_{X\/[Y],Y\/[Y]} @>>> 0\n\\end{CD}\\]\nof $C^*$-algebras whose rows are exact; $C^*_X(Y)$ denotes the ideal of \n$C^*(X)$ of operators supported near $Y$ (which can be identified with \n$C^*(\\OBJcoim [\\iota])$, where $\\OBJcoim [\\iota] = X$ as a set with the \nnonunital coarse structure of entourages of $X$ supported near $Y$; see \nDef.~\\ref{def:Crs-coimage}) and $Q_{X,Y}$ is the quotient $C^*$-algebra (and \nsimilarly for the second row).\n\nNext, one observes that $\\utilde{q}_*$ is an isomorphism \\emph{of \n$C^*$-algebras} hence induces an isomorphism on $K$-theory. Let us specialize \nto the case when $X$ is unital (from which it follows that $Y$ and the quotient \ncoarse spaces are also unital), and examine the consequences. If $Y$ is finite \n(or compact, in the Roe coarse space version) then $X = X\/[Y]$ and $Y = Y\/[Y]$, \nso $\\tilde{q}_*$ and $q_*$ are identity maps on the level of $C^*$-algebras and \nhence the diagram is trivial.\n\nOn the other hand, if $Y$ is infinite, then $Y\/[Y] = |Y|_1$ has the terminal \ncoarse structure and one can show by a standard ``Eilenberg swindle'' (see, \ne.g., \\cite{MR1817560}*{Lem.~6.4.2}) that $K_\\grstar(C^*_{X\/[Y]}(Y\/[Y])) = 0$. \nThus we get a canonical isomorphism\n\\[\n    K_\\grstar(C^*(X\/[Y])) \\isoto K_\\grstar(Q_{X\/[Y],Y\/[Y]})\n        \\isoto K_\\grstar(Q_{X,Y})\n\\]\non $K$-theory. Consequently, using the isomorphism $K_\\grstar(C^*(Y)) \\isoto\nK_\\grstar(C^*_X(Y))$ (which is easy to prove under most circumstances), we get \na long (or six-term) exact sequence\n\\begin{equation}\\label{subsect:quot-Cstar:eq:lx}\n    \\dotsb \\nameto{\\smash{\\die}} K_\\grstar(C^*(Y))\n        \\to K_\\grstar(C^*(X))\n        \\to K_\\grstar(C^*(X\/[Y]))\n        \\nameto{\\smash{\\die}} K_{\\grstar-1}(C^*(Y))\n        \\to \\dotsb \\,.\n\\end{equation}\n\n\\begin{remark}[continuous control]\\label{rmk:ctsctl-quot-Cstar}\nIn the above situation, suppose that $X = \\calO M$ and $Y = \\calO N$ are \ncontinuously controlled open cones, where $N$ is a nonempty closed subspace of \na second countable, compact space $M$ and we abbreviate $\\calO \\defeq \n\\calO_\\TXTtop$. Then there are natural isomorphisms\n\\begin{equation}\\label{rmk:ctsctl-quot-Cstar:eq}\n    K_\\grstar(C^*(\\calO M)) \\cong \\tilde{K}^{1-\\grstar}(C(M))\n        = \\tilde{K}_{\\grstar-1}(M)\n\\quad\\text{and}\\quad\n    K_\\grstar(C^*(\\calO N)) \\cong \\tilde{K}_{\\grstar-1}(N),\n\\end{equation}\nwhere $\\tilde{K}$ is reduced $K$-homology (see, e.g., \n\\cite{MR1817560}*{Cor.~6.5.2}). One can check that there is also a natural \nisomorphism\n\\[\n    K_\\grstar(C^*(\\calO M\/[\\calO N])) \\cong K_{\\grstar-1}(M,N)\n\\]\n(to relative $K$-homology), so that the above long exact sequence \n\\eqref{subsect:quot-Cstar:eq:lx} naturally maps isomorphically to the reduced \n$K$-homology sequence\n\\[\n    \\dotsb \\nameto{\\smash{\\die}} \\tilde{K}_{\\grstar-1}(N)\n        \\to \\tilde{K}_{\\grstar-1}(M)\n        \\to K_{\\grstar-1}(M,N)\n        \\nameto{\\smash{\\die}} K_{\\grstar-2}(N)\n        \\to \\dotsb \\,.\n\\]\nWe have three natural isomorphisms\n\\begin{align*}\n    K_\\grstar(C^*(\\calO M\/[\\calO N])) & \\cong K_\\grstar(C^*(\\calO (M\/N))), \\\\\n    K_\\grstar(C^*(\\calO (M\/N))) & \\cong \\tilde{K}_{\\grstar-1}(M\/N),\n\\qquad\\qquad\\quad\\text{and} \\\\\n    K_{\\grstar-1}(M,N) & \\cong \\tilde{K}_{\\grstar-1}(M\/N),\n\\end{align*}\nfrom Remark~\\ref{rmk:ctsctl-quot}, as in \\eqref{rmk:ctsctl-quot-Cstar:eq} \nabove, and by excision for $K$-homology, respectively; these are mutually \ncompatible in the obvious sense.\n\\end{remark}\n\n\\begin{example}[\\maybeboldmath $K$-theory of $\\calO_\\TXTtop S^n$]\nWe give yet another version of a standard calculation (see, e.g., \n\\cite{MR1817560}*{Thm.~6.4.10}). For $n \\geq 0$, denote the topological \n$n$-sphere by $S^n$ and, for $n \\geq 1$, the closed $n$-disc by $D^n$; recall \nthat $D^n$ has ``boundary'' $S^{n-1}$ and that $D^n\/S^{n-1} \\cong S^n$. Again \nwe abbreviate $\\calO \\defeq \\calO_\\TXTtop$.\n\nFirst, we compute the $K$-theory of $X \\defeq \\calO S^0$. Put $Y \\defeq \\calO \n\\set{-1} \\subseteq X$, $X' \\defeq \\calO \\set{1} \\subseteq X$, and $Y' \\defeq \n\\set{0} \\subseteq Y \\intersect X'$. It is well known that $K_\\grstar(C^*(X')) = \n0$ and $K_\\grstar(C^*_X(Y)) = 0$ (by the aforementioned ``Eilenberg swindle''), \nand that\n\\[\n    K_\\grstar(C^*_{X'}(Y')) = \\begin{cases}\n                \\setZ & \\text{if $\\grstar \\equiv 0 \\AMSdisplayoff\\pmod{2}$,\n                        and} \\\\\n                0 & \\text{otherwise}\n            \\end{cases}\n\\]\n(since $C^*_{X'}(Y')$ is just the compact operators). We have a map of short \nexact sequences\n\\[\\begin{CD}\n    0 @>>> C^*_{X'}(Y') @>>> C^*(X') @>>> Q' @>>> 0 \\\\\n    @. @VVV @VVV @VVV \\\\\n    0 @>>> C^*_{X}(Y) @>>> C^*(X) @>>> Q @>>> 0\n\\end{CD}\\quad.\\]\nBut one checks easily that the map $Q' \\to Q$ is an isomorphism of \n$C^*$-algebras, hence from the $K$-theory long exact sequences we get\n\\begin{multline*}\n    K_\\grstar(C^*(\\calO S^0)) = K_\\grstar(Q) = K_\\grstar(Q') \\\\\n        = K_{\\grstar-1}(C^*_{X'}(Y')) = \\begin{cases}\n                \\setZ & \\text{if $\\grstar \\equiv 1 \\AMSdisplayoff\\pmod{2}$,\n                        and} \\\\\n                0 & \\text{otherwise.}\n            \\end{cases}\n\\end{multline*}\n\nWe proceed to calculate the $K$-theory of $\\calO S^n$, $n \\geq 1$, by \ninduction. Put $X \\defeq \\calO D^n$ and $Y \\defeq \\calO S^{n-1} \\subseteq X$, \nand recall that $X\/[Y] = \\calO (D^n\/S^{n-1}) = \\calO S^n$. Then, by \nRemark~\\ref{rmk:ctsctl-quot-Cstar} above, we have a long exact sequence\n\\[\n    \\dotsb \\nameto{\\smash{\\die}} K_\\grstar(C^*(Y))\n        \\to K_\\grstar(C^*(X))\n        \\to K_\\grstar(C^*(\\calO S^n))\n        \\nameto{\\smash{\\die}} K_{\\grstar-1}(C^*(Y))\n        \\to \\dotsb \\,.\n\\]\nBy another ``Eilenberg swindle'', one shows that $K_\\grstar(C^*(X)) = 0$ and \nhence\n\\[\n    K_\\grstar(C^*(\\calO S^n)) = K_{\\grstar-1}(C^*(Y))\n        = \\begin{cases}\n                \\setZ & \\text{if $\\grstar \\equiv n-1 \\AMSdisplayoff\\pmod{2}$,\n                        and} \\\\\n                0 & \\text{otherwise.}\n            \\end{cases}\n\\]\n\\end{example}\n\n\\begin{example}[\\maybeboldmath suspensions in $K$-homology]%\n        \\label{ex:ctsctl-quot-Khom}\nSuppose that $A$ is a separable $C^*$-algebra. It is known that, for $n \\geq \n0$, elements of the Kasparov $K$-homology group $K^{n+1}(A)$ can be represented \nby (equivalence classes of) $C^*$-algebra morphisms\n\\begin{equation}\\label{ex:ctsctl-quot-Khom:eq:A}\n    \\phi \\from A \\to C^*(\\calO S^n)\n\\end{equation}\n(see \\cites{MR1627621, my-thesis}; I caution that, in my opinion, this is \nprobably not the ``best'' coarse geometric description of $K$-homology, but \nwork remains ongoing). The pairing of $K_m(A)$ with a $K$-homology class \nrepresented by such $\\phi$ is given simply by applying $K$-theory to $\\phi$ \n(and using the computation as in the previous Example).\n\nFix $n \\geq 1$ and suppose that we are given an element of $K^n(\\Sigma A)$, \nwhere $\\Sigma A \\defeq C_0(\\ooitvl{0,1}) \\tensor A$ is the $C^*$-algebraic \nsuspension of $A$, represented by a morphism\n\\begin{equation}\\label{ex:ctsctl-quot-Khom:eq:SA}\n    \\tilde{\\psi} \\from \\Sigma A \\to C^*(\\calO S^{n-1}).\n\\end{equation}\nActually, let us assume something stronger, that we are given a morphism $\\psi$ \nwhich fits into the following commutative diagram whose \\emph{rows} are with \nexact:\n\\[\\begin{CD}\n    0 @>>> \\Sigma A @>>> CA @>>> A @>>> 0 \\\\\n    @. @V{\\psi |_{\\Sigma A}}VV @V{\\psi}VV @VVV \\\\\n    0 @>>> C^*_X(Y)\n        @>>> C^*(X)\n        @>>> Q @>>> 0 \\\\\n    @. @VVV @VVV @VVV \\\\\n    0 @>>> C^*_{X\/[Y]}(Y\/[Y])\n        @>>> C^*(X\/[Y])\n        @>>> Q' @>>> 0\n\\end{CD}\\quad,\\]\nwhere $CA \\defeq C_0(\\coitvl{0,1}) \\tensor A$ is the cone on $A$, $X \\defeq \n\\calO D^n$, and $Y \\defeq \\calO S^{n-1} \\subseteq X$. (In fact, given a \n$\\tilde{\\psi}$, one can find a $\\psi$ such that\n\\[\\begin{CD}\n    K_\\grstar(\\Sigma A) @>{\\tilde{\\psi}}>> K_\\grstar(C^*(Y)) \\\\\n    @V{=}VV @V{\\sim}VV \\\\\n    K_\\grstar(\\Sigma A) @>{\\psi |_{\\Sigma A}}>>\n        K_\\grstar(C^*_X(Y))\n\\end{CD}\\]\ncommutes. This is not easy to prove, and seems to require that $A$ be \nseparable.)\n\nDenote the composition $A \\to Q \\to Q'$ by $\\utilde{\\phi}$. From the previous \nExample, we have natural isomorphisms\n\\[\n    K_\\grstar(\\calO S^{n-1}) = K_\\grstar(C^*_X(Y)) = K_{\\grstar+1}(Q)\n        = K_{\\grstar+1}(Q') = K_{\\grstar+1}(X\/[Y]) = K_{\\grstar+1}(\\calO S^n).\n\\]\nMoreover, since $K_\\grstar(CA) = 0$, we have $K_\\grstar(\\Sigma A) = \nK_{\\grstar+1}(A)$. These isomorphisms are all compatible, in the sense that \n$\\psi$ and $\\utilde{\\phi}$ are naturally equivalent on $K$-theory (with a \ndimension shift).\n\nIn fact, one can ``lift'' the morphism $\\utilde{\\phi}$ to a morphism $\\phi \n\\from A \\to C^*(X\/[Y]) = C^*(\\calO S^n)$ in the weak sense that the composition \n$A \\nameto{\\smash{\\phi}} C^*(X\/[Y]) \\to Q'$ is equal to $\\utilde{\\phi}$ on the \nlevel of $K$-theory. (This is not too difficult, but again seems to require \nthat $A$ be separable.) This provides a map from the $K$-homology group \n$K^n(\\Sigma A)$ (described as classes of morphisms as in \n\\eqref{ex:ctsctl-quot-Khom:eq:SA}) to the group $K^{n+1}(A)$ (described as in \n\\eqref{ex:ctsctl-quot-Khom:eq:A}).\n\\end{example}\n\n\n\n\n\\begin{bibsection}\n\n\\begin{biblist}\n\\bibselect{crscat}\n\\end{biblist}\n\n\\end{bibsection}\n\n\n\n\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn this paper, we study Markov Decision Processes (hereafter MDPs) with arbitrarily varying rewards.\nMDP provides a general mathematical framework for modeling sequential decision making under uncertainty \\cite{bertsekas1995dynamic, howard1960dynamic, puterman2014markov}. In the standard MDP setting, if the process is in some state $s$, the decision maker takes an action $a$ and receives an expected reward $r(s,a)$, before the process randomly transitions into a new state. The goal of the decision maker is to maximize the total expected reward. It is assumed that the decision maker has complete knowledge of the reward function $r(s,a)$, which does not change over time.\n\nOver the past two decades, there has been much interest in sequential learning and decision making in an unknown and possibly \\emph{adversarial} environment.\nA wide range of sequential learning problems can be modeled using the framework of Online Convex Optimization (OCO) \\cite{zinkevich2003online,hazan2016introduction}. In OCO, the decision maker plays a repeated game against an adversary for a given number of rounds. At the beginning of each round indexed by $t$, the decision maker chooses an action $a_t$ in some convex compact set $A$ and the adversary chooses a concave reward function $r_t$, hence a reward of $r_t(a_t)$ is received. After observing the realized reward function, the decision maker chooses its next action $a_{t+1}$ and so on. Since the decision maker does not know how the future reward functions will be chosen, its goal is to achieve a small \\emph{regret}; that is, the cumulative reward earned throughout the game should be close to the cumulative reward if the decision maker had been given the benefit of hindsight to choose one fixed action. We can express the regret after $T$ rounds as\n\\[\n\\text{Regret} (T) = \\max_{a \\in A} \\sum_{t=1}^T r_t(a) - \\sum_{t=1}^T r_t(a_t).\n\\]\nThe OCO model has many applications such as universal portfolios \\cite{cover1991, kalai2002, helmbold1998}, online shortest path \\cite{takimoto2003path}, and online submodular minimization \\cite{hazan2012submodular}. It also has close relations with areas such as convex optimization \\cite{hazan2010optimal, ben2015oracle} and game theory \\cite{cesa2006prediction}. There are many algorithms that guarantee sublinear regret, e.g., Online Gradient Descent \\cite{zinkevich2003online}, Perturbed Follow the Leader \\cite{kalai2005efficient}, and Regularized Follow the Leader \\cite{shalev2007online,abernethy2009competing}.\nCompared with the MDP setting, the main difference is that in OCO there is no notion of states, however the payoffs may be chosen by an adversary. \n\n\n\n\nIn this work, we study a general problem that unites the MDP and the OCO frameworks, which we call the {\\bf Online MDP problem}. More specifically, we consider MDPs where the decision maker knows the transition probabilities but the rewards are dynamically chosen by an adversary. \nThe Online MDP model can be used for a wide range of applications, including multi-armed bandits with constraints \\cite{yu2009markov}, the paging problem in computer operating systems \\cite{even2009online}, the $k$-server problem  \\cite{even2009online}, stochastic inventory control in operations research \\cite{puterman2014markov}, and scheduling of queueing networks \\cite{de2003linear,abbasi2014linear}.\n\n\n\n\\subsection{Main Results}\nWe propose a new computationally efficient algorithm that achieves near optimal regret for the Online MDP problem.\nOur algorithm is based on the linear programming formulation of infinite-horizon average reward MDPs, which uses the occupancy measure of state-action pairs as decision variables. This approach differs from other papers that have studied the Online MDP problem previously, see review in \\S\\ref{subsec:literature}. \n\n\nWe prove that the algorithm achieves regret bounded by $O(\\tau +\\sqrt{\\tau T  (\\ln \\vert S \\vert +\\ln \\vert A \\vert)} \\ln(T) )$, where $S$ denotes the state space, $A$ denotes the action space, $\\tau$ is the mixing time of the MDP, and $T$ is the number of periods. Notice that this regret bound depends \\emph{logarithmically} on the size of state and action space.\nThe algorithm solves a regularized linear program in each period with $poly(|S||A|)$ complexity. The regret bound and the computation complexity compares favorably to the existing methods discussed in \\S\\ref{subsec:literature}. \n\n\nWe then extend our results to the case where the state space $S$ is extremely large so that $poly(|S||A|)$ computational complexity is impractical. We assume the state-action occupancy measures associated with stationary policies are approximated with a linear architecture of dimension $d \\ll |S|$.\nWe design an approximate algorithm combining several innovative techniques for solving large scale MDPs inspired by \\cite{abbasi2019large,abbasi2014linear}.\nA salient feature of this algorithm is that its computational complexity does not depend on the size of the state-space but instead on the number of features $d$.\nThe algorithm has a regret bound $O(c_{S,A}(\\ln|S|+\\ln|A|)\\sqrt{\\tau T}\\ln T)$, where $c_{S,A}$ is a problem dependent constant.\nTo the best of our knowledge, this is the first $\\tilde{O}(\\sqrt{T})$ regret result for large scale Online MDPs.\n\n\\subsection{Related Work}\n\\label{subsec:literature}\n\nThe history of MDPs goes back to the seminal work of Bellman \\cite{bellman1957markovian} and Howard \\cite{howard1960dynamic} from the 1950's. \nSome classic algorithms for solving MDPS include policy iteration, value iteration, policy gradient, Q-learning and their approximate versions (see \\cite{puterman2014markov, bertsekas1995dynamic,bertsekas1996neuro} for an excellent discussion). In this paper, we will focus on a relatively less used approach, which is based on finding the \\textit{occupancy measure} using linear programming, as done recently in \\cite{chen2018scalable,wang2017primal,abbasi2019large} to solve MDPs with \\emph{static}  rewards (see more details in Section \\ref{mdp_via_lp}). To deal with the curse of dimensionality, \\cite{chen2018scalable} uses bilinear functions to approximate the occupancy measures and \\cite{abbasi2019large} uses a linear approximation.\n\n\nThe Online MDP problem was first studied a decade ago by \\cite{yu2009markov,even2009online}.  In \\cite{even2009online}, the authors developed no regret algorithms where the bound scales as $O(\\tau^2 \\sqrt{T \\ln(\\vert A \\vert)})$, where $\\tau$ is the mixing time (see \\S\\ref{sec:mdp_rftl}).  Their method runs an expert algorithm (e.g. Weighted Majority \\cite{littlestone1994weighted}) on every state where the actions are the experts. However, the authors did not consider the case with large state space in their paper. \n In \\cite{yu2009markov}, the authors provide a more computationally efficient algorithm using a variant of Follow the Perturbed Leader \\cite{kalai2005efficient}, but unfortunately their regret bound becomes $O(|S||A|^2\\tau T^{3\/4+\\epsilon})$. \nThey also considered approximation algorithm for large state space, but did not establish an exact regret bound.\n The work most closely related to ours is that from \\cite{dick2014online}, where the authors also use a linear programming formulation of MDP similar to ours. \nHowever, there seem to be some gaps in the proof of their results.\\footnote{In particular, we believe the proof of Lemma 1 in \\cite{dick2014online} is incorrect. Equation (8) in their paper states that the regret relative to a policy is equal to the sum of a sequence of vector products; however, the dimensions of vectors involved in these dot products are incompatible. By their definition, the variable $\\nu_t$ is a vector of dimension $\\vert S \\vert$, which is being multiplied with a loss vector with dimension $\\vert S \\vert \\vert A \\vert$.}\n\n The paper \\cite{ma2015online} also considers Online MDPs with large state space.  Under some conditions,  they show sublinear regret using a variant of approximate policy iteration, but the regret rate is left unspecified in their paper. \\cite{zimin2013online} considers a special class of MDPs called \\textit{episodic} MDPs and design algorithms using the occupancy measure LP formulation. Following this line of work, \\cite{neu2017unified} shows that several reinforcement learning algorithms can be viewed as variant of Mirror Descent \\cite{juditsky2011first} thus one can establish convergence properties of these algorithms. In \\cite{neu2014online} the authors consider Online MDPs with bandit feedback and provide an algorithm based on \\cite{even2009online}'s with regret of $O(T^{2\/3})$.\n \n A more general problem to the Online MDP setting considered here is where the MDP transition probabilites also change in an adversarial manner, which is beyond the scope of this paper. It is believed that this problem is much less tractable computationally \\cite[see discussion in][]{even2005experts}. \\cite{yu2009online} studies MDPs with changing transition probabilities,\nalthough \\cite{neu2014online} questions the correctness of their result, as the regret obtained seems to have broken a lower bound. In \\cite{gajane2018sliding}, the authors use a sliding window approach under a particular definition of regret. \\cite{abbasi2013online} shows sublinear regret with changing transition probabilities when they compare against a restricted policy class.\n\n\n\n\\section{Problem Formulation: Online MDP}\\label{section:online_mdps}\nWe consider a general Markov Decision Process with known transition probabilities but unknown and adversarially chosen rewards. Let $S$ denote the set of possible states, and $A$ denote the set of actions. (For notational simplicity, we assume the set of actions a player can take is the same for all states, but this assumption can be relaxed easily.)\nAt each period $t \\in [T]$, if the system is in state $s_t \\in S$, the decision maker chooses an action $a_t \\in A$ and collects a reward $r_t(s_t,a_t)$. Here, $r_t : S \\times A \\rightarrow [-1,1]$ denotes a reward function for period $t$. \nWe assume that the sequence of reward functions $\\{r_t\\}_{t=1}^T$ is initially unknown to the decision maker. The function $r_t$ is revealed only after the action $a_t$ has been chosen. We allow the sequence $\\{r_t\\}_{t=1}^T$ to be chosen by an \\textit{adaptive adversary}, meaning $r_t$ can be chosen using the history $\\{s_i\\}_{i=1}^{t}$ and $\\{a_i\\}_{i=1}^{t-1}$; in particular, the adversary does \\emph{not} observe the action $a_t$ when choosing $r_t$. \nAfter $a_t$ is chosen, the system then proceeds to state $s_{t+1}$ in the next period with probability $P(s_{t+1}\\vert s_t, a_t)$. \nWe assume the decision maker has complete knowledge of the transition probabilities given by $P(s' \\vert s, a) : S \\times A \\rightarrow S$.\n\nSuppose the initial state of the MDP follows $s_1 \\sim \\nu_1$, where $\\nu_1$ is a probability distribution over  $S$. \nThe objective of the decision maker is to choose a sequence of actions based on the history of states and rewards observed, such that the cumulative reward in $T$ periods is close to that of the optimal offline static policy.\nFormally, let $\\pi$ denote a stationary (randomized) policy: $\\pi:S\\rightarrow \\Delta_A$, where $\\Delta_A$ is the set of probability distributions over the action set $A$. Let $\\Pi$ denote the set of all stationary policies. We aim to find an algorithm that minimizes  \n\\begin{align}\\label{regret_def}\n\\text{MDP-Regret}(T)\\triangleq \\sup_{\\pi \\in \\Pi} R(T,\\pi), \\; \\text{with } R(T,\\pi) \\triangleq \\mathbb{E}[\\sum_{t=1}^T r_t(s^\\pi_t , a^\\pi_t)] - \\mathbb{E}[\\sum_{t=1}^T r_t(s_t,a_t)],\n\\end{align}\nwhere the expectations are taken with respect to random transitions of MDP and (possibly) external randomization of the algorithm.  \n\n\n \\section{Preliminaries}\n\n\nNext we provide additional notation for the MDP. \nLet $P^\\pi_{s,s'} \\triangleq P(s' \\mid s, \\pi(s))$ be the probability of transitioning from state $s$ to $s'$ given a policy $\\pi$. Let $P^\\pi$ be the $\\vert S\\vert \\times \\vert S\\vert$ matrix with entries $P^\\pi_{s,s'} \\, \\forall s,s' \\in S$. \nWe use row vector $\\nu_t \\in \\Delta_S$ to denote the probability distribution over states at time $t$. Let $\\nu^\\pi_{t+1}$ be the distribution over states at time $t+1$ under policy $\\pi$, given by $\\nu^{\\pi}_{t+1} = \\nu_{t} P^\\pi$.  \nLet $\\nu^\\pi_{st}$ denote the stationary distribution for policy $\\pi$, which satisfies the linear equation $\\nu^\\pi_{st} = \\nu^\\pi_{st} P^\\pi$. \nWe assume the following condition on the convergence to stationary distribution, which is commonly used in the MDP literature \\cite[see][]{yu2009markov,even2009online,neu2014online}.\n\n\\begin{assumption}\\label{assumption:mixing}\nThere exists a real number $\\tau \\geq 0$ such that for any policy $\\pi \\in \\Pi$ and any pair of distributions $\\nu,\\nu' \\in \\Delta_S$, it holds that $\\Vert \\nu P^\\pi - \\nu' P^\\pi\\Vert_1 \\leq e^{-\\frac{1}{\\tau}}\\Vert \\nu - \\nu'\\Vert_1$.\n\\end{assumption}\n\nWe refer to $\\tau$ in Assumption~\\ref{assumption:mixing} as the \\emph{mixing time}, which measures the convergence speed to the stationary distribution. In particular, the assumption implies that  $\\nu^\\pi_{st}$ is unique for a given policy $\\pi$.\n\nWe use $\\mu(s,a)$ to denote the proportion of time that the MDP visits state-action pair $(s,a)$ in the long run. We call $\\mu^\\pi \\in \\mathbb{R}^{\\vert S\\vert \\times \\vert A\\vert}$ the \\emph{occupancy measure} of policy $\\pi$. \nLet $\\rho_t^\\pi $ be the long-run average reward under policy $\\pi$ when the reward function is fixed to be $r_t$ every period, i.e., $\\rho^\\pi_t \\triangleq  \\lim_{T\\rightarrow \\infty} \\frac{1}{T} \\sum_{i=1}^T\\mathbb{E}[ r_t(s^\\pi_i,a^\\pi_i) ] $. We define $\\rho_t \\triangleq \\rho_t^{\\pi_t}$, where $\\pi_t$ is the policy selected by the decision maker for time $t$.\n\n\n\\subsection{Linear Programming Formulation for the Average Reward MDP}\\label{mdp_via_lp}\nGiven a reward function $r: S \\times A \\rightarrow [-1,1]$, suppose one wants to find a policy $\\pi$ that maximizes the long-run average reward: $\\rho^*=\\sup_{\\pi}\\lim_{T\\rightarrow \\infty} \\frac{1}{T}\\sum_{t=1}^T r(s^\\pi_t,a^\\pi_t)$.\nUnder Assumption~\\ref{assumption:mixing}, the Markov chain induced by any policy is ergodic and the long-run average reward is independent of the starting state \\cite{bertsekas1995dynamic}. \nIt is well known that \nthe optimal policy can be obtained by solving the Bellman equation, which in turn can be written as a linear program (in the dual form):\n\\begin{align}\n\\rho^* = \\max_\\mu & \\sum_{s\\in S} \\sum_{a \\in A} \\mu(s,a)r(s,a) \\label{eq:LP} \\\\\n\\text{s.t. } &  \\sum_{s\\in S} \\sum_{a\\in A} \\mu(s,a) P(s' \\vert s,a) = \\sum_{a\\in A} \\mu (s',a) \\quad \\forall s' \\in S \\nonumber \\\\\n&\\sum_{s\\in S} \\sum_{a\\in A} \\mu(s,a) = 1,\\quad \\mu(s,a)\\geq 0 \\quad \\forall s\\in S,\\,\\forall a\\in A. \\nonumber \n\\end{align}\nLet $\\mu^*$ be an optimal solution to the LP \\eqref{eq:LP}. We can construct an optimal policy of the MDP by defining $ \\pi^*(s,a) \\triangleq \\frac{\\mu^*(s,a)}{\\sum_{a\\in A} \\mu^*(s,a)}$ for all $s\\in S$ such that $\\sum_{a\\in A} \\mu^*(s,a)>0$;  for states where the denominator is zero, the policy may choose  arbitrary actions, since those states will not be visited in the stationary distribution.\n Let $\\nu^*_{st}$ be the stationary distribution over states under this optimal policy.  \n\nFor simplicity, we will write the first constraint of LP \\eqref{eq:LP}\nin the matrix form as $\\mu^\\top (P-B)=0$, for appropriately chosen matrix $B$. We denote the feasible set of the above LP as $\\Delta_M \\triangleq \\{\\mu\\in \\mathbb{R}: \\mu \\geq 0, \\mu^\\top1=1, \\mu^\\top(P-B)=0 \\}$. \nThe following definition will be used in the analysis later.\n\\begin{definition}\\label{def:delta_0}\nLet $\\delta_0 \\geq 0$ be the largest real number such that for all $\\delta \\in[0,\\delta_0]$, the set $\\Delta_{M,\\delta}\\triangleq\\{ \\mu \\in \\mathbb{R}^{\\vert S \\vert \\times \\vert A \\vert}: \\mu \\geq \\delta, \\mu^\\top 1 = 1, \\mu^\\top(P-B)=0 \\}$ is nonempty.\n\\end{definition}\n\n\\section{A Sublinear Regret Algorithm for Online MDP}\\label{sec:mdp_rftl}\n\nIn this section, we present an algorithm for the Online MDP problem.\n\n\\begin{algorithm}[!htb]\n\\caption{(\\textsc{MDP-RFTL})}\n\\label{alg:MDP-RFTL}\n\\begin{algorithmic}\n  \\STATE {\\bfseries input:} parameter $\\delta>0, \\eta>0$,  regularization term $R(\\mu) = \\sum_{s\\in S} \\sum_{a\\in A} \\mu(s,a) \\ln(\\mu(s,a))$\n  \\STATE {\\bfseries initialization:} choose any $\\mu_1 \\in \\Delta_{M,\\delta} \\subset \\mathbb{R}^{|S|\\times|A|}$\n  \\FOR{$t=1,...T$}  \n  \\STATE observe current state $s_t$\n  \\IF{$\\sum_{a\\in A}\\mu_t(s_t,a) > 0$} \n  \t\\STATE {choose action $a \\in A$ with probability $\\frac{\\mu_t(s_t,a)}{\\sum_{a}\\mu_t(s_t,a)}$.}\n  \\ELSE \n  \t\\STATE{choose action $a\\in A$ with probability $\\frac{1}{|A|}$} \n  \\ENDIF  \n  \\STATE observe reward function $r_t \\in [-1,1]^{\\vert S\\vert \\vert A \\vert}$\n  \\STATE update $\\mu_{t+1}\\leftarrow \\arg \\max_{\\mu \\in \\Delta_{M,\\delta}} \\sum_{i=1}^t \\left[ \\langle r_i , \\mu \\rangle - \\frac{1}{\\eta}R(\\mu) \\right]$\n  \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\nAt the beginning of each round $t\\in [T]$, the algorithm starts with an occupancy measure $\\mu_t$. If the MDP is in state $s_t$, we play action $a\\in A$ with probability $\\frac{\\mu_t(s_t,a)}{\\sum_{a}\\mu_t(s_t,a)}$. If the denominator is 0, the algorithm picks any action in $A$ with equal probability.\nAfter observing reward function $r_t$ and collecting reward $r_t(s_t,a_t)$,  the algorithm changes the occupancy measure to $\\mu_{t+1}$. \n\nThe new occupancy measure is chosen according to the Regularized Follow the Leader (RFTL) algorithm\n \\cite{shalev2007online,abernethy2009competing}. RFTL chooses the best occupancy measure for the cumulative reward  observed so far $\\sum_{i=1}^t r_i$, plus a regularization term $R(\\mu)$. The regularization term forces the algorithm  not to drastically change the occupancy measure from round to round.  In particular, we choose $R(\\mu)$ to be the entropy function.\n\nThe complete algorithm is shown  in Algorithm~\\ref{alg:MDP-RFTL}. \nThe main result of this section is the following.\n\n\\begin{theorem}\\label{theorem:mdp_rftl}\nSuppose $\\{r_t\\}_{t=1}^T$ is an arbitrary sequence of rewards such that $\\vert r_t(s,a)\\vert \\leq 1$ for all $s\\in S$ and $a\\in A$. For $T\\geq \\ln^2({1}\/{\\delta_0})$, the MDP-RFTL algorithm with parameters $\\eta = \\sqrt{\\frac{T \\ln(\\vert S \\vert \\vert A \\vert)}{\\tau}}$, $\\delta = e^{-{\\sqrt{T}}\/{\\sqrt{\\tau}}}$ guarantees\n\\begin{align*}\n\\text{MDP-Regret}(T) \\leq O \\left( \\tau + 4 \\sqrt{\\tau T  (\\ln\\vert S \\vert + \\ln \\vert A \\vert)} \\ln(T) \\right).\n\\end{align*}\n\\end{theorem}\n\nThe regret bound in Theorem~\\ref{theorem:mdp_rftl} is near optimal: a lower bound of $\\Omega(\\sqrt{T\\ln|A|})$ exists for the problem of learning with expert advice \\cite{freund1999adaptive,hazan2016introduction}, a special case of Online MDP where the state space is a singleton.\nWe note that the bound only depends \\emph{logarithmically} on the size of the state space and action space.\nThe state-of-the-art regret bound for Online MDPs is that of \\cite{even2009online}, which is $O(\\tau + \\tau^2 \\sqrt{\\ln(|A|)T})$. Compared to their result, our bound is better by a factor of $\\tau^{3\/2}$. However, our bound has depends on $\\sqrt{\\ln|S|+\\ln|A|}$, whereas the bound in \\cite{even2009online} depends on $\\sqrt{\\ln |A|}$.\nBoth algorithms require $poly(|S||A|)$ computation time, but are based on different ideas:\n The algorithm of \\cite{even2009online} is based on expert algorithms and requires computing $Q$-functions at each time step, whereas our algorithm is based on RFTL. In the next section, we will show how to extend our algorithm to the case with large state space.\n \n\n\n\n\\subsection{Proof Idea for Theorem \\ref{theorem:mdp_rftl}}\n\nThe key to analyze the algorithm is to decompose the regret with respect to policy $\\pi \\in \\Pi$ as follows \n{\n\\medmuskip=1mu\n\\begin{align}\\label{regret_decomposition}\nR(T, \\pi)  =  \\left[\\mathbb{E}[\\sum_{t=1}^T r_t(s^\\pi_t , a^\\pi_t)] - \\sum_{t=1}^T \\rho^\\pi_t\\right] + \\left[\\sum_{t=1}^T \\rho^\\pi_t - \\sum_{t=1}^T \\rho_t \\right] + \\left[  \\sum_{t=1}^T \\rho_t -  \\mathbb{E}[\\sum_{t=1}^T r_t(s_t,a_t)] \\right].\n\\end{align}\n}\nThis decomposition was first used by \\cite{even2009online}. We now give some intuition on why $R(T, \\pi)$ should be sublinear. By the mixing condition in Assumption~\\ref{assumption:mixing}, the state distribution $\\nu^\\pi_t$ at time $t$ under a policy $\\pi$ differs from the stationary distribution $\\nu^\\pi_{st}$ by at most $O(\\tau)$. This result can be used to bound the first term of \\eqref{regret_decomposition}.\n\nThe second term of \\eqref{regret_decomposition} can be related to the online convex optimization (OCO) problem through the linear programming formulation from Section~\\ref{mdp_via_lp}. Notice that $ \\rho^\\pi_t = \\sum_{s\\in S} \\sum_{a\\in A} \\mu^\\pi(s,a) r(s,a) = \\langle \\mu^\\pi, r\\rangle $, and $ \\rho_t = \\sum_{s\\in S} \\sum_{a\\in A} \\mu^\\pi_t(s,a) r(s,a) = \\langle \\mu^{\\pi_t}, r\\rangle$.\nTherefore, we have that \n\\begin{align}\n\\sum_{t=1}^T \\rho^\\pi_t - \\sum_{t=1}^T \\rho_t  = \\sum_{t=1}^T\\langle \\mu^\\pi, r_t\\rangle - \\sum_{t=1}^T \\langle \\mu^{\\pi_t}, r_t \\rangle,\n\\end{align}\nwhich is exactly the regret quantity commonly studied in OCO. \nWe are thus seeking an algorithm that can bound $\\max_{\\mu \\in \\Delta_M}  \\sum_{t=1}^T\\langle \\mu^\\pi, r_t\\rangle - \\sum_{t=1}^T \\langle \\mu^{\\pi_t}, r_t \\rangle$. In order to achieve logarithmic dependence on $|S|$ and $|A|$ in Theorem~\\ref{theorem:mdp_rftl}, we apply the RFTL algorithm, regularized by the negative entropy function  $R(\\mu)$. A technical challenge we faced in the analysis is that $R(\\mu)$ is not Lipschitz continuous over $\\Delta_M$, the feasible set of LP \\eqref{eq:LP}. So we design the algorithm to play in a shrunk set $\\Delta_{M,\\delta}$ for some $\\delta >0$ (see Definition~\\ref{def:delta_0}), in which $R(\\mu)$ is indeed Lipschitz continuous.\n\nFor the last term in \\eqref{regret_decomposition}, note that it is similar to the first term, although more complicated: the policy $\\pi$ is fixed in the first term, but the policy $\\pi_t$ used by the algorithm is varying over time. To solve this challenge, the key idea is to show that the policies do not change too much from round to round, so that the third term grows sublinearly in $T$. To this end, we use the property of the RFTL algorithm with carefully chosen regularization parameter $\\eta>0$.\nThe complete proof of Theorem \\ref{theorem:mdp_rftl} can be found in Appendix \\ref{sec:regret_analysis}.\n\n\\section{Online MDPs with Large State Space}\\label{sec:large_mdps}\nIn the previous section, \nwe designed an algorithm for Online MDP with sublinear regret.\nHowever, the computational complexity of our algorithm\nis $O(poly(\\vert S \\vert \\vert A \\vert))$  per round. In practice, \nMDPs often have extremely large state space $S$ due to the curse of dimenionality \\cite{bertsekas1995dynamic}, so computing the exact solution becomes impractical.\nIn this section we propose an approximate algorithm that can handle large state spaces.\n\n\\subsection{Approximating Occupancy Measures and Regret Definition}\n\nWe consider an approximation scheme introduced in \\cite{abbasi2014linear} for standard MDPs. The idea is to use $d$ feature vectors (with $d \\ll \\vert S \\vert \\vert A \\vert$) to approximate occupancy measures $\\mu \\in \\mathbb{R}^{|S|\\times|A|}$.\nSpecifically, we approximate $\\mu \\approx \\Phi \\theta$ where $\\Phi$ is a given matrix of dimension $\\vert S \\vert \\vert A \\vert \\times d$, and $\\theta \\in \\Theta \\triangleq \\{ \\theta \\in \\mathbb{R}_+^d: \\Vert \\theta \\Vert_1 \\leq W\\}$ for some positive constant $W$.\nAs we will restrict the occupancy measures chosen by our algorithm to satisfy $\\mu = \\Phi \\theta$, the definition of MDP-regret \\eqref{regret_def} is too strong as it compares against all stationary policies.\nInstead, we restrict the benchmark to be the set of policies $\\Pi^\\Phi$ that can be represented by matrix $\\Phi$, where\n\\begin{align*}\n\\Pi^\\Phi \\triangleq \\{ \\pi \\in \\Pi : \\text{ there exists $\\mu^\\pi \\in \\Delta_M$ such that } \\mu^\\pi = \\Phi \\theta  \\text{ for some } \\theta \\in \\Theta \\}.\n\\end{align*}\nOur goal will now be to achieve sublinear $\\Phi$-MDP-regret defined as\n\\begin{align}\\label{eq:regret_def_phi}\n\\text{$\\Phi$-MDP-Regret}(T) \\triangleq \\max_{\\pi \\in \\Pi^\\Phi} \\mathbb{E}[\\sum_{t=1}^T r_t(s^\\pi_t , a^\\pi_t)] - \\mathbb{E}[\\sum_{t=1}^T r_t(s_t,a_t)],\n\\end{align} \nwhere the expectation is taken with respect to random state transitions of the MDP and randomization used in the algorithm. Additionally, we want to make the computational complexity \\emph{independent} of $\\vert S \\vert$ and $\\vert A \\vert$.\n\n\n{\\bf Choice of Matrix $\\Phi$ and Computation Efficiency.}\nThe columns of matrix $\\Phi \\in \\mathbb{R}^{\\vert S \\vert \\vert A \\vert \\times d}$ represent probability distributions over state-action pairs. \nThe choice of $\\Phi$ is problem dependent, and a detailed discussion is beyond the scope of this paper.\n\\cite{abbasi2014linear} shows that for many applications such as the game of Tetris and queuing networks, $\\Phi$ can be naturally chosen as a sparse matrix, which allows\nconstant time access to entries of $\\Phi$ and efficient dot product operations. We will assume such constant time access throughout our analysis.\nWe refer readers to \\cite{abbasi2014linear} for further details. \n\n\n\\subsection{The Approximate Algorithm}\n\nThe algorithm we propose is built on \\textsc{MDP-RFTL}, but is significantly modified in several aspects. In this section, we start with key ideas on how and why we need to modify the previous algorithm, and then formally present the new algorithm. \n\nTo aid our analysis, we make the following definition.\n\\begin{definition}\\label{assumption:large-MDP}\nLet $\\tilde{\\delta}_0 \\geq 0$ be the largest real number such that for all $\\delta \\in[0,\\tilde{\\delta}_0]$ the set $\\Delta^\\Phi_{M,\\delta}\\triangleq\\{ \\mu \\in \\mathbb{R}^{\\vert S \\vert \\vert A \\vert}: \\text{ there exists $\\theta \\in \\Theta$ such that } \\mu = \\Phi \\theta, \\mu \\geq \\delta, \\mu^\\top 1 = 1, \\mu^\\top(P-B)=0 \\}$ is nonempty. We also write $\\Delta_M^\\Phi \\triangleq \\Delta^\\Phi_{M,0}$.\n\\end{definition}\n\nAs a first attempt, one could replace the shrunk set of occupancy measures $\\Delta_{M,\\delta}$ in Algorithm~\\ref{alg:MDP-RFTL} with $\\Delta^\\Phi_{M,\\delta}$ defined above. We then use occupancy measures $\\mu^{\\Phi \\theta^*_{t+1}} \\triangleq \\Phi \\theta^*_{t+1}$ given by the RFTL algorithm, i.e.,\n$ \\theta^*_{t+1} \\leftarrow \\arg \\max_{\\theta \\in \\Delta_{M,\\delta}^\\Phi} \\sum_{i=1}^t \\left[ \\langle r_i, \\mu \\rangle - ({1}\/{\\eta}) R(\\mu) \\right]$. \nThe same proof of Theorem~\\ref{theorem:mdp_rftl} would apply and guarantee a sublinear $\\Phi$-MDP-Regret.\nUnfortunately, replacing $\\Delta_{M,\\delta}$ with $\\Delta^\\Phi_{M,\\delta}$ does not reduce the time complexity of computing the iterates $\\{\\mu^{\\Phi \\theta^*_{t}}\\}_{t=1}^T$, which is still $poly(|S||A|)$.\n\nTo tackle this challenge, we will not apply the RFTL algorithm exactly, but will instead obtain an approximate solution in $poly(d)$ time. We relax the constraints $\\mu \\geq \\delta$ and $\\mu^\\top(P-B) = 0$ that define the set $\\Delta_{M,\\delta}^\\Phi$, and add the following penalty term to the objective function: \n\\begin{equation}\\label{eq:penalty}\n    V(\\theta) \\triangleq -H_t \\Vert (\\Phi \\theta )^\\top (P-B) \\Vert_1 - H_t \\Vert \\min\\{\\delta, \\Phi \\theta \\} \\Vert_1.\n\\end{equation}\nHere, $\\{H_t\\}_{t=1}^T$ is a sequence of tuning parameters that will be specified in Theorem~\\ref{thm:large_mdp_regret}.\nLet $ \\Theta^\\Phi \\triangleq \\{\\theta \\in \\Theta\\ , \\mathbf{1}^\\top (\\Phi \\theta)  = 1 \\}$.\nThus, the original RFTL step in Algorithm~\\ref{alg:MDP-RFTL} now becomes\n\\begin{equation}\\label{eq:c_t_eta}\n\\max_{\\theta \\in \\Theta^\\Phi} \\sum_{i=1}^tc^{t,\\eta}(\\theta), \\quad \\text{where }\nc^{t,\\eta}(\\theta) \\triangleq \\sum_{i=1}^t \\left[\\langle r_i , \\Phi \\theta \\rangle -  \\frac{1}{\\eta} R^\\delta (\\Phi \\theta)\\right] + V(\\theta).\n\\end{equation}\nIn the above function, we use a modified entropy function $R^\\delta(\\cdot)$ as the regularization term, because the standard entropy function has an infinite gradient at the origin.\nMore specifically, let \n$R_{(s,a)}(\\mu)\\triangleq \\mu(s,a) \\ln(\\mu(s,a))$ be the entropy function. We define $ R^\\delta (\\mu) = \\sum_{(s,a)} R^\\delta_{(s,a)}(\\mu (s,a))$, where\n\\begin{align}\\label{eq:def_R_delta}\nR^\\delta_{(s,a)} \\triangleq\n\\begin{cases}\nR_{(s,a)}(\\mu) \\quad &\\text{if } \\mu(s,a) \\geq \\delta \\\\\nR_{(s,a)}(\\delta) +  \\frac{d}{d \\mu(s,a)} R_{(s,a)}(\\delta) (\\mu(s,a)-\\delta) \\quad &\\text{otherwise}.\n\\end{cases}\n\\end{align}\n\n\n\nSince computing an exact gradient for function $c^{t,\\eta}(\\cdot)$ would take $O(\\vert S \\vert \\vert A \\vert)$  time, we solve problem \\eqref{eq:c_t_eta} by stochastic gradient ascent. The following lemma shows how to efficiently generate stochastic subgradients for function $c^{t,\\eta}$ via sampling.\n\\begin{lemma}\\label{stochastic_gradient_of_c}\nLet $q_1$ be any probability distribution over state-action pairs, and $q_2$ be any probability distribution over all states. Sample a pair $(s',a')\\sim q_1$ and $s'' \\sim q_2$. The quantity \n\\begin{align*}\ng_{s',a',s''}(\\theta) &= \\Phi^\\top r_{t} + \\frac{H_t}{q_1(s',a')} \\Phi_{(s',a'),:} \\mathbb{I}\\{\\Phi_{(s',a'),:} \\leq \\delta \\}\\\\\n&\\quad - \\frac{H_t}{q_2(s'')} [(P-B)^\\top \\Phi]_{s'',:} sign([(P-B)^\\top \\Phi ]_{s'',:} \\theta) - \\frac{t}{\\eta q_1(s',a')} \\nabla_\\theta R^\\delta_{(s',a')}(\\Phi \\theta)\n\\end{align*}\nsatisfies $\\mathbb{E}_{(s',a')\\sim q_1, s'' \\sim q_2}[g_{s',a',s''}(\\theta) \\vert \\theta] = \\nabla_\\theta c^{\\eta, t}(\\theta)$ for any $\\theta \\in \\Theta$. Morever, we have $ \\Vert g(\\theta) \\Vert_2 \\leq t \\sqrt{d} + H_t (C_1 + C_2)+ \\frac{t}{\\eta}(1 + \\ln (Wd) + \\vert \\ln(\\delta) \\vert) C_1,\n$ w.p.1, where\n\\begin{equation}\\label{def_of_Cs}\nC_1 = \\max_{(s,a)\\in S \\times A} \\frac{\\Vert \\Phi_{(s,a),:}\\Vert_2}{q_1(s,a)}, \\quad C_2 = \\max_{s\\in S} \\frac{\\Vert (P-B)^\\top_{:,s} \\Phi \\Vert_2}{q_2(s)}.\n\\end{equation}\n\\end{lemma}\n\n\n\n\n\n\nPutting everything together, we present the complete approximate algorithm for large state online MDPs in Algorithm~\\ref{alg:SGD-MDP-RFTL}. \nThe algorithm uses Projected Stochastic Gradient Ascent (Algorithm~\\ref{alg:P-SGA}) as a subroutine, which uses the sampling method in Lemma~\\ref{stochastic_gradient_of_c} to generate stochastic sub-gradients.\n\n\n\\begin{algorithm}[tbh]\n\\caption{(\\textsc{Large-MDP-RFTL})}\n\\label{alg:SGD-MDP-RFTL}\n\\begin{algorithmic}\n  \\STATE {\\bfseries input:} matrix $\\Phi$, parameters: $\\eta, \\delta >0$, convex function $R^\\delta(\\mu)$, SGA step-size schedule $\\{w_t\\}_{t=0}^T$, penalty term parameters $\\{H_t\\}_{t=1}^T$\n  \n  \\STATE {\\bfseries initialize:} $\\tilde{\\theta}_1\\leftarrow $ \\textsc{PSGA}($- R^{\\delta}(\\Phi \\theta) + V(\\theta), \\Theta^\\Phi, w_0, K_0  $)  \n  \\FOR{$t=1,...,T$}\n  \\STATE observe current state $s_t$; play action $a$ with distribution $\\frac{[\\Phi \\tilde{\\theta}_t]_+ (s_t,a)}{\\sum_{a\\in A} [\\Phi \\tilde{\\theta}_t]_+ (s_t,a)}$ \n  \\STATE observe $r_t \\in [-1,1]^{\\vert S\\vert \\vert A \\vert}$\n  \\STATE $\\tilde{\\theta}_{t+1}\\leftarrow $ \\textsc{PSGA}($\\sum_{i=1}^t [ \\langle r_i, \\Phi \\theta \\rangle -\\frac{1}{\\eta} R^\\delta(\\Phi \\theta) ] + V(\\theta), \\Theta^\\Phi, w_t, K_t  $)\n  \\ENDFOR\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\\begin{algorithm}[tbh]\n\\caption{Projected Stochastic Gradient Ascent: \\textsc{PSGA}$(f, X, w, K)$}\n\\label{alg:P-SGA}\n\\begin{algorithmic}\n  \\STATE {\\bfseries input:} concave objective function $f$, feasible set $X$, stepsize $w$, $x_1 \\in X$\n    \\FOR{$k=1,...K$}\n  \\STATE compute a stochastic subgradient $g_k$ such that $\\mathbb{E}[g_k] = \\nabla f(x_k)$ using Lemma \\ref{stochastic_gradient_of_c}\n  \\STATE set $x_{k+1} \\leftarrow P_X(x_k + w g(x_k))$ \n  \\ENDFOR\n  \\STATE {\\bfseries output:} $\\frac{1}{K} \\sum_{k=1}^K x_k$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{Analysis of the Approximate Algorithm}\n\n\n\n\nWe establish a regret bound for the \\textsc{Large-MDP-RFTL} algorithm as follows.\n\n\\begin{theorem}\\label{thm:large_mdp_regret}\nSuppose $\\{r_t\\}_{t=1}^T$ is an arbitrary sequence of rewards such that $\\vert r_t(s,a)\\vert \\leq 1$ for all $s\\in S$ and $a\\in A$. \nFor $T \\geq \\ln^2(\\frac{1}{\\delta_0})$, \\textsc{Large-MDP-RFTL} with parameters $\\eta = \\sqrt{\\frac{T}{\\tau}}, \\delta = e^{-\\sqrt{T}}$, $K(t) = \\left[ W t^2 \\sqrt{d} \\tau^4 (C_1 + C_2) T\\ln(W T) \\right]^2$, $H_t = t \\tau^2 T^{3\/4}$, $w_t=\\frac{W}{\\sqrt{K(t)} (t \\sqrt{d} + H_t (C_1 + C_2) + \\frac{t}{\\eta} C_1)} $ guarantees that \n\\begin{align*}\n\\text{$\\Phi$-MDP-Regret}(T) \\leq O( c_{S,A} \\ln(\\vert S \\vert  \\vert A\\vert) \\sqrt{\\tau T} \\ln(T) ).\n\\end{align*}\nHere $c_{S,A}$ is a problem dependent constant. The constants $C_1, C_2$ are defined in Lemma~\\ref{stochastic_gradient_of_c}.\n\\end{theorem}\n\nA salient feature of the \\textsc{Large-MDP-RFTL} algorithm is that its computational complexity in each period is independent of the size of state space $|S|$ or the size of action space $|A|$, and thus is amenable to large scale MDPs. In particular, in Theorem~\\ref{thm:large_mdp_regret}, the number of SGA iterations, $K(t)$, is $O(d)$ and independent of $|S|$ and $|A|$.\n\nCompared to Theorem~\\ref{theorem:mdp_rftl}, we achieve a regret with similar dependence on the number of periods $T$ and the mixing time $\\tau$. The regret bound also depends on $\\ln(|S|)$ and $\\ln(|A|)$, with an additional constant term $c_{S,A}$. The constant comes from a projection problem (see details in Appendix~\\ref{sec:regret_analysis_large_mdp}) and may grow with $|S|$ and $|A|$ in general. But for some classes of MDP problem, $c_{S,A}$ is bounded by an absolute constant: an example is the Markovian Multi-armed Bandit problem \\cite{whittle1980multi}.\n\n{\\bf Proof Idea for Theorem~\\ref{thm:large_mdp_regret}.} Consider the \\textsc{MDP-RFTL} iterates ,\\{$\\theta^*_t\\}_{t=1}^T$, and the occupancy measures $\\{\\mu^{\\Phi \\theta^*_t}\\}_{t=1}^T$ induced by following policies $\\{\\Phi \\theta^*_t\\}_{t=1}^T$. Since $\\theta^*_t \\in \\Delta_{M,\\delta}^\\Phi$ it holds that $\\mu^{\\Phi \\theta^*_t} = \\Phi \\theta^*_t$ for all $t$. Thus, following the proof of Theorem \\ref{theorem:mdp_rftl}, we can obtain  the same $\\Phi$-MDP-Regret bound in Theorem \\ref{theorem:mdp_rftl} if we follow policies $\\{\\Phi \\theta^*_t\\}_{t=1}^T$. However, computing $\\theta^*_t$ takes $O(poly(\\vert S \\vert \\vert A \\vert))$ time. \n\nThe crux the proof of Theorem \\ref{thm:large_mdp_regret} is to show that the $\\{\\Phi \\tilde{\\theta}_{t}\\}_{t=1}^T$ iterates in Algorithm~\\ref{alg:SGD-MDP-RFTL} induce occupancy measures $\\{\\mu^{\\Phi \\tilde{\\theta}_{t}}\\}_{t=1}^T$ that are close to $\\{\\mu^{\\Phi \\theta^*_t}\\}_{t=1}^T$. Since the algorithm has relaxed constraints of $\\Delta_{M,\\delta}^\\Phi$, in general we have $\\tilde{\\theta}_{t} \\notin \\Delta_{M,\\delta}^\\Phi$ and thus $\\mu^{\\Phi \\tilde{\\theta}_{t}} \\neq \\Phi \\tilde{\\theta}_{t}$. So we need to show that the distance between $\\mu^{\\Phi \\theta^*_{t+1}}$, and $\\mu^{\\Phi \\tilde{\\theta}_{t+1}}$ is small. Using triangle inequality we have \n\\begin{align*}\n\\Vert \\mu^{\\Phi \\theta_t^*} - \\mu^{\\Phi \\tilde{\\theta}_t}  \\Vert_1 &\\leq \\Vert \\mu^{\\Phi \\theta_t^*} - P_{\\Delta_{M,\\delta}^\\Phi}(\\Phi \\tilde{\\theta}_t)  \\Vert_1  +  \\Vert P_{\\Delta_{M,\\delta}^\\Phi}(\\Phi \\tilde{\\theta}_t) - \\Phi \\tilde{\\theta}_t  \\Vert_1 + \\Vert \\Phi \\tilde{\\theta}_t - \\mu^{\\Phi \\tilde{\\theta}_t} \\Vert_1, \n\\end{align*}\nwhere $P_{\\Delta_{M,\\delta}^\\Phi}(\\cdot)$ denotes the Euclidean projection onto $\\Delta_{M,\\delta}^\\Phi$. We then proceed to bound each term individually. We defer the details to Appendix \\ref{sec:regret_analysis_large_mdp} as bounding each term requires lengthy proofs. \n\n\\section{Conclusion}\n\nWe consider Markov Decision Processes (MDPs) where the transition probabilities are known but\nthe rewards are unknown and may change in an adversarial manner.  We provide an online algorithm, which applies Regularized Follow the Leader (RFTL) to the linear programming formulation of the average reward MDP. The algorithm achieves a\nregret bound of $O( \\sqrt{\\tau (\\ln|S|+\\ln|A|)T}\\ln(T))$, where $S$ is the state space, $A$ is the action space, $\\tau$ is the mixing time of the MDP, and $T$ is the number of periods. The algorithm's computational complexity  is polynomial in $|S|$ and $|A|$ per period. \n\nWe then consider a setting often encountered in practice, where the state space of the MDP is too large to allow for exact solutions. We approximate the state-action occupancy measures with a linear architecture of dimension $d\\ll|S||A|$. We then propose an approximate algorithm which relaxes the constraints in the RFTL algorithm, and solve the relaxed problem using stochastic gradient descent method. \nA salient feature of our algorithm is that its computation time complexity is independent of the size of state space $|S|$ and the size of action space $|A|$. \nWe prove a regret bound of $O( c_{S,A} \\ln(\\vert S \\vert  \\vert A\\vert) \\sqrt{\\tau T} \\ln(T))$ compared to the best static policy approximated by the linear architecture, where $c_{S,A}$ is a problem dependent constant.\nTo the best of our knowledge, this is the first $\\tilde{O}(\\sqrt{T})$ regret bound for large scale MDPs with changing rewards.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nIn the Hilbert space formalism of quantum mechanics (QM), a quantum probability function is a function $\\pee$ from the lattice $L(\\h)$ of projection operators acting on the Hilbert space $\\h$, to the unit interval $[0,1]$, which satisfies the rules\n\\begin{itemize}\n\\item $\\pee(1)=1$,\n\\item $\\pee(P_1\\vee P_2\\vee\\ldots)=\\pee(P_1)+\\pee(P_2)+\\ldots$ \\quad whenever $P_iP_j=0$ for all $i\\neq j$.\n\\end{itemize}\nThis characterization of quantum probabilities is reminiscent (at least in form) of the classical structure, where a probability function is specified by a pair $(\\mathcal{F},\\pee)$ where $\\mathcal{F}$ denotes a $\\sigma$-algebra of subsets of some space $\\Omega$ and $\\pee:\\mathcal{F}\\to[0,1]$ is a function which satisfies\n\\begin{itemize}\n\\item $\\pee(\\Omega)=1$,\n\\item $\\pee(\\Delta_1\\cup\\Delta_2\\cup\\ldots)=\\pee(\\Delta_1)+\\pee(\\Delta_2)+\\ldots$ \\quad whenever $\\Delta_i\\cap\\Delta_j=\\varnothing$ for all $i\\neq j$.\n\\end{itemize}\n\nBecause of the resemblance in structure between the two definitions, it is tempting to think that one must be able to find resemblances between admissible interpretations of the probabilities.\nHowever, as indicated in \\cite{Wilce12}, such a program has to face the problem of explaining the non-Boolean nature of $L(\\h)$.\nThis issue may best be understood by contrasting the quantum structure with the classical structure of $\\sigma$-algebras.\nBy far the majority of philosophical approaches to classical probability agree on one thing.\nNamely, that the probability $\\pee(\\Delta)$ can be understood as the probability that some proposition $S_\\Delta$ codified by $\\Delta$ is true. \nFor example, if $\\Omega$ is understood as the set of all possible worlds, then $\\Delta$ corresponds to the subset of all possible worlds in which $S_\\Delta$ is true.  \nThe question for the quantum case is now straightforward: if the elements of $L(\\h)$ correspond to propositions, what do these propositions express?\n\nThe most prominent early contribution to this question is undoubtedly the work of Birkhoff and von Neumann \\cite{BirkhoffNeumann36}.\nThe backbone of this paper already appeared in \\cite{Neumann55} where the idea of taking projections as propositions was introduced.\nIt is roughly motivated as follows.\nIf $A$ is an observable, and $\\Delta$ a (measurable) subset of the reals, then the proposition $A\\in\\Delta$ is taken to be true if and only if the probability that a measurement of $A$ reveals some value in $\\Delta$ equals one.\nThis is then the case if and only if the state of the system lies in $P_A^{\\Delta}\\mathcal{H}$, with $\\mathcal{H}$ the Hilbert space describing the system, and $P_A$ the projection valued measure generated by $A$.\nOr, in terms of density operators, if and only if $\\mathrm{Tr}(\\rho P_A^\\Delta)=1$.\n\nOf course this raises the question of what is meant by ``a measurement of $A$ reveals some value in $\\Delta$''.\nThe most tempting answer is the one suggested by the notation $A\\in\\Delta$, i.e., that a measurement of $A$ reveals some value in $\\Delta$ if and only if the observable $A$ has some value in $\\Delta$, independent of the measurement.\nA straightforward adoption of this realist reading is bound to run into difficulties posed by results such as the Kochen-Specker theorem \\cite{KS67} or the Bell inequalities \\cite{Bell64,Clauser69}.\nA possible escape is to deny that $A\\in\\Delta$ expresses a proposition (or has a truth value) for every pair $(A,\\Delta)$ at every instant.\nThis can be done by, for example, adopting the strong property postulate ($A$ has a value if and only if the system is in an eigen state of $A$).\nAnother possibility is to deny the bijection between observables and self-adjoint operators such as in Bohmian mechanics, where position observables play a privileged role.\\footnote{In this approach, observables are associated with functions on the configuration space of particle positions. Every proposition $A\\in\\Delta$ then reduces to a proposition on particle positions. This reduction is contextual in a sense; the function associated with $A$ is not the same for every experimental setup used to measure $A$.}\nOr one may suggest that these propositions do not obey the laws of classical logic.\nThis last option was the one suggested by Putnam in \\cite{Putnam69}.\nHe argued that, apart from coinciding with a set of propositions about values possessed by observables, the lattice $L(\\h)$ also describes the correct logic for these propositions.\nHowever, this realist interpretation of the quantum logic $L(\\h)$ is known to lead to problems \\cite{Dummett76,Stairs83} and the consensus seems to be that there is no hope for this direction \\cite{Maudlin05}. \n\nIn contrast to these difficulties encountered in realist approaches to understanding $L(\\h)$, it is often thought that an operationalist interpretation is unproblematic, and perhaps even straightforward:\n\\begin{quote}\nIf we put aside scruples about `measurement' as a primitive term in physical theory, and accept a principled distinction between `testable' and non-testable properties, then the fact that $L(\\h)$ is not Boolean is unremarkable, and carries no implication about logic per se. \nQuantum mechanics is, on this view, a theory about the possible statistical distributions of outcomes of certain measurements, and its non-classical `logic' simply reflects the fact that not all observable phenomena can be observed simultaneously. -- Wilce \\cite[\\S 2]{Wilce12}\n\\end{quote}\nThe idea is that the task of interpreting the structure of $L(\\h)$ is tied up with explicating the notion of measurement.\nThen, since the operationalist takes the notion of measurement as primitive, the structure of $L(\\h)$ may also be taken as primitive.\nThis line of reasoning seems unsatisfactory to me and I think that the questions of what the propositions in $L(\\h)$ express, and what the logic is that governs them, are also meaningful and require study from an operationalist point of view.\nAnd as a guide to such a logic one may take into account Bohr's demand that \n\\begin{quote}\nall well-defined experimental evidence, even if it cannot be analyzed in terms of classical physics, must be expressed in ordinary language making use of common logic. -- Bohr \\cite{Bohr48}\n\\end{quote}\nIn this paper I continue the work done in \\cite{Hermens12} to meet this demand.\nSpecifically, in section \\ref{SQMsection} I first introduce a lattice of experimental propositions $S_{QM}$ which is then extended to an intuitionistic logic $L_{QM}$ in section \\ref{LQMsection}.\nThese results comprise a summary of some of the results in \\cite{Hermens12}.\nThen, also in section \\ref{LQMsection}, this logic is extended to a classical logic $CL_{QM}$ for experimental propositions.\nIn section \\ref{QProbsec} R\\'enyi's theory of conditional probability spaces \\cite{Renyi55} is applied to $CL_{QM}$ and it is shown that the quantum probabilities as given by the Born rule can be reproduced.\nThis is followed by a short discussion on the issue that non-quantum probabilities are also allowed in this framework.\nFinally, in section \\ref{conclusionsection}, the results of this paper are evaluated and some ideas on what further role they may play in foundational debates are given.\n\n \n \n \n\n\\section{The lattice of experimental propositions}\\label{SQMsection}\nThe idea to focus on the experimental side of QM is reminiscent of the approach of Birkhoff and von Neumann in \\cite{BirkhoffNeumann36}.\nThey start by introducing `experimental propositions' which are identified with subsets of `observation spaces'.\nAn observation space is the Cartesian product of the spectra of a set of pairwise commuting observables.\nHowever, once such a subset is associated with a projection operator in the familiar way, the experimental context (i.e., the set of pairwise commuting observables) is no longer being considered.\nFor example, one may have that $P_{A_1}^{\\Delta_1}=P_{A_2}^{\\Delta_2}$ even though $A_1$ and $A_2$ may be incompatible.\nThis indicates that whatever is encoded by propositions in $L(\\h)$, they do not refer to actual experiments.\n\nTo avoid a cumbersome direct discussion of $L(\\h)$, it is then appropriate to instead develop a logic of propositions that \\emph{do} encode these experimental contexts.\nAs an elementary example of what Bohr may have had in mind when speaking of ``well-defined experimental evidence'' I take propositions of the form\n\\begin{equation}\\label{reading}\n  M_A(\\Delta) =\\text{``$A$ is measured and the result lies in $\\Delta$.''}\n\\end{equation}\nOf course, no logical structure can be derived for these propositions without some structural assumptions.\nI will adopt the following two assumptions which I believe to be quite innocent\\footnote{These assumptions are the ones also adopted in \\cite{Hermens12} although there the second was implicit and the first was formulated to also apply to other theories.}:\n\\begin{itemize}\n\\item[\\textbf{LMR}](Law-Measurement Relation) If $A_1$ and $A_2$ are two observables that can be measured together, and if $f$ is a function such that whenever $A_1$ and $A_2$ are measured together the outcomes $a_1$ and $a_2$ satisfy $a_1=f(a_2)$ (i.e., $f$ represents a law), then a measurement of $A_2$ alone also counts as a measurement of $A_1$ with outcome $f(a_2)$.\\footnote{This assumption is reminiscent of the FUNC rule adopted in the Kochen-Specker theorem (c.f. \\cite[p.121]{Redhead87}). However, FUNC is a much stronger assumption stating that whenever measurement outcomes for a pair of observables satisfy a functional relation, the values possessed by these observables must satisfy the same relationship. That is, FUNC is the conjunction of LMR together with the idea that measurements reveal pre-existing values.}\n\\item[\\textbf{IEA}](Idealized Experimenter Assumption) Every experiment has an outcome, i.e., for every observable $A$ $M_A(\\varnothing)$ is understood as a contradiction.\n\\end{itemize}\nThe following lemma describes the structure arising from these assumptions for QM. \nA more elaborate exposition may be found in \\cite{Hermens12}.\n\n\\begin{lemma}\nUnder the assumptions LMR and IEA the set of (equivalence classes of) experimental propositions for quantum mechanics lead to the lattice \n\\begin{equation}\n  S_{QM}:=\\{(\\A,P)\\:;\\:\\A\\in\\mathfrak{A},P\\in L(\\A)\\backslash\\{0\\}\\}\\cup\\{\\bot\\},\n\\end{equation}\nwhere $\\mathfrak{A}$ denotes the set of all Abelian operator algebras on $\\h$ and $L(\\A)$ the Boolean lattice $\\A\\cap L(\\h)$.\nThe partial order is given by\n\\begin{equation}\n  (\\mathcal{A}_1,P_1)\\leq(\\mathcal{A}_2,P_2)~\\text{iff}~P_1=0~\\text{or}~\\mathcal{A}_1\\supset\\mathcal{A}_2,P_1\\leq P_2.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nA proposition $M_{A_1}(\\Delta_1)$ implies $M_{A_2}(\\Delta_2)$ if and only if a measurement of $A_1$ implies a measurement of $A_2$ and every outcome in $\\Delta_1$ implies an outcome in $\\Delta_2$ for the measurement of $A_2$ (or if $M_{A_1}{\\Delta_1}$ expresses a contradiction).\nBy LMR, this is the case if and only if the Abelian algebra $\\A_2$ generated by $A_2$ is a subalgebra of the Abelian algebra $\\A_1$ generated by $A_1$ and $P_{A_1}^{\\Delta_1}\\leq P_{A_2}^{\\Delta_2}$ (or $P_{A_1}^{\\Delta_1}=0$).\nThis also establishes that $M_{A_1}(\\Delta_1)$ and $M_{A_2}(\\Delta_2)$ are equivalent if and only if $(\\mathcal{A}_1,P_{A_1}^{\\Delta_1})=(\\mathcal{A}_2,P_{A_2}^{\\Delta_2})$ (or both express a contradiction). \n\\end{proof}\n\nThe meet on $S_{QM}$ is given by\n\\begin{equation}\n  (\\mathcal{A}_1,P_1)\\wedge(\\mathcal{A}_2,P_2)=\\begin{cases}(\\mathpzc{Alg}(\\mathcal{A}_1,\\mathcal{A}_2),P_1\\wedge P_2),&\\text{if}~[\\mathcal{A}_1,\\mathcal{A}_2]=0,P_1\\wedge P_2\\neq0,\\\\ \\bot,&\\text{otherwise,}\\end{cases}\n\\end{equation}\nwhere $P_1\\wedge P_2$ is the meet on $L(\\h)$.\nIt is consistent with the interpretation given to $M_A(\\Delta)$ in the sense that this meet `distributes' over the `and' in ``$A$ is measured and the result lies in $\\Delta$''.\nIn particular, it assigns a contradiction to a simultaneous measurement of two observables whenever their corresponding operators do not commute.\n\nIt is harder to interpret the join on this lattice as a disjunction.\nWhen restricting to joins of two elements of $S_{QM}$ with the same algebra, the results seem intuitively correct.\nIn this case one has \n\\begin{equation}\\label{disjcom}\n  (\\mathcal{A},P_1)\\vee(\\mathcal{A},P_2)=(\\mathcal{A},P_1\\vee P_2).\n\\end{equation}\nAnd indeed, ``$A$ is measured and the measurement reveals some value in $\\Delta_1\\cup\\Delta_2$'' sounds like a good paraphrase of ``$A$ is measured and the measurement reveals some value in $\\Delta_1$ or $A$ is measured and the measurement reveals some value in $\\Delta_2$''.\nBut in more general cases problems arise.\nFor example, one has $(\\mathcal{A}_1,1)\\vee(\\mathcal{A}_2,1)=(\\mathcal{A}_1\\cap\\mathcal{A}_2,1)$.\nThen, if the two algebras are completely incompatible (i.e. $\\mathcal{A}_1\\cap\\mathcal{A}_2=\\mathbb{C}1$), this equation states that ``$A_1$ is measured or $A_2$ is measured'' is a tautology even though one can consider situations in which neither is measured.\n\nThe example can further be used to show that $S_{QM}$ is not distributive.\nLet $A_3$ be a third observable that is totally incompatible with both $A_1$ and $A_2$.\nOne then has\n\\begin{equation}\n  (\\mathcal{A}_3,1)\\wedge((\\mathcal{A}_1,1)\\vee(\\mathcal{A}_2,1))\n  =(\\mathcal{A}_3,1)\\neq\\bot\n  =((\\mathcal{A}_3,1)\\wedge(\\mathcal{A}_1,1))\\vee((\\mathcal{A}_3,1)\\wedge(\\mathcal{A}_2,1)).\n\\end{equation}\nHence, when it comes to providing a logic that fares well with natural language, $S_{QM}$ doesn't perform much better than the original $L(\\h)$.\n\n \n \n \n\n\\section{Deriving the logics \\texorpdfstring{$L_{QM}$}{LQM} and \\texorpdfstring{$CL_{QM}$}{CLQM}}\\label{LQMsection}\nTo solve the problem of non-distributivity of $S_{QM}$ the lattice has to be extended by formally introducing disjunctions which in general are stronger than the join on $S_{QM}$.\nThe outcome of this process is described by the following theorem.\\footnote{Again, more details may be found in \\cite{Hermens12}.}\n\n\\begin{theorem}\nFormally introducing disjunctions to $S_{QM}$ while respecting \\eqref{disjcom} results in the complete distributive lattice \n\\begin{equation}\n  L_{QM}=\\left\\{S:\\mathfrak{A}\\to L(\\mathcal{H})\\:;\\:\\substack{S(\\mathcal{A})\\in L(\\A)\\\\ S(\\A_1)\\leq S(\\A_2)~\\text{whenever}~\\A_1\\subset\\A_2}\\right\\}\n\\end{equation} \nwith  partial order on $L_{QM}$ defined by\n\\begin{equation}\n  S_1\\leq S_2~\\text{iff}~S_1(\\mathcal{A})\\leq S_2(\\mathcal{A})\\forall\\mathcal{A}\\in\\mathfrak{A}\n\\end{equation} \ngiving the meet and join\n\\begin{equation}\\label{operations}\n\\begin{split}\n  (S_1\\vee S_2)(\\mathcal{A})&=S_1(\\mathcal{A})\\vee S_2(\\mathcal{A}),\\\\\n  (S_1\\wedge S_2)(\\mathcal{A})&=S_1(\\mathcal{A})\\wedge S_2(\\mathcal{A}).\n\\end{split}\n\\end{equation}\nThe embedding of $S_{QM}$ into $L_{QM}$ is given by\\footnote{For notational convenience $(\\A,0)$ is identified with $\\bot\\in S_{QM}$.}\n\\begin{equation}\\label{imbed}\n  i:(\\mathcal{A},P)\\mapsto S_{(\\mathcal{A},P)},~S_{(\\mathcal{A},P)}(\\mathcal{A}'):=\\begin{cases}P,&\\mathcal{A}\\subset\\mathcal{A}',\\\\0,&\\text{otherwise}.\\end{cases}\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nThe key to the extension of $S_{QM}$ to $L_{QM}$ lies in noting that the embedding \\eqref{imbed} satisfies the relation\n\\begin{equation}\\label{relation}\n  S=\\bigvee_{\\A\\in\\mathfrak{A}}i(\\A,S(\\A)).\n\\end{equation}\nThis gives the intended reading of elements of $L_{QM}$, namely, as a disjunction of experimental propositions.\nBy LMR it follows that $i$ preserves the interpretation of elements of $S_{QM}$.\nThe conjunctions on $L_{QM}$ further agree with those on $S_{QM}$:\n\\begin{equation}\n  S_{(\\A_1,P_1)}\\wedge S_{(\\A_2,P_2)}= S_{(\\A_1,P_1)\\wedge(\\A_2,P_2)}.\n\\end{equation}\nFurther, $L_{QM}$ respects \\eqref{disjcom} while introducing `new' elements for other disjunctions, i.e.,\n\\begin{equation}\n  S_{(\\A_1,P_1)}\\vee S_{(\\A_2,P_2)}\\leq S_{(\\A_1,P_1)\\vee(\\A_2,P_2)},\n\\end{equation}\nwith equality if and only if either $P_1$ or $P_2$ equals $0$, or $\\A_1=\\A_2$. \nFinally, that $L_{QM}$ is a complete distributive lattice follows from \\eqref{operations} and observing that $L(\\mathcal{A})$ is a complete distributive lattice for every $\\mathcal{A}\\in\\mathfrak{A}$.\n\\end{proof}\n\nThe lattice $L_{QM}$ is turned into a Heyting algebra by obtaining the relative pseudo-complement in the usual way:\n\\begin{equation}\n  S_1\\to S_2=\\bigvee\\left\\{S\\in L_{QM}\\:;\\:S\\wedge S_1\\leq S_2\\right\\}.\n\\end{equation}\nAnd negation is introduced by $\\neg S:=S\\to\\bot$ thus obtaining an intuitionistic logic for reasoning with experimental propositions.\nThe logic $L_{QM}$ is also a proper Heyting algebra and, in fact, is radically non-classical.\nThat is, the law of excluded middle only holds for the top and bottom element.\nThis is easily checked by observing that $S(\\mathbb{C}1)=1$ if and only if $S=\\top$. \nIt then follows that $S\\vee\\neg S=\\top$ if and only if $S=\\top$ or $\\neg S=\\top$.\n\nThe strong non-classical behavior can be understood by noting that the logic is entirely built up from propositions about performed measurements.\nThe negation of $M_A(\\Delta)$ is identified with other propositions about measurements that exclude the possibility of $M_A(\\Delta)$ but leave open the option of no experiment having been performed:\n\\begin{equation}\n  \\neg S_{(\\A,P)}=\\bigvee \\left\\{S_{(\\A',P')}\\:;\\: [\\A',\\A]\\neq 0~\\text{or}~P'\\wedge P=0\\right\\}.\n\\end{equation}\nAn unexcluded middle thus presents itself as the proposition $\\neg M_A$=``$A$ is not measured''.\nIt was suggested in \\cite[\\S2]{Hermens12} that incorporating such propositions could lead to a classical logic.\nI will now show that this is indeed the case.\n\nIt is tempting to start the program for a classical logic for experimental propositions by formally adding the suggested propositions $\\neg M_A$ and building on from there.\nHowever, it turns out that it is much more convenient to introduce new elementary experimental propositions, namely\n\\begin{equation}\\label{prov}\n  M_A^!(\\Delta)=\\text{``$A$ is measured with result in $\\Delta$, and no finer grained measurement has been performed''}.\n\\end{equation}\nThe notion of fine graining here relies on LMR.\nAnother way of expressing the conjunct added is to state that ``only $A$ and all the observables implied to be measured by the measurement of $A$ have been measured''.\nWith this concept the following can now be shown.\\footnote{In the remainder of this paper it is tacitly assumed that $\\h$ is finite-dimensional.}\n\n\\begin{theorem}\nThe classical logic for experimental propositions is given by\\footnote{The exclamation mark is merely a notational artifact introduced to discern the elements of $S^!_{QM}$ from those of $S_{QM}$. That is, without the exclamation mark, $S^!_{QM}$ would be a subset of $S_{QM}$.}\n\\begin{equation}\n  CL_{QM}=\\mathcal{P}(S^!_{QM}),~S^!_{QM}:=\\left\\{(\\A^!,P)\\:;\\:\\A\\in\\mathfrak{A}, P\\in L_0(\\A)\\right\\},\n\\end{equation}\nwhere $L_0(\\A)$ denotes the set of atoms of the lattice $L(\\A)$. \n\\end{theorem}\n\\begin{proof}\nBeing defined as a power set, it is clear that $CL_{QM}$ is a classical propositional lattice. \nSo it only has to be shown that every element signifies an experimental proposition, and that it is rich enough to incorporate the propositions already introduced by $L_{QM}$. \nThe first issue is readily established by the identification\n\\begin{equation}\\label{ident}\n  M_A^!(\\Delta)\\mapsto \\{(\\A^!,P)\\in S^!_{QM}\\:;\\:P\\leq \\mu_A(\\Delta)\\},\n\\end{equation}\nwith $\\A$ the algebra generated by $A$.\nIdentifying $\\A$ with $A$ is again justified by LMR, and IEA is incorporated by the fact that, whenever $\\Delta$ contains no elements in the spectrum of $A$, $M_A^!(\\Delta)$ is identified with the empty set.\nThus \\eqref{ident} ensures that every element of $CL_{QM}$ is understood as a disjunction of experimental propositions, i.e., every singleton set in $CL_{QM}$ encodes a proposition of the form $M_A^!(\\{a\\})$.\n\nThat $CL_{QM}$ is indeed rich enough follows by observing that the map\n\\begin{equation}\n  c:L_{QM}\\to CL_{QM},~S\\mapsto\\bigcup_{\\A\\in\\mathfrak{A}}\\left\\{(\\A^!,P)\\in S^!_{QM}\\:;\\:P\\leq S(\\A)\\right\\}\n\\end{equation}\nis an injective complete homomorphism.\n\\end{proof}\n\nSome additional reflections are helpful to get a grip on $CL_{QM}$.\nFirst one may note that the map $(c\\circ i):S_{QM}\\to CL_{QM}$, encoding propositions of the form $M_A(\\Delta)$, acts as\n\\begin{equation}\\label{ident2}\n  c\\circ i:(\\A,P)\\mapsto \\left\\{(\\A'^!,P')\\in S^!_{QM}\\:;\\:(\\A',P')\\leq (\\A,P)\\right\\}.\n\\end{equation}\nThus the provision ``and no finer grained measurement has been performed'' that distinguishes $M_A^!(\\Delta)$ from $M_A(\\Delta)$ precisely excludes all the $(\\A'^!,P')\\in (c\\circ i)(\\A,P) $ with $\\A'\\neq\\A$, i.e., the finer grained measurements (c.f. \\eqref{ident} and \\eqref{ident2}). \n\nThe non-classicality of $L_{QM}$ is also explicated by $CL_{QM}$.\nFor any $(\\A,P)\\in S_{QM}$, the complement of \n\\begin{equation}\n  c(S_{(\\A,P)}\\vee\\neg S_{(\\A,P)})\n\\end{equation}\nconsists of all the $(\\A'^!,P')\\in S^!_{QM}$ with $\\A'\\subsetneq\\A$ and $P'\\leq P$.\nThis is because only the added provision in \\eqref{prov} makes $(\\A'^!,P')$ exclude the measurement of $A$, while within $L_{QM}$ there is nothing to denote a possible incompatibility for $A$ and $A'$.\nA proposition in $CL_{QM}$ of special interest in this context is the singleton $\\{(\\mathbb{C}1,1)\\}$ which expresses that only the trivial measurement is performed.\nIn light of IEA this is just the same as saying that no measurement is performed; the most typical proposition in $CL_{QM}$ not present in $L_{QM}$.  \n\nTo conclude this section I compare the constructions of $L_{QM}$ and $CL_{QM}$ to another way of extending $S_{QM}$ to a distributive lattice, namely, by using the Bruns-Lakser theory of distributive hulls \\cite{BrunsLakser70} as advocated in \\cite{Coecke02}.\nIn this approach $S_{QM}$ is embedded into the lattice of its distributive ideals\\footnote{A distributive ideal is a subset $I\\subset S_{QM}$ that is downward closed and further contains the join $\\bigvee\\{(\\A,P)\\in J\\}$ for every subset $J\\subset I$ that satisfies $\\left(\\bigvee_{(\\A,P)\\in J}(\\A,P)\\right)\\wedge(\\A',P')= \\bigvee_{(\\A,P)\\in J}\\left((\\A,P)\\wedge(\\A',P')\\right)$ for all $(\\A',P')\\in S_{QM}$.} $\\mathcal{DI}(S_{QM})$ by taking each element to its downward set, i.e.,\n\\begin{equation}\n  (\\A,P)\\mapsto\\downarrow(\\A,P):=\\{(\\A',P')\\in S_{QM}\\:;\\:(\\A',P')\\leq(\\A,P)\\}.\n\\end{equation}\n\nGenerally, the abstract nature of the extension $\\mathcal{DI}(L)$ for some non-distributive lattice $L$ prevents a straightforward identification of distributive ideals with propositions.\nIn the present situation, for example, it is not immediately clear whether the resulting lattice operations behave consistently with respect to the intended reading of the elements $\\downarrow(\\A,P)$ given by \\eqref{reading}. \nFortunately, this issue can be resolved for $S_{QM}$ by noting that the resulting lattice of its distributive ideals is isomorphic to the power set of the atoms in $S_{QM}$:\n\\begin{equation}\n  \\mathcal{DI}(S_{QM})\\simeq\\mathcal{P}(S^*_{QM}),~S^*_{QM}:=\\{(\\A,P)\\in S_{QM}\\:;\\:\\A\\in \\mathfrak{A}_{\\mathrm{max}},P\\in L_0(\\A)\\},\n\\end{equation}\nwhere $\\mathfrak{A}_{\\mathrm{max}}$ is the set of maximal Abelian algebras.\nHere then one sees that the proposition identified with $(\\A,P)\\in S_{QM}$ is understood in  $\\mathcal{P}(S^*_{QM})$ as a disjunction over all possible \\emph{complete} measurements compatible with a measurement of $A$. \nThat is, $M_A(\\Delta)$ in this setting may be rephrased as ``some complete measurement $A'$ has been performed with outcome $a'$ which implies $M_A(\\Delta)$''.\nAlthough this provides a consistent reading, the underlying presupposition that every measurement in fact constitutes a complete measurement seems somewhat reckless.\n\n \n \n \n\n\\section{Quantum logic and probability}\\label{QProbsec}\nIf the logic $CL_{QM}$ is to play any significant role in quantum mechanics, it has to be connected with quantum probability theory.\nThat is, one has to know how the Born rule can be expressed using the language of $CL_{QM}$.\nIdeally this consists of two parts.\nFirst, to show that probability functions on $CL_{QM}$ can be introduced that are in accordance with the Born rule.\nAnd secondly, to show that the Born rule comes out as a necessary consequence holding for all probability functions on $CL_{QM}$ (i.e., a result similar to Gleason's theorem for $L(\\h)$ (\\cite{Gleason57})).  \nIn this section some first investigations in this direction are made.\n\nThe first approach for probabilities on $CL_{QM}$ undoubtedly is to note that $(S^!_{QM},CL_{QM})$ is a measure space.\nHence, it is tempting to state that all probability functions are simply all probability measures on this space.\nHowever, this does not do justice to the quantum formalism.\nFor example, one possible probability measure is the Dirac measure peaked at $\\{(\\mathbb{C}1,1)\\}$ expressing certainty that no measurement will be performed.\nThis is likely to be too minimalistic even for the hardened instrumentalist. \nThat is, although one may adhere to the idea that `unperformed experiments have no results' \\cite{Peres78}, a central idea in QM is that possible outcomes for measurements do have probabilities irrespective of whether the measurements are performed.\nIndeed, it is a trait of QM that certain conditional probabilities have definite values even if the propositions on which one conditions have prior probability zero.\nThe Dirac measure does not provide these conditional probabilities.\n\nTo get around this it is natural to take conditional probabilities as primitive in the quantum case.\\footnote{Quantum mechanics has even been used as an example to advocate that conditional probability should be considered more fundamental than unconditional probability, c.f. \\cite{Hajek03}.}\nFor this approach the axiomatic scheme of R\\'enyi is an appropriate choice.\\footnote{C.f., \\cite{Renyi55}. A similar axiomatic approach was developed independently by Popper \\cite[A *ii--*v]{Popper59}. The present presentation is taken from \\cite{Spohn86}.}\n\n\\begin{definition}\nLet $(\\Omega,\\mathcal{F})$ be a measurable space and $\\mathcal{C}\\subset\\mathcal{F}\\backslash\\{\\varnothing\\}$ be a non-empty set, then $(\\Omega,\\mathcal{F},\\mathcal{C},\\pee)$ is called a \\emph{conditional probability space (CPS)} if $\\pee:\\mathcal{F}\\times\\mathcal{C}\\to[0,1]$ satisfies\n\\begin{enumerate}\n\\item for every $C\\in\\mathcal{C}$ the function $A\\mapsto\\pee(A|C)$ is a probability measure on $(\\Omega,\\mathcal{F})$,\n\\item for all $A,B,C\\in\\mathcal{F}$ with $C,B\\cap C\\in\\mathcal{C}$ one has\n\\begin{equation}\n  \\pee(A\\cap B|C)=\\pee(A|B\\cap C)\\pee(B|C).\n\\end{equation}\n\\end{enumerate}\nIf in addition to these criteria $\\mathcal{C}$ is closed under finite unions, the space is called an \\emph{additive CPS}, and if $\\mathcal{C}=\\mathcal{F}\\backslash\\{\\varnothing\\}$ it is called a \\emph{full CPS}.\n\\end{definition}\n\nIn the present situation it is straightforward to take $(S^!_{QM},CL_{QM})$ as the measure space.\nFor the set of conditions a modest choice is to take the set of propositions expressing the performance of a measurement: \n\\begin{equation}\n  \\mathcal{C}_{QM}:=\\left\\{F_{\\A}\\:;\\:\\A\\in\\mathfrak{A}\\right\\},~F_{\\A}:=(c\\circ i)(\\A,1).\n\\end{equation}\nFor this set of conditions the Born rule can easily be captured with the CPS $(S^!_{QM},CL_{QM},\\mathcal{C}_{QM},\\pee_{\\rho})$ with the probability function given by\n\\begin{equation}\\label{qmcpss}\n  \\pee_\\rho(F|F_{\\A})\n  :=\\sum_{\\left\\{\\substack{P\\in L(\\h)\\:;\\\\ (\\A^!,P)\\in F}\\right\\}}\\mathrm{Tr}(\\rho P),\\\\\n\\end{equation}\nfor $F\\in CL_{QM}$ and with $\\rho$ a density operator on $\\h$.\\footnote{Another possibility is to take for the conditions the set of propositions expressing the performance of a measurement excluding the possibility of a finer grained measurement: \n\\begin{equation*}\n\\mathcal{C}_{QM}^!:=\\left\\{F^!_{\\A}\\:;\\:\\A\\in\\mathfrak{A}\\right\\},~F^!_{\\A}:=\\{(\\A^!,P)\\in S^!_{QM}\\:;\\:P\\in L(\\A)\\}.\n\\end{equation*}\nAlso in this setting the Born rule can be recovered by taking the CPS $(S^!_{QM},CL_{QM},\\mathcal{C}^!_{QM},\\pee^!_{\\rho})$ with the probability function given by\n\\begin{equation*}\n  \\pee_\\rho^!(F|F^!_{\\A})\n  :=\\sum_{\\left\\{\\substack{P\\in L(\\h)\\:;\\\\ (\\A^!,P)\\in F}\\right\\}}\\mathrm{Tr}(\\rho P)=\\pee_\\rho(F|F_{\\A}).\n\\end{equation*}\nMuch of the remainder of this section can also be phrased in terms of this CPS.\nI will however restrict myself to \\eqref{qmcpss} for the sake of clarity.} \n\nIt would however be more interesting to take $\\mathcal{C}=CL_{QM}\\backslash\\{\\varnothing\\}$ for the set of conditions.\nAt the present it is not clear to me if this is possible, as extending the set of conditions seems a non-trivial matter.\nThere is a positive result on this matter however, namely, the CPS\n$(S^!_{QM},CL_{QM},\\mathcal{C}_{QM},\\pee_{\\rho})$ can be extended to\nan additive CPS.\nThe proof for this claim relies on the following definition and theorem.\n\n\\begin{definition}\nA family of measures $(\\mu_i)_{i\\in I}$ on $(\\Omega,\\mathcal{F})$ is called a \\emph{generating family of measures for the CPS}  $(\\Omega,\\mathcal{F},\\mathcal{C},\\pee)$ if and only if for all $C\\in\\mathcal{C}$ there exists $i\\in I$ such that $0<\\mu_i(C)<\\infty$ and for all $F\\in\\mathcal{F}$, for all $C\\in\\mathcal{C}$ and for all $i\\in I$ with $0<\\mu_i(C)<\\infty$\n\\begin{equation}\n  \\pee(F|C)=\\frac{\\mu_i(F\\cap C)}{\\mu_i(C)}.\n\\end{equation}\nThe family is called \\emph{dimensionally ordered} if and only if there is a total order $<$ on $I$ such that for all $F\\in\\mathcal{F}$ and $i,j\\in I$: $i<j$ $\\mu_j(F)=0$ whenever $\\mu_i(F)<\\infty$. \n\\end{definition}\n\n\\begin{theorem}[Cs\\'asz\\'ar 1955 \\cite{Csaszar55}]\nA CPS $(\\Omega,\\mathcal{F},\\mathcal{C},\\pee)$ can be extended to an additive CPS $(\\Omega,\\mathcal{F},\\mathcal{C}',\\pee')$ (i.e., $\\mathcal{C}\\subset\\mathcal{C}'$, and $\\pee'$ acts as $\\pee$ when restricted to $\\mathcal{F}\\times\\mathcal{C}$) if and only if it is generated by a dimensionally ordered family of measures on $(\\Omega,\\mathcal{F})$.\n\\end{theorem}\n\nThus to show that the CPS given by $\\pee_\\rho$ in \\eqref{qmcpss} can be extended to an additive CPS it suffices to give a dimensionally ordered family of measures  that generates it.\nThe construction of this family relies on the following dimensionality function\n\\begin{equation}\n\\begin{split}\n  d:\\mathfrak{A}\\to\\mathbb{N},&~d(\\A):=\\# L_0(\\A),\\\\\n  d:CL_{QM}\\to\\mathbb{N},&~d(F):=\\min\\{d(\\A)\\:;\\:(\\A^!,P)\\in F\\}.\n\\end{split}\n\\end{equation}\nThat is, $d$ picks out the coarsest measurements in $F$. \nIn particular, $d(F_{\\A})=d(\\A)$.\nThe family of measures $\\left(\\mu_i^\\rho\\right)_{i\\in\\mathbb{N}}$ is now defined as\n\\begin{equation}\n   \\mu_i^\\rho(F):=\\delta_{id(F)}\\sum_{\\left\\{\\substack{P\\in L(\\h)\\:;\\:\\exists\\A\\in\\mathfrak{A}\\text{ s.t.}\\\\(\\A^!,P)\\in F, d(\\A)=i}\\right\\}}\\mathrm{Tr}(\\rho P),\n\\end{equation}\nwhere $\\delta_{id(F)}$ denotes the Kronecker delta.\nThis Kronecker delta warrants that the family is indeed dimensionally ordered where the order is the usual one on $\\mathbb{N}$.\nUsing that\n\\begin{equation}\n   \\mu_i^\\rho(F_{\\A})=\\delta_{id(\\A)}\n\\end{equation}\none may easily check that this is indeed a generating family for the CPS $(S^!_{QM},CL_{QM},\\mathcal{C}_{QM},\\pee_\\rho)$.\n\nAnother open issue with the CPS approach to quantum probabilities is that of deriving the Born rule, i.e., to derive the conditions under which $\\pee$ is of the form $\\pee_{\\rho}$.\nA particular difficulty here is that quantum probabilities are non-contextual, a property that is not expected to hold in general for an admissible probability function on $(S^!_{QM},CL_{QM})$.\nThat is, one may have that for $\\A_1,\\A_2\\supset\\A$ the equality\n\\begin{equation}\\label{ineqB}\n  \\pee\\left(\\{(\\A^!,P)\\}\\middle|F_{\\A_1}\\right)=\\pee\\left(\\{(\\A^!,P)\\}\\middle|F_{\\A_2}\\right),\n\\end{equation}\ndoes not hold, although it is satisfied for the Born rule \\eqref{qmcpss}.\n\nConversely, if in addition to \\eqref{ineqB} the equality\n\\begin{equation}\n  \\pee\\left(\\{(\\A_1^!,P)\\}\\middle|F_{\\A}\\right)=\\pee\\left(\\{(\\A_2^!,P)\\}\\middle|F_{\\A}\\right)\n\\end{equation}\nwere also to hold whenever $\\A_1,\\A_2\\subset\\A$, the Born rule would follow.\\footnote{Provided that the dimension of $\\h$ is greater than $2$.}\nThis is because then the function\n\\begin{equation}\n  \\pee_{QM}:L(\\h)\\to[0,1],~\\pee_{QM}(P):=\\pee\\left(\\{(\\A^!,P)\\}\\middle|F_{\\A'}\\right)\n\\end{equation}\nwith $\\A'\\supset\\A$ is well defined (i.e., independent of the choice of $\\A$ and $\\A'$) and satisfies\n\\begin{itemize}\n\\item $\\pee_{QM}(1)=1$,\n\\item $\\pee_{QM}(P_1\\vee P_2\\vee\\ldots)=\\pee(P_1)+\\pee(P_2)+\\ldots$ \\quad whenever $P_iP_j=0$ for all $i\\neq j$.\n\\end{itemize} \nHence, Gleason's theorem applies.\n\nNow, before coming to an end it is interesting to take a quick look at the connection between possible extensions of the CPSs and Bell inequalities.\nConsider the standard EPR-Bohm situation where Alice can choose between two possible measurements $A^A_1$  and $A^A_2$, whilst Bob can choose between two possible measurements $A^B_1$ and $A^B_2$. \nLet $\\A_{ij}$ denote the algebra generated by $\\mathcal{A}^A_i$ and $\\mathcal{A}^B_j$. \nIt is further presupposed that each measurement has two possible outcomes associated with the projections $P^X_{i0}$ and $P^X_{i1}$ in $\\A^X_ i$.\n\nThe CPSs given by \\eqref{qmcpss} aren't rich enough to evaluate possible derivations for a Bell inequality.\nFor that, the set of conditions has to be extended to\n\\begin{equation}\n  \\mathcal{C}_{QM}':=\\left\\{F_{(\\A,P)}\\:;\\:(\\A,P)\\in S_{QM}\\right\\},~F_{(\\A,P)}:=(c\\circ i)(\\A,P).\\\\\n\\end{equation}\nWith these conditions the requirements of outcome independence (OI) and parameter independence (PI) from which the Bell inequality can be derived \\cite{Jarrett84,Shimony84} can be stated:\n\\begin{equation}\n\\begin{split}\n  \\pee\\left(\\{(\\A^{A!}_i,P^A_{i0})\\}\\middle|F_{(\\A_{i1},1)}\\right)\n  &=\n  \\pee\\left(\\{(\\A^{A!}_i,P^A_{i0})\\}\\middle|F_{(\\A_{i2},1)}\\right)\\\\\n  \\pee\\left(\\{(\\A^{B!}_i,P^B_{i0})\\}\\middle|F_{(\\A_{1i},1)}\\right)\n  &=\n  \\pee\\left(\\{(\\A^{B!}_i,P^B_{i0})\\}\\middle|F_{(\\A_{2i},1)}\\right) \n\\end{split} \\tag{PI}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n  \\pee\\left(\\{(\\A^{A!}_i,P^A_{i0})\\}\\middle|F_{(\\A_{ij},1)}\\right)\n  &=\n  \\pee\\left(\\{(\\A^{A!}_i,P^A_{i0})\\}\\middle|F_{(\\A_{ij},P^B_{jk})}\\right)\\\\\n  \\pee\\left(\\{(\\A^{B!}_i,P^B_{i0})\\}\\middle|F_{(\\A_{ji},1)}\\right)\n  &=\n  \\pee\\left(\\{(\\A^{B!}_i,P^B_{i0})\\}\\middle|F_{(\\A_{ji},P^A_{jk})}\\right).\n\\end{split} \\tag{OI}\n\\end{equation}\nThe violation of Bell inequalities in QM shows that OI must fail for any extension of $\\pee_{\\rho}$ to $\\mathcal{C}_{QM}'$.\nThis may be indicative of the idea that the development of \\emph{any} full CPS $(S^!_{QM},CL_{QM},CL_{QM}\\backslash\\{\\varnothing\\},\\pee)$ is a non-trivial matter.\\footnote{Note that I'm thinking here of interesting cases. A trivial CPS is easily constructed. Just fix one $(\\A !,P)\\in S^!_{QM}$ and take $\\pee(F|C)=\\begin{cases}1&(\\A^!,P)\\in F,\\\\0& \\text{otherwise},\\end{cases}$ \\quad for all $C\\in CL_{QM}\\backslash\\{\\varnothing\\}$.}\n\n \n \n \n\n\\section{Conclusion and outlook}\\label{conclusionsection}\nWith the construction of $CL_{QM}$ a concrete formalism is provided that adheres to the Bohrian view that also for quantum mechanics experimental reports must be communicable in unambiguous common language.\nHowever, the association with the Copenhagen doctrine ends there.\nAlthough the notion of measurement is taken as primitive here, no demand is made for the necessity of this.\nThat is, the current program should not be seen as a method to dismiss the measurement problem, but merely as a way of making sense of the formalism without endorsing a particular solution to the problem.\nIn this paper one obstacle in such investigations has been overcome, namely, the non-distributivity of quantum logic.\nIn return a less abstract obstacle has emerged, namely, the justification of the Born rule in quantum mechanics.\nTo get a further grip on this issue it may be fruitful future work to seek connections with other programs in quantum foundations such as generalized probability theory \\cite{Barrett07} or the Bayesian approach to quantum probabilities \\cite{Fuchs10}.\n\n \n \n \n\n\\section*{Acknowledgments}\nI would like to thank the people at the QPL 2012 conference in Brussels and at the first Quantum Toposophy workshop in Nijmegen for useful questions and discussions. \nAnd further J. W. Romeijn for insisting on the use of classical logic.\nThis work was supported by the NWO (Vidi project nr. 016.114.354).\n \n\n\\bibliographystyle{eptcs}    \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Hochschild homology and cyclic homology}\\label{1}\n\\markboth{\\hfil{\\sc When the theories meet }\\hfil}\n{\\hfil{\\sc Khovanov homology as Hochschild homology of links}\\hfil}\n\nWe recall in this section definition of Hochschild homology and cyclic \nhomology and we sketch two classical calculations for tensor algebras \nand symmetric tensor algebras.\nMore calculations are  reviewed in Section 4 in which we use our main \nresult, Theorem 3.1, to obtain new results in Khovanov homology, \nin particular solving some conjectures from \\cite{H-P-R}.\\ \n We follow \\cite{Lo} in our exposition of Hochschild homology.\n\nLet $k$ be a commutative ring and $\\A$ a $k$-algebra (not necessarily \ncommutative). \nLet $\\M$ be a bimodule over $\\A$ that is a $k$-module on \nwhich $\\A$ operates linearly on the left and on the right in such a way \nthat $(am)a'=a(ma')$ for $a,a'\\in \\A$ and $m \\in \\M$. The actions of $\\A$ \nand $k$ are always compatible (e.g. \n$m(\\lambda a)= (m\\lambda)a=\\lambda (ma)$). When $\\A$ has a unit element \n$1$ we always assume that $1m=m1=m$ for all $m \\in \\M$. Under this \nunital hypothesis, the bimodule $\\M$ is equivalent to a right \n$\\A\\otimes \\A^{op}$-module via $m(a'\\otimes a)=ama'$. Here $\\A^{op}$ \ndenotes the opposite algebra of $\\A$ that is $\\A$ and $\\A^{op}$ are \nthe same as sets but the product $a\\cdot b$ in $\\A^{op}$ is the \nproduct $ba$ in $\\A$. The product map of $\\A$ is usually denoted \n$\\mu: \\A \\otimes \\A \\to A$,\\ $\\mu (a,b)=ab$.\\\\  \n\nIn this paper we work only with unital algebras. We also assume, \nunless otherwise stated, that $\\A$ is a free $k$-module, however \nin most cases, it suffices to assume that $\\A$ is $k$-projective, or less \nrestrictively, that $\\A$ is flat over $k$. \\ Throughout the paper the \ntensor product $\\A \\otimes \\B$ denotes the tensor product over $k$, \nthat is $\\A \\otimes_k \\B$. \n\n\\begin{definition}[\\cite{Hoch,Lo}]\\label{1.1}\\ \nThe Hochschild chain complex $C_*(\\A,\\M)$ is defined as:\n$$ \\ldots\n\\stackrel{b}{\\to} \\M\\otimes \\A^{\\otimes n} \\stackrel{b}{\\to} \n\\M\\otimes \\A^{\\otimes n-1} \\stackrel{b}{\\to} \\ldots \n\\stackrel{b}{\\to} \\M\\otimes \\A \\stackrel{b}{\\to} \\M$$\nwhere $C_n(\\A,\\M)= \\M\\otimes \\A^{\\otimes n}$ and the Hochschild \nboundary is the $k$-linear map $b: \\M\\otimes \\A^{\\otimes n} \\to \n\\M\\otimes \\A^{\\otimes n-1}$ given by the formula \n$b= \\sum_{i=0}^n (-1)^id_i$, where the face maps $d_i$ are given by\\\\\n$d_0(m,a_1,\\ldots, a_n)= (ma_1,a_2,\\ldots, a_n),$\\\\\n$d_i(m,a_1,\\ldots, a_n)= (m,a_1,\\ldots , a_ia_{i+1},\\ldots , a_n)$ for \n$1\\leq i \\leq n-1$, \\\\\n$d_n(m,a_1,\\ldots, a_n)= (a_nm,a_1,\\ldots ,a_{n-1})$.\\\\\nIn the case when $\\M=\\A$ the Hochschild complex is called the \n{\\it cyclic bar complex}.\\\\\nBy definition, the $n$th Hochschild homology group of the unital \n$k$-algebra $\\A$ with coefficients in the $\\A$-bimodule $\\M$ is \n the $n$th homology group of the Hochschild chain complex \ndenoted by $H_n(\\A,\\M)$. In the particular case $\\M=\\A$ we write \n$C_*(\\A)$ instead of $C_*(\\A,\\A)$ and $HH_*(\\A)$ instead of $H_*(\\A,\\A)$.\n\\end{definition}\n\nThe algebra $\\A$ acts on $C_n(\\A,\\M)$ by \n$a\\cdot (m,a_1,...,a_n)=(am,a_1,...,a_n)$. \nIf $\\A$ is a commutative algebra then the action commutes with \nboundary map $b$, therefore $H_n(\\A,\\M)$ (in particular, $HH_*(\\A)$) is \nan $\\A$-module. \n\nIf $\\A$ is a graded algebra and $\\M$ a coherently graded \n$\\A$-bimodule, and the boundary maps are grading preserving, then \nthe Hochschild chain complex is a bigraded chain complex with \n($b: C_{i,j}(\\A,\\mathbb M) \\to C_{i-1,j}(\\A,\\mathbb M))$, and \n$H_{**}(\\A,\\mathbb M)$ is a bigraded $k$-module. In the case of \nabelian $\\A$ and $\\A$-symmetric $\\mathbb M$ (i.e. $am=ma$), \n$H_{**}(\\A,\\mathbb M)$ is bigraded $\\A$-module. The main examples \ncoming from the knot theory are $\\A_m=Z[x]\/(x^m)$ and $\\mathbb M$ the \nideal in $\\A_m$ generated by $x^{m-1}$.\n\nWe complete this survey by describing, after \\cite{Lo},\n two classical results in \nHochschild homology -- the computation of Hochschild homology for \na tensor algebra and for a symmetric tensor algebra.\n\\begin{theorem}\\label{1.2}\\ \\\\\nLet $V$ be any $k$-module and let $\\A =T(V)= k \\oplus V \n\\oplus V^{\\otimes 2} \\oplus ...$ be its tensor algebra. \nWe denote by $\\tau_n : V^{\\otimes n} \\to V^{\\otimes n}$ the cyclic \npermutation, $\\tau_n(v_1,...,v_{n-1},v_n) = (v_n,v_1,...,v_{n-1})$.\nThen the Hochschild homology of $\\A=T(V)$ is:\\\\ \n$HH_0(\\A)= \\oplus_{i\\geq 0} V^{\\otimes i}\/(1-\\tau_i)  $,\\\\\n$HH_1(\\A)= \\oplus_{i\\geq 1} (V^{\\otimes i})^{\\tau_i}$, where \n$(V^{\\otimes i})^{\\tau_i}$ is the space of invariants, that is \nthe kernel of $1-\\tau_i$.\\\\\n$HH_n(\\A)= 0$ for $n \\geq 2$.\n\\end{theorem}\nThe main idea of the proof is to show that the Hochschild chain complex \nof $T(V)$ is quasi-isomorphic\\footnote{Two chain complexes $C$ and $C'$ are \ncalled quasi-isomorphic if there is a chain map $f: C \\to C'$ which \ninduces an isomorphism on homology, $f_*: H_n(C) \\to H_n(C')$, for all $n$.\nThe map $f$ is called a quasi-isomorphism, chain equivalence, or homologism.}\n to the ``small\" chain complex:\n$$ C^{small}(T(V)):\\ \\ \\ ... \\to 0 \\to \\A\\otimes V \\stackrel{\\hat b}{\\to} \\A $$ \nwhere the module $\\A$ is in degree $0$ and where the map $\\hat b$ is \ngiven by $\\hat b(a\\otimes v)= av - va$. Therefore $\\hat b$ restricted \nto $V^{\\otimes n-1}\\otimes V$ is precisely \n$(1-\\tau_n): V^{\\otimes n} \\to V^{\\otimes n}$ and Theorem 1.2 follows.\n See Proposition 3.1.2 and Theorem 3.1.4 of \\cite{Lo}.\n\n\\begin{theorem}\\label{1.3} Symmetric and polynomial algebras. \\\\\n Let $V$ be a module\nover $k$ and let $S(V)$ be the symmetric tensor algebra over $V$;\n$S(V)= k \\oplus V\n\\oplus S^2(V) \\oplus ...$. If $V$ is free of dimension $n$ generated by\n$x_1,...,x_n$ then $S(V)$ is the polynomial algebra $k[x_1,...,x_n]$.\nAssume that $V$ is a flat $k$-module (e.g. a free module). Then there\nis an isomorphism\n$$S(V)\\otimes \\Lambda^nV \\cong  HH_n(S(V))$$\nwhere $\\Lambda^*V$ is the exterior algebra of $V$.\nIf $V$ is free of dimension $n$ generated by\n$x_1,...,x_n$ then $\\Lambda^mV$ is a free $n\\choose m$ $k$-module with\na basis $v_{i_1} \\wedge v_{i_2} \\wedge ...\\wedge v_{i_m}$, $i_1 < i_2 <...\n<i_m$.\n\\end{theorem}\nThe above theorem is a special case of Hochschild-Konstant-Rosenberg \ntheorem about Hochschild homology of smooth algebras \\cite{HKR}, \nwhich we discuss in section 4. Here we stress, after Loday, that \nthe isomorphism $\\varepsilon_*: S(V)\\otimes \\Lambda^*V \\to  HH_*(S(V))$\nis induced by a chain map, that is not true in general for smooth algebras.\n$S(V)\\otimes \\Lambda^*V$ is a chain complex with the zero boundary maps.\nThe chain map $\\varepsilon: S(V)\\otimes \\Lambda^*V \\to C_n(S(V),S(V))$ \nis given by $\\varepsilon (a_0\\otimes a_1 \\wedge ...\\wedge a_n) = \n\\varepsilon_n (a_0,a_1,...,a_n)$ where $\\varepsilon_n$ is the \nantisymmetrization map given as the sum \n$\\sum_{\\sigma \\in S_n}sgn(\\sigma)\\sigma((a_0,a_1,...,a_n))$ and \nthe permutation $\\sigma \\in S_n$ acts by $\\sigma((a_0,a_1,...,a_n)) = \n(a_0,a_{\\sigma^{-1}(1)},..., a_{\\sigma^{-1}(n)})$. \n\n\n\nIn the second section we describe Khovanov homology of links and its \n(``comultiplication-free\") version for graphs \nintroduced in \\cite{H-R-2}. In order \nto compare them with Hochschild homology we offer various generalizations,\nrelaxing a condition that underlying algebra needs to be commutative \n(in the case of a polygon or a line graph), and \nallowing an $\\M$-reduced case. \nAnother innovation in our presentation is that we start with \ncohomology of a functor from the category \nof subsets of a fixed set to the category of modules. In this setting we \ndescribe graph homology introduced in \\cite{H-R-2} and its generalizations \nto $\\mathbb M$-reduced cohomology and to cohomology of a polygon or \na line graphs with a noncommutative underlying algebra. \nA set in this case is the set of edges of a graph. \nFinally, we describe a generalization \nto ``supersets\" which allows homology for signed graphs, link diagrams \nand, in some cases, links (with the classical Khovanov \nhomology as the main example). In examples of functors from ``supersets\" \nwe utilize comultiplication in the underlying algebra $\\A$, after \nKhovanov \\cite{Kh-1}, we assume that $\\A$ is a Frobenius algebra \n\\cite{Abr,Kock}.\n\nIn the third section we prove our main result relating \n Hochschild homology $HH_*(\\A)$ to graph cohomology and \nKhovanov homology of links, and the homology $H_n(\\A,\\M)$ \nto reduced ($\\M$-reduced, \nmore precisely)  cohomology of graphs and links.\n\nIn the fourth section we use our main result to describe graph cohomology \nof polygons for various algebras solving, in particular, several \nproblems from \\cite{H-P-R}.\n\nWe envision the connection between Connes cyclic homology and Khovanov \ntype homology, possibly when analyzing symmetry of graphs and links.\nFor the idea of cyclic homology it is the best to quote from J-L. Loday\n\\cite{Lo}:\\\\ \n``...in his search for a non-commutative analogue of de Rham homology\ntheory, A. Connes discovered in 1981 the following striking phenomenon: \\\\\n- the Hochschild boundary map $b$ is still well defined when one factors\nout the module $A\\otimes A^{\\otimes n} = A^{\\otimes n+1}$ by the action\nof the (signed) cyclic permutation of order $n+1$.\\\\\nHence a new complex was born whose homology is now called (at least in\ncharacteristic zero) {\\it cyclic homology}.\"\n\n\\section{Khovanov homology}\n\nWe start this section with a very abstract definition based \non Khovanov construction but, initially, devoid of topological \nor geometric context.\nIn this setting \nwe recall and generalize the concept of graph cohomology \\cite{H-R-2}\nand of classical Khovanov \nhomology for unoriented framed links. We follow, in part, the exposition in \n\\cite{H-R-1,H-R-2} and \\cite{Vi}.\nWe review, after \\cite{H-P-R}, the connection between Khovanov homology \nof links and graph cohomology of associated Tait graphs.\nWe define homology of link diagrams \nrelated to graph cohomology for any commutative algebra\\footnote{We can extend \nthe definition to noncommutative algebras in the case of $(2,n)$-torus \nlink diagrams.}.\n\n\\subsection{Cohomology of a functor on sets}\\label{sub2.1}\\ \\\\ \nLet $k$ be a commutative ring and $E$ a finite (or countable) set.\n\\begin{definition}\nLet $\\Phi$ be a functor from the category of subsets of $E$\n(i.e. subsets of $E$ are\nobjects and inclusions are morphisms) to the category of $k$-modules.\nWe define the ``Khovanov cohomology\" of the functor, $H^i(\\Phi)$,\n as follows. \nWe start from the graded $k$-module $\\{C^i(\\Phi)\\}$, where $C^i(\\Phi)$ \nis the direct sum of $\\Phi(s)$ over all\n$s\\in E$ of $i$ elements ($|s|=i$). \nTo define $d: C^i(\\Phi) \\to C^{i+1}(\\Phi)$ we first define \nface maps $d_e(s)= \\Phi(s\\subset s\\cup e)(s)$ for $e \\in E-s$ (notice that \n$s\\subset s\\cup e$ is the unique morphism in $Mor(s,s\\cup e)$). \nNow, as usually, $d(s)= \\sum_{e \\notin s} (-1)^{t(s,e)}d_e(s)$,\nwhere $t(s,e)$ requires ordering of elements of $E$ and is equal the\nnumber of elements of $s$ smaller then $e$. Because $\\Phi$ is a functor, \ntherefore we have $d_{e_2}d_{e_1}(s) = d_{e_1}d_{e_2}(s)$ for any \n$e_1,e_2 \\notin s$. The sign convention guarantees now that $d^2=0$ and \n$(\\{C^i(\\Phi)\\},d)$ is a cochain complex.\nNow we define, in a standard way, cohomology as\n $H^i(\\Phi)= ker (d(C^i(\\Phi)\\to C^{i+1}(\\Phi))\/d(C^{i-1}(\\Phi))$.\nThe standard argument shows that $H^i(\\Phi)$ is independent on \nordering of $E$.\n\\end{definition}\nIn the case when $E$ are edges of a graph $G$ we can define\nspecific functors in various ways taking into account a structure of $G$. \nWe construct below our main example: a generalization \nof a graph cohomology, defined in \\cite{H-R-2}, to $\\M$-reduced case \nand its translation to \nhomology of alternating diagrams. In the case of the \nalgebra $\\A_2=Z[x]\/(x^2)$,\nthis homology agrees partially with the classical Khovanov homology (see \nTheorem 2.7). To deal with all link diagrams we will later  \nexpand Definition 2.1 to ``supersets\" (Definition 2.4) in a construction \nwhich can incorporate multiplication and comultiplication \nin $\\A$ (Example 2.5).\n\n\\begin{definition}[Cohomology introduced in \\cite{H-R-2} and extended to \n$\\M$-reduced case]\\ \\\n\\begin{enumerate}\n\\item[(1)] We define here $\\M$-reduced cohomology denoted by by \n$H^*_{\\A,\\mathbb M}(G,v_1)$.\nIf we assume $\\mathbb M = \\A$ we obtain\n(comultiplication-free) cohomology of graphs, $H^*_{\\A}(G)$,\n  defined in \\cite{H-R-2}.\\\\\nLet $G$ be a graph with an edge set $E=E(G)$, and a chosen base vertex $v_1$.\nFix a commutative $k$-algebra $\\A$ and an $\\A$-module $\\mathbb M$. \nWe define a functor $\\Phi$ on a category of subsets of $E$ as follows:\n\\begin{enumerate}\n\\item[(Objects)] To define the functor $\\Phi$ on objects $s\\subset E$ we \ndefine it more generally on any subgraph $H \\subset G$, starting \nfrom a connected $H$, to be\n$\\Phi(H)=\\mathbb M$ if $v_1 \\in H$, and $\\Phi(H)=\\A$ if $v_1 \\notin H$.\nIf $H$ has connected components $H_1,...,H_k$ then we define \n$\\Phi(H)= \\Phi(H_1)\\otimes ... \\otimes \\Phi(H_k)$. Finally, \n$\\Phi(s)= \\Phi([G:s])$ where $[G:s]$ is a subgraph of $G$ containing \nall vertices of $G$ and edges $s$. \\\\\nIn what follows $k(s)$ is the number of components of $[G:s]$.\n\\item[(Morphisms)] It suffices to define $\\Phi(s \\subset s\\cup e)$ \nwhere $e\\notin s$. The definition depends now on the role of $e$ in \n$[G:s]$ as follows:\n\\begin{enumerate}\n\\item[(i)] Assume that $e$ connects different components of $[G:s]$:\\\\\n(i') If $e$ connects components $u_i$ and $u_{i+1}$ not containing $v_1$ \nthen we define \n$\\Phi(s \\subset s\\cup e)(m,a_1,...,a_i,a_{i+1},...,a_{k(s)-1})=\n(m,a_1,...,a_ia_{i+1},...,a_{k(s)-1})$.\\\\\n(i'') If $e$ connects a component of $[G:s]$ containing $v_1$ with another \ncomponent of $[G:s]$, say $u_1$, then we put \n$\\Phi(s \\subset s\\cup e)(m,a_1,...,a_{k(s)-1})= (ma_1,...,a_{k(s)-1})$.\n\\item[(ii)] Assume that $e$ connects vertices of the same component \nof $[G:s]$, then $\\Phi(s \\subset s\\cup e)$ is the identity map \non $\\Phi(s)=\\Phi(s\\cup e)= \\mathbb M \\otimes \\A^{\\otimes k(s)-1}$ \n\\end{enumerate}\nIn the proof that $\\Phi$ is a functor commutativity of $\\A$ is important \n(compare (3)). \n\\end{enumerate}\n\\item[(2)] If we modify the functor $\\Phi$ from (1) to a new functor,\n $\\hat\\Phi$ which differs from $\\Phi$ only in the rule $(1)(ii)$ \nthat $\\hat\\Phi(s \\subset s\\cup e)$ is a zero map if $e$ \nconnects vertices of the same component of $[G:s]$. \nWe denote the cohomology yielded by the functor $\\hat\\Phi$ by \n$\\hat H^*_{\\A,\\mathbb M}(G,v_1)$.  \nFor $\\mathbb M = \\A$ this cohomology, $\\hat H^*_{\\A}(G)$,\n is introduced in \\cite{H-R-2}.  \n\\item[(3)]  For a polygon or a line graph the cohomology \n$\\hat H^*_{\\A,\\mathbb M}(G,v_1)$ is defined also for \na noncommutative algebra $\\A$ and any $\\A$-bimodule $\\M$. \nWe consider a polygon or a line graph as a directed graph: from left \nto right (in the case of a line graph, Fig. 3.1) and in the anti-clockwise \norientation (in the case of a polygon).\nIn the formula for the morphism, $\\hat\\Phi(s\\subset s \\cup e)$, of \nDefinition 2.2(2)\nwe use the product $xy$ if $x$ is the weight of the initial point of the\ndirected edge $e$ connecting different components of $[G:s]$.\\\n We use this graph cohomology of the directed polygon when\ncomparing graph cohomology with Hochschild homology.\n\\end{enumerate} \n\\end{definition} \nNotice that cohomology described in (1) and (2) coincide to certain degree.\nNamely $H^i_{\\A,\\mathbb M}(G,v_1) = \\hat H^i_{\\A,\\mathbb M}(G,v_1)$ \nfor all $i < \\ell -1$ where $\\ell$ is \nthe length of the shortest cycle in $G$.\\\\\nFurthermore, if $k$ is a principal ideal domain (e.g. $k=Z$)\n and $\\A$ and $\\M$ are free \n$k$-modules then $Tor(H^i_{\\A,\\mathbb M}(G,v_1)) = \nTor(\\hat H^i_{\\A,\\mathbb M}(G,v_1))$ for $i=\\ell-1$ (compare Theorem 2.7).\n\n\\begin{remark} \n\\begin{enumerate}\n\\item[(i)] One can generalize\\footnote{We are motivated here by \n \\cite{Sto}.} construction in Definition 2.2(1) and (2)\n by choosing the sequence of elements $f_1,f_2,f_3,...,f_{|E|}$ \nin $\\A$ and modifying functors $\\Phi$ and $\\hat \\Phi$ on morphisms \nto get the functors $\\Phi'$ and $\\hat \\Phi'$. \nWe put $\\Phi'(s \\subset s\\cup e)=f_{|s|+1}\\Phi(s \\subset s\\cup e)$ \nand, similarly,  \n$\\hat\\Phi'(s \\subset s\\cup e)=f_{|s|+1}\\hat\\Phi(s \\subset s\\cup e)$.  \n\\item[(ii)] If $\\A$ is not commutative and we work with cohomology \nof a line graph or a polygon (as in Definition 2.2(3))\nwe have to assume, in order to have $d^2=0$,\n that $f_i$'s are in the center of $\\A$. \nWe define \n$f\\cdot(m,a_1,...,a_{k(s)-1})(s)=(fm,a_1,...,a_{k(s)-1})(s)$.\n\\end{enumerate}\n\\end{remark}\n\nWe can define Khovanov cohomology on an alternating link diagram, $D$ \nby considering associated plane graph, G(D) (Tait graph; compare Fig. 2.1) \nand its cohomology described in Definition 2.2 and Remark 2.3. \n\nIn order to define Khovanov cohomology on any link diagram (and \ntake both multiplication and comultiplication into account) we \nhave to define cohomology on any signed planar graph. \n We can start, as in Definition  2.1, from the \nvery general setting (again cohomology of a functor) and to produce \n specific examples of a cohomology of  signed planar graphs using \n a coherent algebra and coalgebra structures (Frobenius algebra).\n\\begin{definition}\\label{2.4}\nLet $k$ be a commutative ring and $E=E_+ \\cup E_-$ a finite set divided \ninto two disjoint subsets (positive and negative sets). We consider the \ncategory of subsets of $E$ \n($E\\supset s=s_+\\cup s_-$ where $s_{\\pm}= s\\cap E_{\\pm}$). the set $Mor(s,s')$ \nis either empty or has one element if $s_- \\subset s'_-$ and \n$s_+ \\supset s'_-$. Objects are graded by $\\sigma(s)=|s_-|- |s_+|$. \nLet us call this category the {\\it superset category}\n (as the set $E$\n is initially $Z_2$-graded). We define ``Khovanov cohomology\" for every functor,\n$\\Phi$, from the superset category to the category of $k$-modules.\nWe define cohomology of $\\Phi$ in the similar way as for a functor \nfrom the category of sets (which corresponds to the case $E=E_-$).\nThe cochain complex corresponding to $\\Phi$ is defined to be\n$\\{C^i(\\Phi)\\}$ where $C^i(\\Phi)$ is the direct sum of $\\Phi(s)$ over all\n$s\\in E$ with $\\sigma(s)=i$. To define $d: C^i(\\Phi) \\to C^{i+1}(\\Phi)$ \nwe first define face maps $d_e(s)$ where $e=e_- \\notin s_-$ ($e_- \\in E_-$) \nor $e=e_+ \\in s_+$. In such a case \n$d_{e_-}(s)= \\Phi(s\\subset s\\cup e_-)(s)$  and \n$d_{e_+}(s)= \\Phi(s\\supset s-e_+)$. We define  \n$d(s)= \\sum_{e \\notin s} (-1)^{t(s,e)}d_e(s)$,\nwhere $t(s,e)$ requires ordering of elements of $E$ and is equal the\nnumber of elements of $s_-$ smaller then $e$ plus the number of elements \nof $s_+$ bigger than $e$.\nWe obtain the cochain complex whose cohomology does not \ndepend on ordering of $E$.\n\\end{definition}\n\\begin{example}\\label{2.5}\nLet $G$ be a signed plane graph with an edge set \n$E=E_+ \\cup E_-$ were $E_+$ is the \nset of positive edges and $E_-$ is the set of negative edges. We  \ndefine the functor from the superset category $\\mathbb E$ using the \nfact that $G$ is the (signed) Tait graph of a link diagram $D(G)$ \n(with white infinite region). See Figure 2.1 for conventions).\n\\\\ \\ \\\\\n\\centerline{\\psfig{figure=TaitGr,height=1.2cm}}\n\\centerline{ Figure 2.1} \\ \\\\\n\nTo define the functor $\\Phi$ we fix \na Frobenius algebra $\\A$ with multiplication $\\mu$ and comultiplication \n$\\Delta$ (the main example used by Khovanov\n is the algebra of truncated polynomials $\\A_m=Z[x]\/(x^m)$ with a \ncoproduct $\\Delta (x^k)=\\sum_{i+j=m-1+k}x^i \\otimes x^j$).\nTo get  our functor on objects, $\\Phi(s)$,\n we consider the Kauffman state defined by $s$ (so also \ndenoted by $s$) and we associate $\\A$ to every circle of $D_s$ obtained \nfrom $D(G)$ by smoothing every crossing according to $s$ and \nthen taking tensor product of these copies of $\\A$ (compare \\cite{Vi}).\nTo get $\\Phi$ on morphisms, \nwithout going into details, we say succinctly that \nif Mor$(s,s')\\neq \\emptyset$ \n and $\\sigma(s')=\\sigma(s)+1$ then $\\Phi(Mor(s,s'))$ is defined \nusing product or coproduct depending on whether $D_s$ has more or less \ncircles than $D_{s'}$. For $A_2$ we obtain the classical Khovanov homology.  \n\\end{example}\n\\begin{remark} \\label{2.6} One can also extend Example 2.5 to \ninclude the concept of $M$-reduced cohomology. We can consider, for example \n$M$ to be an ideal in $\\A$ with $\\Delta(\\M) \\subset M\\otimes M$. An example, \nconsidered by Khovanov, is $A_m$ with $M$ generated by $x^{m-1}$ \n(in which case $\\Delta(x^{m-1})= x^{m-1}\\otimes x^{m-1}$). \n\\end{remark}\nOne can build more delicate (co)homology theory for ribbon graphs \n(flat vertex graph) using the fact that they embed uniquely \ninto the closed surface.\nFor $A=A_2$ it can be achieved using the approach presented \nin \\cite{APS}, while for more general (Frobenius) algebras\nit is not yet done (most likely one should not use Frobenius algebra \nalone but its proper enhancement like in $A_2$ case).\n\nIn \\cite{H-P-R} we proved the following relation between \ngraph cohomology and classical Khovanov homology \nof alternating links.\n\\begin{theorem}\\label{Let $D$ be the diagram}\nLet $D$ be the diagram of an unoriented framed alternating link\nand let $G$ be its Tait graph. Let $\\ell$ be the length of the\nshortest cycle in $G$. For all $i<\\ell-1$, we have\n\n\\[\nH^{i,j}_{\\A_2}(G) \\cong H_{a,b}(D)\n\\]\nwith $\\left \\{\\begin{array}{l}\na=E(G)-2i,\\\\\nb=E(G)-2V(G)+4j.\n\\end{array}\n\\right.$\n\nwhere $H_{a,b}(D)$ are the Khovanov homology groups \nof the unoriented framed link defined by $D$, as explained in\n\\cite{Vi}.\n\nFurthermore, $Tor(H^{i,j}_{\\A_2}(G)) = Tor(H_{a,b}(D))$ for $i=\\ell-1$.\n\\end{theorem}\n\nWe also speculate that for other $sl(m)$ Khovanov-Rozansky \nhomology \\cite{Kh-R-1,Kh-R-1} the graph cohomology of \na polygon with $\\A=\\A_m=Z[x]\/(x^m)$ keeps \nessential information on $sl(m)$-homology of a torus link.\n\n\\section{Relation between Hochschild homology and Khovanov homology}\n\nThe main goal of our paper is to demonstrate relation between \nKhovanov homology and Hochschild homology. Initially I observed \nthis connection \nfor a commutative algebra $\\A$ by showing that for every commutative\nunital algebra $\\A$ the graph cohomology of \n$(n+1)$-gon, $H^{i}_{\\A}(P_{n+1})$, is isomorphic to Hochschild homology \nof $\\A$, $H_{n-i}(\\A)$; $0<i<n$. From this, via Theorem 2.7, relation \nbetween classical Khovanov homology of $(2,n+1)$ torus link and \nHochschild homology of $\\A_2=Z[x]\/(x^2)$, follows. This relation was \nalso observed independently by Magnus Jacobson \\cite{Jac}.\n  \nIn this paper  we prove more general result. In order to \nformulate it we use an extended version of (Khovanov type) \ngraph cohomology (working with noncommutative algebras and \n${\\mathbb M}$-reduced cohomology):  \n\\begin{enumerate}\n\\item[(i)] We can work with a noncommutative algebra, because, as mentioned \nin Section 2 (Definition 2.2(3)), for a polygon the graph cohomology \nis defined also for noncommutative algebras.\n\n\\item[(ii)] We can fix an $\\A$-bialgebra ${\\mathbb M}$ \nand compare ${\\mathbb M}$-reduced Khovanov type graph cohomology with \nthe Hochschild homology of ${\\A}$ with coefficients in \n${\\A}$-bimodule ${\\mathbb M}$.\n\\end{enumerate}\nOne can observe that we generalize notion of graph (co)homology while \nwe keep the original definition of Hochschild homology. Our point is \nthat the graph (co)homology is the proper generalization of \nHochschild homology: from a polygon to any graph. We have this \ninterpretation only for a commutative $\\A$. It seems to be, that if \none work with general graphs and not necessary commutative algebras then \nthese algebras should satisfy some ''multiface\" properties. \nVery likely that \nplanar algebras or operads provide the proper \nframework\\footnote{The author's idea of working with directed graphs (quivers)\nseems to work, as observed by Y.Rong, only for line graphs and polygons.}.\n\nThe interpretation of Hochschild homology as\na homology of $\\A$ treated as an algebra over \n$\\A\\otimes \\A^{op}$ allows us to use the standard tool of homological algebra,\nthat is  we \nfind appropriate (partial) free resolution of $\\A\\otimes \\A^{op}$ \nmodule $\\A$ using graph cochain complex of a line graph (Figure 3.1). \nThe graph cohomology of \nthe polygon is the cohomology obtained from this resolution. In Theorem 3.1 \nwe use cohomology $ \\hat H^i_{\\A,\\mathbb M}(P_{n+1})$ because we \ndo not assume that $\\A$ is commutative. \n\n\\begin{theorem}\\label{3.1} Let $\\A$ be a unital \nalgebra\\footnote{We assume in this paper that $\\A$ is a \nfree $k$-module,\nbut we could relax the condition to have $\\A$ to be projective or, \nmore generally, flat over a commutative ring with identity $k$; \ncompare \\cite{Lo}. We require $\\A$ to be a unital algebra in order to \nhave an isomorphism $\\M \\otimes_{\\A^{\\epsilon}} \\A^{\\otimes n+2} = \n\\M \\otimes \\A^{\\otimes n}$; the isomorphism is given by \n$\\M \\otimes_{\\A^{\\epsilon}} \\A^{\\otimes n+2} \\ni(m,a_0,a_1,...,a_n,a_{n+1}) \n\\to (a_{n+1}ma_0,1,a_1,...,a_n,1)$ which we can write succinctly as \n$(a_{n+1}ma_0,a_1,...,a_n) \\in \\M \\otimes \\A^{\\otimes n}$. We should stress \nthat in $\\M \\otimes_{\\A^{\\epsilon}} \\A^{\\otimes n+2}$ the tensor product \nis taken over $\\A^{\\epsilon}=\\A\\otimes \\A^{op}$ while in \n$\\M \\otimes \\A^{\\otimes n}$ the tensor product is taken over $k$.}, \n$\\mathbb M$ an $\\A$-bimodule and \n$P_{n+1}$ -- the $(n+1)-gon$. Then for $0< i \\leq n$ we have:\n$$ \\hat H^i_{\\A,\\mathbb M}(P_{n+1}) = H_{n-i}(\\A,\\mathbb M).$$\nFurthermore, if ${\\A}$ is a graded algebra and ${\\mathbb M}$ a \ncoherently graded module then $ \\hat H^{i,j}_{\\A,\\mathbb M}(P_{n+1}) = \nH_{n-i,j}(\\A,\\mathbb M)$, for $0< i \\leq n$ and every $j$.\n\\end{theorem}\n\\begin{corollary}\n$ \\hat H^{i,j}_{\\A}(P_{n+1}) =\nHH_{n-i,j}(\\A)$, for $0< i \\leq n$ and every $j$.\nFurthermore, for a commutative $\\A$, \n$H^{i,j}_{\\A}(P_{n+1}) = \\hat H^{i,j}_{\\A}(P_{n+1})$ , for $0< i <n$ and \n$H^{n,*}_{\\A}(P_{n+1}) = 0$, $\\hat H^{n,j}_{\\A}(P_{n+1})=  \nHH_{0,*}(\\A)=\\A$.\\ \nFor a general $\\A$,  $\\hat H^{n,*}_{\\A}(P_{n+1}) =HH_{0,*}(\\A)=\\A\/(ab-ba)$).\n\\end{corollary}\n{\\bf Poof of Theorem 3.1}\\\\\nWe consider graph cohomology for a unital, possibly noncommutative\nalgebra $\\A$ and any $\\A$-bimodule $\\mathbb M$. \nThere is no difference in the proof \nbetween commutative and noncommutative case except that we have to \nprove some property of cohomology given in \\cite{H-R-2} for \na commutative $\\A$ (Lemma 3.3).\\\\\nThe main idea of our proof is to interpret the graph cochain complex \nof a line graph as a (partial) resolution of $\\A$. It was proved \nin \\cite{H-R-2} that $H^i_{\\A}$(line graph)$=0$, $i>0$  \nfor a commutative algebra $\\A$. We give here the proof \nfor any unital algebra $\\A$.\nLet $L_n$ be the (directed) line graph of $n+1$ vertices ($v_0,...,v_n$)\n and $n$ edges ($e_1,...,e_n$); Fig. 3.1. \n\\\\ \\ \\\\\n\\centerline{\\psfig{figure=Ln-dir,height=1.6cm}}\n\\centerline{ Figure 3.1} \\ \\\\\n\n\\begin{lemma}\nThe graph cochain complex of $L_n$:\\\\\n$C^*_{\\A}(L_n):\\ \\   C^{0} \\stackrel{d^0}{\\to} C^{1} \\stackrel{d^1}{\\to} C^{2} \n...\\to  C^{n-1} \\stackrel{d^{(n-1)}}{\\to} C^{n} $, \nis acyclic, except for the first term. \nThat is, $\\hat H^{i}_{\\A}(L_{n})=0$ for $i>0$ \nand $\\hat H^{0}_{\\A}(L_{n})$ is usually nontrivial\\footnote{From the \nfact that chromatic polynomial of $L_n$ is equal to $\\lambda(\\lambda -1)^n$ \nfollows that  rank($\\hat H^{0}_{\\A}(L_{n})=$ \nrank($\\A)$ $(rank(\\A)-1)^n)$, we assume here that $k$ is a principal ideal \ndomain. It was proven in \\cite{H-R-2} that for \na commutative $\\A$ decomposable into \n$k1 \\oplus \\A\/k$ one has that $H^*_{\\A}(L_n)= H^0_{\\A}(L_n)= \n\\A \\otimes (\\A\/k)^{\\otimes n}$.}. \n\\end{lemma}\nFor a line graph $\\hat H_{\\A} = H_{\\A}$ so we will use $H_{\\A}$ to simplify \nnotation. \\ \nWe prove Lemma 3.3 by induction on $n$. For $n=0$, $L_0$ is \nthe one vertex graph, thus $H^*_{\\A}= H^0_{\\A}= \\A$ and \nthe Lemma 3.3 holds. Assume that the lemma holds for $L_k$ with $k<n$.\nIn order to perform inductive step we construct the long exact sequence\nof cohomology of line graphs. For a commutative ${\\A}$ it is a special \ncase\\footnote{One can construct an exact sequence of functor cohomology \nimitating deleting-contracting exact sequence. One have to define properly \ntwo functors on $E\\cup e$, one ``covariant\" and one ``contravariant\" and \nthe exact sequence will be based on a functor on subsets of $E$ \nand these two additional functors. We will discuss this idea\n in a sequel paper.}\n of the exact sequence of graph cohomology in \\cite{H-R-2} which in turn \nresemble the skein exact sequence of Khovanov homology \\cite{Vi}:\n$$ 0 \\to H^{0}_{\\A}(L_{n}) \\to \nH^{0}_{\\A}(L_{n})\\otimes {\\A} \\stackrel{\\partial}{\\to}\nH^{0}_{\\A}(L_{n-1}) \\to H^{1}_{\\A}(L_{n}) \\to...$$\n$$\\to H^{i-1}_{\\A}(L_{n-1}) \\to \nH^{i}_{\\A}(L_{n}) \\to \nH^{i}_{\\A}(L_{n})\\otimes {\\A}\\to...$$\nsuch that $\\partial:H^{0}_{\\A}(L_{n}) \\otimes {\\A} \\to \nH^{0}_{\\A}(L_{n-1})$ is an epimorphism. \nFrom this exact sequence the inductive step follows.\\\\\nTo construct the above exact sequence we consider the short \nexact sequence of chain complexes (for the notation see Figure 3.1):\n$$ 0 \\to C^{i-1}(L_n\/e_{n}) \\stackrel{\\alpha}{\\to}  C^{i}(L_n) \n\\stackrel{\\beta}{\\to} \nC^{i}(L_{n}-e_n) \\to 0.$$\nThis exact sequence is constructed in the same way as in the case of \ncommutative ${\\A}$, that is ${\\alpha}(a_0\\otimes a_1\\otimes ...\n\\otimes a_{n-i})(e_{j_1},...,e_{j_{i-1}})= (a_0\\otimes a_1\\otimes ...\n\\otimes a_{n-i})(e_{j_1},...,e_{j_{i-1}},e_n)$. Further, \n$\\beta$ is defined in such a way that if $e_n\\in s$ then $\\beta (S)=0$,\nand if  $e$ is not in $s$ then $\\beta$ is the identity \nmap (up to $(-1)^{|s|}$).\n\n\nExactness of the sequence \nfollows from the definition.  This exact sequence leads to the long \nexact sequence of cohomology:\n$$0 \\to H^{0}_{\\A}(L_{n}) \\to\nH^{0}_{\\A}(L_{n}-e_n) \\stackrel{\\partial}{\\to}\nH^{0}_{\\A}(L_{n}\/e_n) \\to $$\n$$...\\to H^{i-1}_{\\A}(L_{n}\/e_n) \\to\nH^{i}_{\\A}(L_{n}) \\to\nH^{i}_{\\A}(L_{n}-e_n) \\to...$$\nNow $L_{n-1}=L_{n}\/e_n$ and $L_{n}-e_n$ is $L_{n-1}$ with an additional \nisolated vertex, therefore by a K\\\"unneth formula (see for example \n1.0.16 of \\cite{Lo}) we have $H^{i}_{\\A}(L_{n}-e_n)= \nH^{i}_{\\A}(L_{n})\\otimes {\\A}$. From this we get \nthe exact sequence used in the proof of Lemma 3.3. To see the \nepimorphism of $\\partial$ notice that the map \n$H^{0}_{\\A}(L_{n}-e_n) \\to\nH^{0}_{\\A}(L_{n-1})$ is an epimorphism almost by the definition \n(we can think of decorating the last vertex of $L_n$ by $1$, to see \nthe epimorphism. \\\\\nThere is chain map epimorphism $\\partial_c: C^i_{\\A}(L_{n}-e_n) \\to\nC^{i}_{\\A}(L_{n-1})$ which is obtained by multiplying the \nweight of component containing $v_{n-1}$ by the weight of $v_n$ which \ndescends to $\\partial$. This map has a chain map section\n$\\partial_c^{-1}: C^{i}_{\\A}(L_{n-1}) \\to C^i_{\\A}(L_{n}-e_n)$ \nsuch that in the image $v_n$ has always weight $1$. Because \n$\\partial_c\\partial_c^{-1} = Id$ therefore, on the \ncohomology level $\\partial$ is an epimorphism.\n\nWe can continue now with the proof of Theorem 3.1. \nThe (partially) acyclic chain complex of Lemma 3.3 is the chain complex \nof ${\\A}^e={\\A}\\otimes {\\A}^{op}$ modules. It is a \n(partial) free resolution of the ${\\A}^e$-module ${\\A}$. \nUpon tensoring this resolution with ${\\mathbb M}$ considered as a right \nmodule over ${\\A}^e$ we obtain the cochain complex, \n$$\\{\\mathbb M \\otimes_{{\\A}^e} C^i\\}_{i=0}^{n-1} : \\ \n\\M \\otimes_{{\\A}^e} C^0  \\stackrel{\\partial^0}{\\to}\n\\mathbb M \\otimes_{{\\A}^e} C^1 \\stackrel{\\partial^1}{\\to} ...  \\to\n\\mathbb M \\otimes_{{\\A}^e} C^{n-2} \\stackrel{\\partial^{n-1}}{\\to}\n\\mathbb M \\otimes_{{\\A}^e} C^{n-1} \\to 0$$\n whose cohomology \n(except possibly $H^0$) are the Hochschild homology of ${\\A}$ \nwith coefficients in ${\\mathbb M}$ (compare for example \\cite{Wei}).\nHaving in mind relation between  indexing we get that \n$H^i = H_{n-i}({\\A},{\\mathbb M})$ for $i>0$. \nTo get exactly the chain complex of the ${\\mathbb M}$-reduced\n (directed) graph homology of $P_{n}$, $\\hat H^*_{{\\A},{\\mathbb M}}(P_n)$,\n we extend this chain complex to \n$$\\{\\mathbb M \\otimes_{{\\A}^e} C^i\\}_{i=0}^{n}\\ :\\ \\\n\\mathbb M \\otimes_{{\\A}^e} C^0  \\stackrel{\\partial^0}{\\to}\n\\mathbb M \\otimes_{{\\A}^e} C^1 \\stackrel{\\partial^1}{\\to} ...  \\to\n\\mathbb M \\otimes_{{\\A}^e} C^{n-1} \\stackrel{\\partial^{n-1}}{\\to}\n\\mathbb M \\otimes_{{\\A}^e} C^{n}=\\mathbb M \\to 0$$\nwhere the homomorphism\n$\\partial^{n-1}$ is the zero map. \n\n\nTo complete the proof of Theorem 3.1 we show that this complex is \nexactly the same as the cochain complex of the ${\\mathbb M}$-reduced\n (directed) graph cohomology of $P_{n}$. We consider carefully the \nmap $\\mathbb M \\otimes_{{\\A}^e} C^j \\stackrel{\\partial^j}{\\to} \n\\mathbb M \\otimes_{{\\A}^e} C^{j+1}$. In the calculation \nwe follow  the proof of Proposition 1.1.13 of \\cite{Lo}.\nThe idea is to ``bend\" the line graph $L_n$ to the polygon $P_n$ and \nshow that it corresponds to tensoring, over ${\\A}^e$ with \n$\\mathbb M$; compare Figure 3.2.\n\\\\ \\ \\\\\n\\centerline{\\psfig{figure=bending,height=3.4cm}}\n\\centerline{ Figure 3.2} \\ \\\\\n\nLet us order components of $[G:s]$ ($G$ is equal to $P_n$ or $L_n$) \nin the anticlockwise orientation, \nstarting from the component containing $v_0$ (decorated by an element of $\\M$ \nif $G=P_n$). The element of $C^j_{\\A}(L_n)$ will be denoted by \n$(a_{i_0},a_{i_1},...,a_{i_{n-j-1}},a_{i_{n-j}})(s)$ and of \n$C^j_{\\A,\\M}(P_n)$ by $(m,a_{i_1},...,a_{i_{n-j-1}})(s)$. In the \nisomorphism $\\M \\otimes_{{\\A}^e} C^{j}_{\\A}(L_n) \\to C^j_{\\A,\\M}(P_n)$ \n($j<n$) the element $(m,a_{i_0},a_{i_1},...,a_{i_{n-j-1}},a_{i_{n-j}})(s)$ \nis send \nto $(a_{i_{n-j}}ma_{i_0},a_{i_1},...,a_{i_{n-j-1}})(s)$. \nOne can easily check that it yields a cochain map (compare \\cite{Lo}) so \nit induces the isomorphism on cohomology. Note that  \n$(m,a_{i_0},a_{i_1},...,a_{i_{n-j-1}},a_{i_{n-j}}) = \n(a_{i_{n-j}}ma_{i_0},1,a_{i_1},...,a_{i_{n-j-1}},1)$ in \n$\\M \\otimes_{{\\A}^e} C^{j}_{\\A}(L_n)$.\nThe proof of Theorem 3.1 is completed.\n \n\\section{Calculations and speculations}\n\nThere is an extensive literature on Hochschild homology and a lot of \ningenious methods of computing them (e.g. \\cite{Lo,Wei,Ros,Kon}).\nOur main result, Theorem 3.1, allows us to use these methods \nto compute graph cohomology  \nfor polygons and, to some extend, to other graphs (using for example an \nobservation that some properties of a cohomology of a polygon propagate \nto graphs containing it (compare \\cite{A-P,H-P-R}). Properties of \nHochschild homology (and, equally well, cyclic homology) should \neventually shed light on Khovanov type homology of links.\n\nWe start from adapting Theorem 1.3 about Hochschild homology of \nsymmetric tensor algebra. The simplest case of one variable\npolynomials $\\A=\\A_{\\infty}=Z[x]$ allows us to extend Theorem 27 \nof \\cite{H-P-R} from the triangle to any polygon. \n$\\A_{\\infty}$ is a graded algebra with \n$x^i$ being of degree $i$. Consequently Hochschild homology of \n$\\A_{\\infty}$  is a bigraded\nmodule. We treat $\\A_{\\infty}$ as a $Z$-module (an abelian group) and\nto simplify description of homology we use the Poincar\\'e polynomial\nof $HH_{**}(\\A_{\\infty})$ to describe the free part of homology.\nRecall that the Poincar\\'e polynomial (or series) of\nbigraded finitely generated $Z$-modules $H_{**}$\nis $PP(t,q)=PP(H_{**})(t,p) =\n\\sum_{i,j}a_{i,j}t^iq^j$ where $a_{i,j}$ is the rank of the group $H_{i,j}$.\n\\begin{corollary}\nFor an $n$-gon $P_n$ the graph cohomology groups \n$H^{i,j}_{\\A_{\\infty}}(P_n)$ \nare free abelian with Poincar\\'{e} polynomial $(q+q^2+q^3+...)^3 + \nt^{n-2}(q+q^2+q^3+...)=$ $ (\\frac{q}{1-q})^3 + t^{n-2}\\frac{q}{1-q}$.\n\\end{corollary}\n\\begin{proof} From Theorem 3.1 we obtain $H^{i,j}(P_n)$ for $0<i<n-1$.\n It was observed in \\cite{H-R-2}  that $H^{i,j}(P_n)=0$ for $i \\geq n-1$.\nTo find $H^{0,*}(P_n)$ we use the fact that the chromatic polynomial \nof $P_n$ is equal to $(\\lambda -1)^n + (-1)^n(\\lambda -1)$ and \nthe graph cohomology categorify the chromatic polynomial. That is, \nif we substitute $t=-1$ and $1+q+q^2+q^3+...=\\lambda$ in the Poincar\\'{e} \npolynomial we obtain the chromatic polynomial \\cite{H-R-2}.\n\\end{proof}\nAnother illustration of the power of our connection is for the \nalgebra ${\\A}={\\A}_{p(x)}= Z[x]\/(p(x))$ where $p(x)$ is a polynomial in \n$Z[x]$. We discuss the general case later, here let us notice that two \nspecial cases of $p(x)=x^m$ and $p(x)=x^m-1$ are of great interest in \nknot theory (in Khovanov-Rozansky homology \\cite{Kh-R-1} \nand its deformations \\cite{Gor}).\nLet us apply first the knowledge of Hochschild homology for \n$\\A_m = Z[x]\/(x^m)$ (compare \\cite{Lo}) and solving Conjectures 30 and 31 \nof \\cite{H-P-R}.\n\\begin{theorem}\\label{4.2}\\ \\\\ \n(Free)\\ \\ The Poincar\\'e polynomial of $HH_{**}(\\A_m)$ is equal to:\\\\\n$(1+q+...+q^{m-1}) + t(q+q^2+...+q^{m-1}) + \n(t^2+t^3)(q+q^2+...+q^{m-1})q^{m} + (t^4+t^5)(q+q^2+...+q^{m-1})q^{2m}+...+ \n(t^{2i}+t^{2i+1})(q+q^2+...+q^{m-1})q^{im}+...$.\n\\ \\  (Torsion)\\ \\\n$Tor (H_{**}(\\A_m))= \\bigoplus_{i=1}^{\\infty}H_{2i-1,im}(\\A_m)$ \nwhere each summand is isomorphic to $Z_m$.\n\\end{theorem}\n We solve Conjectures 30 and 31 \n of \\cite{H-P-R} by applying Theorems 4.2 and 3.1.\n\\begin{corollary}\\label{4.3}\n\\begin{enumerate}\n\\item[(Odd)] For $n=2g+1$ we have: \\\\\n$Tor(H^{*,*}_{{\\A}_{m}}(P_{2g+1}))=\n H^{v-2,m}_{{\\A}_{m}}(P_{2g+1})\\oplus\nH^{v-4,2m}_{{\\A}_{m}}(P_{2g+1})\\oplus ... \\oplus\nH^{1,gm}_{{\\A}_{m}}(P_{2g+1})$ with each summand\nisomorphic to $Z_m$.\\\\\nThe Poincar\\'{e} polynomial of $H^{*,*}_{{\\A}_{m}}(P_{2q+1})$ is\nequal to\\\\\n$ (q+...+q^{m-1})^v +$ \\\\\n$(q+...+q^{m-1})(t^{v-2}+ (t^{v-3}+t^{v-4}q^m +\n(t^2+t)q^{m(g-1})$. \n\\item[(Even)]\nFor $n=2g+2$ we have: \\\\\n$Tor(H^{*,*}_{{\\A}_{m}}(P_{2g+2}))=\nH^{v-2,m}_{{\\A}_{m}}(P_{2g+1})\\oplus\nH^{v-4,2m}_{{\\A}_{m}}(P_{2g+1})\\oplus ... \\oplus\nH^{2,gm}_{{\\A}_{m}}(P_{2g+2})$ with each summand\nisomorphic to $Z_m$.\\\\\nThe Poincar\\'{e} polynomial of $H^{*,*}_{{\\A}_{m}}(P_{2q+2})$ is\nequal to\\\\\n$(q+...+q^{m-1})^n + q^{m(n\/2)-1}(q+...q^{m-1})+$ \\\\\n$(q+...+q^{m-1})(t^{n-2}+ (t^{n-3}+t^{n-4})q^m +\n(t^3+t^2)q^{m(g-1)}+ tq^{mg})$.\n\\end{enumerate}\n\\end{corollary}\nAssume that $m=2$ in Corollary 4.3. Then, using Theorem 2.7 we can \nrecover Khovanov computation of homology of the torus \nlink $T_{2,n}$ \\cite{Kh-1,Kh-2}. In particular we get:\n\\begin{corollary}\\cite{Kh-2}\\  \nLet $T_{2,-n}$ be a left-handed \ntorus link  of type $(2,-n)$, $n>2$. Then the torsion part of the \nKhovanov homology of $T_{2,-n}$ is given by (in the description \nof homology we use notation of \\cite{Vi} treating $T_{2,-n}$ as \na framed link):\\\\  \n(Odd)\\ \\  For $n$ odd, all the torsion of $H_{**}(T_{2,-n})$ is \nsupported by \\\\ \\ \\  \n$H_{n-2,3n-4}(T_{2,-n})= H_{n-4,3n-8}(T_{2,-n})=...= \nH_{-n+4,-n+8}(T_{2,-n})=Z_2$.\\\\\n(Even)\\ \\ For $n$ even, all the torsion of $H_{**}(T_{2,-n})$ is\nsupported by\\\\\n\\ \\ $H_{n-4,3n-8}(T_{2,-n})= H_{n-6,3n-12}(T_{2,-n})=...=\nH_{-n+4,-n+8}(T_{2,-n})=Z_2$.\\\\\n\nFor a right-handed torus link  of type $(2,n)$, $n>2$, we can  \nuse the formula for the mirror image (Khovanov duality theorem; \nsee for example\\\\\n \\cite{A-P,APS}:\\ \\ \\  $H_{-i,-j}({\\bar D})= \nH_{ij}(D)\/Tor(H_{ij}(D)) \\oplus Tor(H_{i-2,j}(D))$.\n\\end{corollary}\n\nThe result on Hochschild homology of symmetric algebras has a \nmajor generalization to the large class of algebras called \n{\\it smooth algebras}.\n\\begin{theorem}(\\cite{Lo,HKR})\\label{4.1}\nFor any smooth algebra $\\A$ over $k$, the antisymmetrization map \n$\\varepsilon_*: \\Lambda^*_{\\A|k} \\to HH_n(\\A)$ is an isomorphism of \ngraded algebras. Here $\\Omega^n_{\\A|k}= \\Lambda^n \\Omega^1_{\\A|k}$ \nis an $\\A$-module of differential $n$-forms.\n\\end{theorem}\nWe refer to \\cite{Lo} for a precise definition of a smooth algebra, \nhere we only \nrecall that the following are examples of smooth\nalgebras:\\\\\n (i) Any finite extension of a perfect field $k$ (e.g. a field\nof characteristic zero). \\\\\n(ii) The ring of algebraic functions on a nonsingular\nvariety over an algebraically closed field $k$, e.g. $k[x]$, $k[x_1,...,x_n]$,\n$k[x,y,z,t]\/(xt-yz-1)$ \\cite{Lo}. \\\\ \nNot every quotient of a polynomial algebra is a smooth algebra.\nFor example, $C[x,y])(x^2y^3)$ or $Z[x]\/(x^m)$ are not smooth.\nThe broadest, to my knowledge, treatment of Hochschild homology of \nalgebras $C[x_1,...,x_n]\/(Ideal)$ is given by Kontsevich in \\cite{Kon}.\nFor us the motivation came from one variable polynomials, Theorem 40 \nof \\cite{H-P-R}. In particular we generalize Theorem 40(i) from a \ntriangle to any polygon that is we compute the graph cohomology \nof a polygon for truncated\npolynomial algebras and their deformations. Thus, possibly, we can \napproximate Khovanov-Rozansky $sl(n)$ homology and their deformations. \n\n\\begin{theorem}\\label{4.6}\n\\begin{enumerate}\n\\item[(i)] $HH_i({\\A}_{p(x)}) = Z[x]\/(p(x),p'(x))$ for $i$ odd and \n$HH_i({\\A}_{p(x)}) = \\{[q(x)]\\in Z[x]\/(p(x)) \\ | \\ \\ \nq(x)p'(x)$  is divisible by $p(x)$ \\}, \nfor $i$ even $i \\geq 2$. In both cases the $Z$ rank of the group \nis equal to the degree of gcd$(p(x),p'(x))$.\n\\item[(ii)] In particular, for $p(x)= x^{m+1}$, we obtain homology of \nthe ring of truncated polynomials, ${\\A}_{m+1}=Z[x]\/x^{m+1}$ \nfor which:\\  \\ \n$HH_i({\\A}_{m+1})=Z_{m+1} \\oplus Z^m$ for odd\\ $i$ and \n$HH_i({\\A}_{m+1})= Z^m$ for for $i$ even, $i\\geq 2$. \n$HH_0({\\A}_{m+1})= \\A = Z^{m+1}$.\n\\item[(iii)] The graph cohomology of a polygon $P_n$, $H^i_{{\\A}_{p(x)}}(P_n)$, \nis given by:\\\\\n $H^{n-2i}_{{\\A}_{p(x)}}(P_n)= {\\A}_{p(x)}\/(p'(x))$ for $1\\leq i \\leq \\frac{v-1}{2}$,\\ \nand \\\\\n$H^{n-2i-1}_{{\\A}_{p(x)}}(P_n)= ker({\\A}_{p(x)} \n\\stackrel{p'(x)}{\\to} {\\A}_{p(x)})$ \nfor $1\\leq i \\leq \\frac{v-2}{2}$.\\\\\nFurthermore,  $H^{k}_{{\\A}_{p(x)}}(P_n)= 0$ for $k\\geq n-1$ and \n$H^{0}_{{\\A}_{p(x)}}(P_n)$ is a free abelian group of rank \n$(d-1)^n + (-1)^n(d-1)$ for $n$ even ($d$ denotes the degree of $p(x)$) and\n  it is of rank $(d-1)^n + (-1)^n(d-1) - rank(H^{1}_{{\\A}_{p(x)}}(P_n))$  \nif $n$ is odd (notice that $(d-1)^n + (-1)^n(d-1)$ \nis the Euler characteristic of $\\{H^{i}_{{\\A}_{p(x)}}(P_n)\\}$). \n \n\\end{enumerate}\n\\end{theorem}\n\n\\begin{proof} Theorem 4.6(i)\n is proven by considering a resolution of ${\\A}_{p(x)}$ as \nan ${\\A}_{p(x)}^e={\\A}_{p(x)}\\otimes \n{\\A}^{op}_{p(x)}$ \nmodule:\n$$ \\cdots \\to {\\A}_{p(x)}\\otimes {\\A}_{p(x)} \n\\stackrel{u}{\\to} \n{\\A}_{p(x)}\\otimes {\\A}_{p(x)} \\stackrel{v}{\\to}\n{\\A}_{p(x)}\\otimes {\\A}_{p(x)} \\stackrel{u}{\\to} \\cdots\n\\to {\\A}_{p(x)} $$ \nwhere $u= x\\otimes 1 - 1\\otimes x$ and $v=\\Delta (p(x))$ is a coproduct \ngiven by $\\Delta (x^{i+1}) = x^i\\otimes 1 + x^{i-1}\\otimes x +...+ \nx \\otimes x^{i-1} + 1\\otimes x^i$.\n\\end{proof}\n Curious but not accidental observation \nis that by choosing coproduct $\\Delta (1)= v$ \nwe define a Frobenius algebra structure on $\\A$. \nIn Frobenius algebra $(x\\otimes 1)\\Delta (1) = (1 \\otimes x)\\Delta (1)$ \nwhich makes $uv=vu=0$ in our resolution. Furthermore the distinguished \nelement of the Frobenius algebra $\\mu\\Delta (1)= p'(x)$. \n\\\\\n\n\\iffalse\n\\ \\\\\nEND OF THE PAPER (ideas from letters below will be developed in \nfuture papers, hopefully).\n\n\\section{Letter to Mietek; August 16, 2005}\nI will use the general definition Def. 2.4 of the paper:\nand extend Example 2.5 from plane signed graphs to signed graphs with\nedges along every vertex ordered, i.e. flat vertex graphs\n (such a graph defines the unique embedding into a closed surface).\n\n\\begin{definition}\\label{Text2.4}\nLet $k$ be a commutative ring and $E=E_+ \\cup E_-$ a finite set divided\ninto two disjoint subsets (positive and negative sets). We consider the\ncategory of subsets of $E$\n($E\\supset s=s_+\\cup s_-$ where $s_{\\pm}= s\\cap E_{\\pm}$). the set $Mor(s,s')$\nis either empty or has one element if $s_- \\subset s'_-$ and\n$s_+ \\supset s'_-$. Objects are graded by $\\sigma(s)=|s_-|- |s_+|$.\nLet call this category the superset category (as the set $E$\n is initially bigraded). We define ``Khovanov cohomology\" for every functor,\n$\\Phi$, from the superset category to the category of $k$-modules.\nWe define cohomology of $\\Phi$ in the similar way as for a functor\nfrom the category of sets (which corresponds to the case $E=E_-$).\nThe cochain complex corresponding to $\\Phi$ is defined to be\n$\\{C^i(\\Phi)\\}$ where $C^i(\\Phi)$ is the direct sum of $\\Phi(s)$ over all\n$s\\in E$ with $\\sigma(s)=i$. To define $d: C^i(\\Phi) \\to C^{i+1}(\\Phi)$\nwe first define face maps $d_e(s)$ where $e=e_- \\notin s_-$ ($e_- \\in E_-$)\nor $e=e_+ \\in s_+$. Namely, in such cases\n$d_{e_-}(s)= \\Phi(s\\subset s\\cup e_-)(s)$  and\n$d_{e_+}(s)= \\Phi(s\\supset s-e_+)$. Now, as usually\n$d(s)= \\sum_{e \\notin s} (-1)^{t(s,e)}d_e(s)$,\nwhere $t(s,e)$ requires ordering of elements of $E$ and is equal the\nnumber of elements of $s_-$ smaller then $e$ plus the number of elements\nof $s_+$ bigger than $e$.\nAs before we obtain the cochain complex whose cohomology does not\ndepend on ordering of $E$.\n\\end{definition}\n\nConstruction of (Khovanov type) graph homology for signed \"flat vertex\"\ngraphs.\nLet $G$ be a connected signed plane graph with an edge set\n$E=E_+ \\cup E_-$ were $E_+$ is the\nset of positive edges and $E_-$ is the set of negative edges.\nFurthermore we assume that at every vertex adjacent edges are ordered, loops\nare listed twice\n(or equivalently the neighborhood of each vertex is placed on the plane).\n[[Such a structure defines a 2-cell embedding of $G$ into a closed\noriented surface, $F_G$. $2$-cell embedding means that $F_G -G$\nis a collection of open discs.]]\nWe define the functor from the superset category $\\mathbb E$ using the\nfact that $G \\subset F_G$ defines a link diagram $D(G)$ on $F_G$.\n\nTo define the functor $\\Phi$ we fix\na Frobenius algebra $A$ with a multiplication $\\mu$ and a comultiplication\n$\\Delta$ (the main example used by Khovanov,\n is the algebra of truncated polynomials $\\A_m=Z[x]\/x^m$ with a\ncoproduct $\\Delta (x^k)=\\sum_{i+j=m-1+k}x^i \\otimes x^j$). We assume that\n$\\A$ is commutative and cocommutative. The main property of Frobenius\nalgebra we use is: $(\\mu \\otimes Id)(Id \\otimes \\Delta)= \\Delta\\mu$.\nFor a functor on objects $s \\subset S$, we consider the Kauffman state\nassociated to $s$, that is every crossing is decorated by a marker.\nIf $e \\in s_+$ or $e \\in E_- - s_-$ then we choose a positive marker,\notherwise (i.e. $e \\in E_+ -s_+$ or $e\\in s_-$) we choose negative\nmarker (Figure ??). We denote by $D_s$ the set of circles obtained from\n$D_G$ by smoothing crossings of $D_G$ along markers yielded by $s$.\n$k(s)$ denotes the number of circles in $D_s$. Now we define\n$\\Phi(s)= \\A^{\\otimes k(s)}$ and we interpret it as having $\\A$\nattached to every circle of $D_s$.\nTo define $\\Phi$ on morphisms pairs of objects\n$s,s'$ such that $Mor(s,s')\\neq \\emptyset$\n and $\\sigma(s')=\\sigma(s)+1$ and define $\\Phi(Mor(s,s')$ first.\nWe have two cases to consider:\\\\\n(1) $k(s')= k(s)-1$. In this case we use the product in $\\A$.\nIf two circles of $D_s$, say $u_i,u_{i+1}$, are combined into one circle\nin $D_(s')$ then $\\Phi(Mor(s,s'))(a_1\\otimes a_2 \\otimes ...\\otimes a_i\n\\otimes a_{i+1})\\otimes ...\\otimes a_{k(s}) =\n(a_1\\otimes a_2 \\otimes ...\\otimes \\mu(a_i\\otimes a_{i+1})\\otimes\n...\\otimes a_{k(s)})$.\\\\\n(2) $k(s')= k(s)-1$. In this case we use the coproduct in $\\A$.\nIf a circle of $D_s$, say $u_i$ is split into two circles in \n$D_{s'}$ then $\\Phi(Mor(s,s'))(a_1\\otimes a_2 \\otimes ...\\otimes a_i\n\\otimes ...\\otimes a_{k(s}) =\n(a_1\\otimes a_2 \\otimes ...\\otimes \\Delta (a_i)\\otimes ...\\otimes a_{k(s})$.\n\\\\\n\\ \\\\\n$\\Phi$ can be now extended to all morphisms is superset category\ndue to Frobenius algebra axioms.\n\nCohomology of our functor $H^i(\\Phi)$ are cohomology of\nour signed, flat vertex graph. If $G$ is a disjoint union of $G_1$\nand $G_2$ then we define $H^{*}(G) = H^{*}(G_1) \\otimes H^{*}(G_2)$.\nSo we use K\\\"unneth formula as a definition.\n\n[[Because our graph is uniquely embedded into a closed surface we \ncan produce more delicate (co)homology theory by distinguishing \nnontrivial from trivial curves on the surface and following \\cite{APS} \nin the construction; at least it will work for the algebra $A_2$ but \nwe do not use a product and coproduct as the only information but \nextend Frobenius algebra rules lead by the topology of the surface.]]\n\\section{From e-mails to Laure and Yongwu: August 12, 13, 2005}\n\n{\\bf Long exact sequence of functor homology}\\\\\n\\ \\\\\nAssume that $E' \\subset E$ and consider three functors;\n$\\Phi_1$ and $\\Phi_2$  on the category of subsets of $E'$ and\n$\\Phi$ on the category of subsets of $E$.\nWe say that $\\Phi_1$ and $\\Phi$ are covariantly coherent if\n$\\Phi_1(s)= \\Phi (s\\cup E\")$ where $E''= E-E'$ and\n$\\Phi_1(s_1 \\subset s_2) = \\Phi (s_1\\cup E\" \\subset s_2 \\cup E\")$.\n\nClaim 1. If $\\Phi_1$ and $\\Phi$ are covariantly coherent than we have\na cochain map $\\alpha: C^*(\\Phi_1) \\to C^{*+|E\"|}(\\Phi)$. In the definition\nof $\\alpha$ we use the isomorphism $\\Phi_1(s) \\to \\Phi (s\\cup E\")$ for\nevery $s \\in E$.\n\nConsequently we have the map on cohomology:\n$\\alpha_*: H^*(\\Phi_1) \\to H^{*+|E\"|}(\\Phi)$.\n\nWe say that $\\Phi_2$ and $\\Phi$ are cotravariantly coherent if\nfor $s \\in E'$ we have $\\Phi_2(s)= \\Phi (s)$ and for\n$s_1\\subset s_2 \\in E'$ we have $\\Phi_2(s_1 \\subset s_2) =\n\\Phi(s_1 \\subset s_2)$.\n\nClaim 2. If $\\Phi_2$ and $\\Phi$ are cotravariantly coherent than we have\na cochain map $\\beta: C^*(\\Phi) \\to C^{*}(\\Phi_2)$.\nWe define $\\beta: C^i(\\Phi) \\to C^{i}(\\Phi_2)$ as follows.\nIf $s \\subset E'$ then $\\beta$ is the identity on $\\Phi(s)$, if\n$s \\notin E'$ then $\\beta$ restricted to $\\Phi(s)$ is the zero map.\n\nOur goal is to use these maps to construct the long exact sequence of\ncohomology involving $H^*(\\Phi_1), H^*(\\Phi)$ and $H^*(\\Phi_2)$.\n\nTheorem. Let $E'=E-\\{e\\}$ and $\\Phi_1,\\Phi_2$ are functors on $E'$\nand $\\Phi$ is a functor on $E$ such that\n$\\Phi_1$ and $\\Phi$ are covariantly coherent and\n$\\Phi_2$ and $\\Phi$ are cotravariantly coherent.\nThen\nWe have the short exact sequence of cochain complexes:\n$$ O \\to C^i(\\Phi_1) \\to C^{i+1}(\\Phi) \\to  C^{i+1}(\\Phi_2) \\to 0$$\nleading  to the long exact sequence of (co)homology.\n\n\\fi\n\\ \\\\\n\\ \\\\\n\\ \\\\\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLinear response theory for nonequilibrium systems is slowly emerging from a great variety of formal approaches --- see \\cite{njp} for a recent review. It \nremains however very important in nonequilibrium to concentrate more on the physical--operational meaning of \nthe response expressions. Obviously, it is very practical to have experimental access to the various terms in a \nresponse formula and to learn in general to recognize facts of the unperturbed system that are responsible \nfor the particular response. \nThat at least is what has made the fluctuation-dissipation theorem so useful in equilibrium. \nFor example, transport properties as summarised in the mobility or conductivity, can be obtained from the diffusion \nin the unperturbed equilibrium system. In other words, not only is there a unifying response relation in equilibrium, \nit also possesses a general meaning in terms of fluctuations and dissipation.  Such is not yet quite the situation for \nnonequilibrium systems and extra examples, in particular for spatially extended systems will therefore be useful.\\\\\n\n\nThe present paper gives the response systematics for the zero range process.\nThe zero range process regularly appears in nonequilibrium studies and has the simplifying structure that its stationary distribution is \nsimple (and remains a product distribution even away from equilibrium) while it shows a rich and quite realistic phenomenology. \nWe refer to \\cite{shuts,EH} for a general introduction and nonequilibrium study of the model.   We refer to Section 3 in \\cite{EH} for a review \nof applications, in particular for the correspondence\nwith shaken granular gases.  We will repeat the set-up in \nSection \\ref{zrp}. Interestingly, the time-reversed zero range process has an external field and particle currents directed \nopposite to the density profile. To start however we repeat in the next section some more formal aspects of the nonequilibrium \nlinear response.  \n Our point of view is to look in particular for the \ndecomposition of the response into a frenetic and an entropic contribution. The entropic part is expressed in terms of \n(time-antisymmetric) currents and the frenetic part gets related to the (time-symmetric) dynamical activity.  The latter \nrefers to the number of exits and entrances of particles at the boundaries of the system.\nSection \\ref{rzrp} performs that decomposition for the boundary driven zero range process, and gives a number of response formul{\\ae} for density and current.  There we find our main results, in particular modified Green-Kubo relations.\nFinally, in Section \\ref{eqd} we treat some special cases which bring the nonequilibrium response to resemble \nthe equilibrium Kubo formula.  That opens the separate theme of trying to understand under what physical conditions nonequilibrium \nfeatures remain largely absent.\n\n\\section{Nonequilibrium response}\\label{nonr}\nWe restrict ourselves to open systems  connected to various different equilibrium reservoirs.  Their nonequilibrium is passive \nin the sense that they do not affect the reservoirs directly and that all nonequilibrium forcing works directly on the \nparticles of the system.   \n\nThe state of the open system is described by values $x$ for some reduced variables, e.g. (some) particle positions.  In the course of time $[0,t]$ there is a \npath or trajectory $\\omega := (x_s, 0\\leq s \\leq t)$ for which we have a well-defined entropy flux $S(\\omega)$, that is \nthe change of entropy in the equilibrium reservoirs. That $S(\\omega)$ typically depends on the elementary changes in $x$ and \nhow that affects the energy and particle number of the reservoirs.  Of course $S(\\omega)$ also depends on parameters such as \nthe temperature, the chemical potentials of the reservoir and coupling coefficients of interaction.  It will thus change under \nperturbations.\\\\Similarly, for every state $x$ there is a notion of reactivity, like the escape rate from $x$. \nFor the path $\\omega$ there will then be a dynamical activity $D(\\omega)$ which reflects the expected amount of changes \nalong the path $\\omega$, again function of the system and reservoir parameters.\\\\\nConsider now a perturbation of that same system in which parameters are changed. Clearly, for any path $\\omega$ the entropy \nflux $S$ and the dynamical activity $D$ will change.  We can look for the linear excess, that is the amount by which the perturbation \nhas changed these observables to first order.  \nWe refer to \\cite{bam09,bmw1} for the general introduction, and to \\cite{mnw08} for complementary aspects to entropy.  \nThe linear response for a path-observable $O(\\omega)$ is the difference in \nexpectation $\\langle \\cdot\\rangle^h$  between the perturbed process (with small time-dependent amplitude $h_s, s\\in [0,t]$) and the \noriginal steady expectation $\\langle\\cdot\\rangle$.  It has the form\n\\begin{equation}\n\\langle O(\\omega)\\rangle^h - \\langle O(\\omega)\\rangle = \\frac {1}{2}\\langle \\mbox{Ent}^{\\left[ 0,t \\right]}(\\omega)\\,O(\\omega)\\rangle\n- \\langle \\mbox{Esc}^{\\left[ 0,t\\right] }(\\omega)\\,O(\\omega )\\rangle\n\\label{eq:xsusc}\n\\end{equation}\nwhere Ent$^{\\left[ 0,t\\right] }$ is the excess in entropy flux per $k_B$ over the trajectory\ndue to the perturbation and Esc$^{\\left[ 0,t\\right] }$ \nis the excess in dynamical activity over the trajectory.\nThe latter and second term on the right-hand side of \\eqref{eq:xsusc} is \nthe frenetic contribution \\footnote{the response formula \\eqref{eq:xsusc} can be written in \nseveral equivalent ways: in the second term often there is a \nfactor $1\/2$ which here we include in the \ndefinition of the dynamical activity term in brackets; section \\ref{zrp} \nwill treat some specific formulations. }. \nIn many nonequilibrium situations \nthe physical challenge is to learn to guess \nor to find and evaluate that Esc$^{\\left[ 0,t \\right] }$ from partial information on the dynamics. \nThe present paper takes the opportunity to explore this question and to make such task more specific for the zero range process. \nLet us however first give the more general formal structure, restricting ourselves to Markov jump processes.  \nFor a more general review of various recent approaches, see \\cite{njp}.\\\\\n\n\nOn the finite state space $K$ we consider transition rates\n$k(x,y), x,y\\in K$.  We assume irreducibility so that there is exponentially fast convergence to a unique stationary\ndistribution $\\rho(x), x\\in K$, satisfying\n\\[\n\\sum_{y\\in K} [\\rho(x)\\,k(x,y) - \\rho(y)\\,k(y,x)] = 0, \\quad x\\in\nK \\] \nStill, in general, there are nonzero currents of the form\n$j(x,y) := \\rho(x)\\,k(x,y) - \\rho(y)\\,k(y,x)\\neq 0$ for some pairs $x\\neq\ny\\in K$, so that the stationary process is not time-reversible.\\\\\nFor physical models the rates carry a specific meaning.  Following the condition of local detailed balance the ratio\n\\[\n\\log\\frac{k(x,y)}{k(y,x)} =\\sigma(x,y)\n\\]\nshould be the entropy flux (in units of $k_B$) in the transition $x\\rightarrow y$.  \nConsider now again the path $\\omega := (x_s, 0\\leq s\\leq t)$.  It consists of jumps (transitions) at specific times $s_i$ \nand waiting times over $s_{i+1} - s_i$.  The total entropy flux $S$ (in units of $k_B$) is\n\\begin{equation}\\label{set}\nS(\\omega) = \\sum_{s_i} \\sigma(x_{s_i^-},x_{s_{i}}) \n\\end{equation}\nwhere the sum takes the two states of the transition $x_{s_i^-} \\longrightarrow x_{s_{i}}$, with $x_{s_i^-}$ being the state \njust before the jump time $s_i$ to $x_{s_i}$.\\\\\nFor the dynamical activity we need a reference process. Writing \n\\[\n\tk(x,y) = \\psi(x,y) e^{\\sigma(x,y)\/2} \\mbox{ with } \\psi(x,y) =\\psi(y,x) \\mbox{ and } \\sigma(x,y)=-\\sigma(y,x),\n\\]\nwe take the reference rates $k_o(x,y) =1$ whenever $\\psi(x,y)\\neq 0$ and zero otherwise. That reference process corresponds to an infinite \ntemperature limit but it will not matter \nin the end. With respect to that reference we do not only have a change \nin ``potential barrier'' $-\\log 1 = 0 \\rightarrow -\\log \\psi(x,y)$ for each transition, but also a change in the escape \nrates for each state $x$:\n\\[\n\\xi(x) = \\sum_{y: \\psi(x,y) > 0} [k(x,y)-1]\n\\]\nWe then take the dynamical activity $D$ over the path $\\omega$  be the combination\n\\begin{equation}\\label{setd}\nD(\\omega) = \\int_0^t\\id s \\,\\xi(x_s) - \\sum_{s_i} \\log \\psi(x_{s_i^-},x_{s_{i}}).\n\\end{equation}\nPerturbations change $S$ and $D$.  Let us look at a specific example of\nperturbed transition rates considered in\n\\cite{diez,mprf}:\n\\begin{equation}\\label{general}\nk_s(x,y) =\nk(x,y)\\,e^{h_s[b V(y)-a V(x)]}, \\quad t>s\\geq 0\n\\end{equation}\nwhere the $a,b \\in \\bbR$ are independent of the perturbing potential $V$ and \nthe $h_s \\ll 1$ is small. The coresponding perturbed Master \nequation for the \ntime-dependent probability law $\\rho_t$ is\n\\[\n\\frac{\\id}{\\id t} \\rho_t(x) = \\sum_y \\big[k_t(y,x) \\rho_t(y) - k_t(x,y) \\rho_t(x)\\big];\n\\]\nwhile the unperturbed equations of motion are obtained by making $h_s=0$.\nOne standard possible choice of perturbation is\ntaking $a= b = 1\/2T$ where $T$ is the temperature of the environment which exchanges the energy $V$ with the system. \nIn general $a,b$ could be \narbitrary; however, for the perturbed rates in \\eqref{general} to satisfy \nthe condition of local detailed balance, one requires that $a+b=1\/T$.\n\nWe continue however with the more general perturbation \\eqref{general}.\nIt is instructive to rewrite the perturbation \\eqref{general} as\n\\begin{eqnarray}\nk_s(x,y) &=&\nk(x,y)\\,e^{h_s\\frac{b-a}{2}(V(x) + V(y))}\\,e^{h_s \\frac{a+b}{2}(V(y) - V(x))}\\nonumber\\\\\n&=& \\big[\\psi(x,y)e^{h_s\\frac{b-a}{2}(V(x) + V(y))}\\big]\\;e^{\\sigma(x,y)\/2 + h_s (a+b)(V(y) - V(x))\/2}\\label{ins}\n\\end{eqnarray}\nagain being split in a symmetric prefactor (between square brackets) and an anti-symmetric part in the exponential.  \nFrom here it is easy to see the excess for the entropy flux at a transition $x\\rightarrow y$ to be\n\\begin{equation}\\label{se}\n h_s (a+b)(V(y) - V(x))\n\\end{equation}\nWe use \\eqref{se} to find the perturbation to $S$ in \\eqref{set} yielding\n\\begin{eqnarray}\\label{en}\n\\mbox{Ent}^{\\left[ 0,t\\right] }(\\omega) &=& (a+b)\\, \\sum_{s_i} h_{s_i} [V(x_{s_{i}}) - V(x_{s_i^-}) \\nonumber\\\\\n&=& (a+b) \\{h_t V(x_t) - h_0 V(x_0) - \\int_0^t \\id s \\,\\dot{h}_s V(x_s)\\}.\n\\end{eqnarray}\nFor the dynamical activity we should use again the reference process with rates $k_o(x,y)$ as above.  Then,\nat least for the change in escape rates (first term in \\eqref{setd}) at state $x$,\n\\begin{eqnarray}\\label{dy}\n\\sum_y [k_s(x,y) - k(x,y)] &=&  h_s \\sum_y k(x,y)\\{\\frac{b-a}{2}(V(x) + V(y)) + \\frac{a+b}{2}(V(y) - V(x))\\}\\nonumber\\\\\n&=& h_s \\,\\sum_y k(x,y)[bV(y) - aV(x)]\n\\end{eqnarray}\nto first order in $h_s$.\nThe total change to $D$ of \\eqref{setd} is thus\n\\begin{equation}\\label{es}\n\\mbox{Esc}^{\\left[ 0,t\\right] }(\\omega) = \\int_0^t\\id s\\, h_s\\,\\sum_y k(x_s,y)\\,[bV(y) - aV(x_s)] + \\frac{a-b}{2}\\sum_{s_i} h_{s_i} [V(x_{s_{i}}) + V(x_{s_i^-})]\n\\end{equation}\nwhere the last term corresponds to the change in the second term of $D$ in \\eqref{setd}, as from \\eqref{ins}.\nIn all, the expressions \\eqref{en} and \\eqref{es} completely specify the response \\eqref{eq:xsusc} for the example \\eqref{general}.\\\\\n\n  \nWe can still rewrite the previous formul{\\ae}, loosing somewhat the physical interpretation but gaining somewhat formal elegance.  To start, let us restrict ourselves to the more simple situation where the observable $O$  is just a state function $O(x), x\\in K$.  The response then investigates the change\n\\[\n\\langle O(x_t)\\rangle^h - \\langle O(x_t)\\rangle = \\langle O(x_t)\\rangle^h - \\langle O\\rangle\n\\]\nto first order in the $h_s$, where the first expectation $\\langle \\cdot\\rangle^h$ is under the perturbed Markov dynamics ($s\\geq 0$) and the second $\\langle \\cdot \\rangle$ is the original steady expectation.  To say it differently, linear response  wants to compute the generalized susceptibility $R(t,s)$ in\n\\[\n\\langle O(x_t)\\rangle^h = \\langle O \\rangle +\n\\int_0^t\\id s\\,\nh_s\\, R(t,s) + o(h)\\]\nThe nonequilibrium answer can be written in a variety of ways, many of which are rather formal, \nbut they should in the end all coincide with \\eqref{eq:xsusc} for \\eqref{en}--\\eqref{es}.  For example,\nin terms of  the backward generator $L$ of the jump process,\n\\[\nLf(x) = \\left.\\frac{\\id}{\\id s}\\right|_{s=0}\\langle\nf(x_s)\\rangle_{x_0=x}\\; = \\;\\sum_y k(x,y)[f(y)-f(x)]\n\\]\nwe have  for \\eqref{en} that\n\\begin{eqnarray}\n\\langle  \\{h_t V(x_t) - h_0 V(x_0) - \\int_0^t \\id s \\,\\dot{h}_s V(x_s)\\} \\,O(x_t)\\rangle &=&\n \\int_0^t \\id s \\,h_s \\frac{\\id}{\\id s}\\langle V(x_s) \\,O(x_t)\\rangle\\nonumber\\\\\n&=& -\\int_0^t \\id s \\,h_s \\langle V(x_s) \\,LO(x_t)\\rangle\\nonumber\n \\end{eqnarray}\nOn the other hand, for \\eqref{es},\n \\[\n \\mbox{Esc}^{\\left[ 0,t\\right] }(\\omega) = \nb\\,\\int_0^t\\id s\\, h_s\\, LV(x_s) + (b-a)\\{\\int_0^t\\id s\\, h_s\\, \n\\sum_y k(x_s,y)\\,V(x_s) - \\sum_{s_i} h_{s_i} \\frac{V(x_{s_{i}}) + V(x_{s_i^-})}{2}\\}\n \\]\nWe must substitute that expression together with \\eqref{en} into \\eqref{eq:xsusc}, which leads to\n\\begin{equation}\\label{stationary}\n R(t,s) = a\\frac{\\partial}{\\partial s}\\langle V(x_s)O(x_t)\\rangle - b\\langle LV(x_s)\\,O(x_t)\\rangle\n\\end{equation}\nfor all times $0\\leq s <t$, which recovers a result of \\cite{mprf}.\n Of course, these expectations are stationary and only depend on the time-difference $t-s$.\nFor the case $b=a=1\/(2T)$ in \\eqref{general}, the response formula thus becomes\n\\[ \nR(t,s) = \\frac{1}{2T}\n\\frac{\\partial}{\\partial s}\\langle V(x_s)\\,O(x_t)\\rangle_{\\mu}-\\frac{1}{2T}\\langle LV(x_s)\\,O(x_t)\\rangle\\]\nIn equilibrium, i.e., under stationary time-reversal\nsymmetry where all currents $j(x,y)=0$, we have for $t>s$\n\\[\n\\langle LV(x_s)\\,O(x_t)\\rangle = \\langle V(x_s)\\,LO(x_t)\\rangle = \\frac{\\partial}{\\partial t}\\langle V(x_s)O(x_t)\\rangle \n\\]\nand hence the two terms in the right-hand side of\n\\eqref{stationary}\ncoincide and we recover the Kubo-formula, \\cite{kubo66},\n\\begin{equation}\\label{eqform}\nR^{\\mbox{eq}}(t,s) = \\frac{1}{T}\\,\\frac{\\partial}{\\partial s}\\left<\nV(s)\\,O(t)\n \\right>_{\\textrm{eq}},\\quad 0 < s <t\n \\end{equation}\n  whenever $a+b = 1\/T$.\\\\\nSuch formal systematics in nonequilibrium  as in \\eqref{stationary} is useful as it is generally available, but its physical \ninterpretation relies on the equivalence (as discussed above for the example \\eqref{general}) with \\eqref{eq:xsusc}, \nmuch in the same way as, for equilibrium, the Kubo-formula can be called a fluctuation--{\\it dissipation} relation. \nNote that in nonequilibrium, from \\eqref{eq:xsusc}, the response formula has become a fluctuation--{\\it dissipation--activity} relation.\nWe refer to \\cite{bam09,bmw1,mprf,njp} for more details.\\\\\n  \nAnother (again more formal) possibility of writing the linear response formula \\eqref{stationary} uses the adjoint $L^* $ of \nthe backward generator with respect to the stationary distribution $\\rho$: $\\langle (Lf)\\,g\\rangle =\\langle f\\,(L^*g)\\rangle$ or\n\\begin{equation}\\label{ster}\nL^*g(x) = \\sum_y k(y,x) \\frac{\\rho(y)}{\\rho(x)}\\,[g(y)-g(x)]\n\\end{equation}\nwhich generates the time-reversed stationary process.\n With this notation, for $s<t$ and in the stationary regime,\n\\begin{equation}\\label{stra}\n\\frac{\\partial}{\\partial s}\\langle V(x_s)\\,O(x_t)\\rangle = \n- \\frac{\\partial}{\\partial t}\\langle V(x_s)\\,O(x_t)\\rangle \n= - \\langle V(x_s)\\,LO(x_t)\\rangle = - \\langle L^*V(x_s)\\,O(x_t)\\rangle\n\\end{equation}\nso that\n \\[\n \\frac{\\partial}{\\partial s}\\langle V(x_s)O(x_t)\\rangle + \\langle LV(x_s)\\,O(x_t)\\rangle = \n \\langle (L-L^*)V(x_s) \\,O(x_t)\\rangle\n \\]\nTherefore, referring to the response \\eqref{stationary},\n\\[\n a\\frac{\\partial}{\\partial s}\\langle V(x_s)O(x_t)\\rangle - b\\langle LV(x_s)\\,O(x_t)\\rangle =\n(a+b)\\, \\frac{\\partial}{\\partial s}\\langle V(x_s)O(x_t)\\rangle - b \\langle (L-L^*)V(x_s) \\,O(x_t)\\rangle\n \\]\n or\n \\begin{equation}\\label{rrr}\n R(t,s) =(a+b)\\, \\frac{\\partial}{\\partial s}\\langle V(x_s)O(x_t)\\rangle - b \\langle (L-L^*)V(x_s) \\,O(x_t)\\rangle\n  \\end{equation}\nThe first term is the Kubo expression \\eqref{eqform}.  The second term will be useful whenever we \nknow more about the time-reversed process.  If wanted, one can still substitute there\n\\[\n((L- L^*)f)(x) = 2\\sum_y\\frac{j(x,y)}{\\rho(x)}\\,[f(y)-f(x)]\n\\]\nto obtain the interpretation of \\cite{gaw} in terms of the moving frame.\\\\\n\nLet us finally mention the Agarwal-Kubo procedure for arriving at a linear response expression, see e.g. formula 13 in \\cite{njp}.  \nThat is first order perturbation theory on the level of forward generators.  We consider the unperturbed forward generator \n\\[\nL^+ g(x) = \\sum_y [k(y,x)g(y) - k(x,y)g(x)] \n\\]\nand its perturbation is denoted by ${L^h}^+$.  Then, we have in general\n\\begin{equation}\\label{ak}\n R(t,s) =\\langle \\frac{({L^h}^+-L^+)\\rho}{\\rho}(x_s)\\,O(x_t\\rangle)\n \\end{equation}\n The obvious disadvantage here is that one should know the stationary density $\\rho$; in contrast, all observables in \n\\eqref{stationary} are explicit and known and in \\eqref{eq:xsusc} they even have a meaning.\n \n \n\\section{Formal elements of the zero range process}\n\\label{zrp}\n\\subsection{Steady state}\nOn the lattice interval $\\{1,2,\\ldots,N\\}$, each site $i$ carries a number $x(i) \\in \\mathbb{N}$ of indistinguishable particles.  \nThe dynamics is characterized by the rate $w(k)$ at which a particle jumps from site $i$ when $x(i)=k$, \nand parameters $\\alpha,\\beta,\\gamma,\\delta$ for the rates of exits and entrances at the boundaries.   \nMore specifically, a particle moves from $i$ to a neighboring site $j=i\\pm 1$ at rate $w(x(i))$. \nWe need that $w(0)=0$ and $w(k)>0$ for $k>0$. At the boundary site $i=1$ a particle is added at rate $\\alpha$ and at $i=N$ \nis added at rate $\\delta$, while a particle moves out from $i=1$ at rate $\\gamma\\, w(x(1))$ and moves out from $i=N$ \nat rate $\\beta\\, w(x(N))$.  As reference for more details using mostly the same notation, we refer to \\cite{EH}.\\\\\nIt is well-known that the product distribution $\\rho = \\rho_{N,\\alpha,\\beta,\\gamma,\\delta}$ is invariant,\n\\begin{eqnarray}\\label{ste}\n\\rho(x) &=&  \\prod_{i=1}^N \\nu_i(x(i)), \\quad \\nu_i(k) = \\frac{z_i^{k}}{{\\cal Z}_i}\\,\\frac 1{w(1)\\,w(2)\\,\\ldots w(k)},\\; k>0\\nonumber\\\\\n{\\cal Z}_i &=& 1 + \\sum_{k=1}^\\infty\\frac{z_i^{k}}{w(1)\\,w(2)\\,\\ldots w(k)}\n\\end{eqnarray}\nThe ``fugacities'' $z_i$ are of the form $z_i = Ci + B = z_1 + C(i-1)$ where\n\\[\nB :=\\frac{\\alpha + (1-\\gamma)C}{\\gamma},\\quad C:=\\frac{\\delta \\gamma - \\beta \\alpha}{\\beta \\gamma N + \\beta(1-\\gamma) + \\gamma}=\n\\frac{e^{\\mu_r\/T} - e^{\\mu_\\ell\/T}}{N}\\,\\big(1 + \\frac{\\beta +\\gamma-\\beta \\gamma}{\\beta\\gamma N}\\big)^{-1} \n\\]\nWe have introduced ``chemical potentials'' $\\mu_\\ell := T\\log \\alpha\/\\gamma$ and $\\mu_r := T\\log \\delta\/\\beta$ with \n$T$ the environment temperature.\nWhen $\\mu_\\ell=\\mu_r, \\alpha\/\\gamma = \\delta\/\\beta$, then $C=0, B=z_i=\\alpha\/\\gamma$; and detailed balance is satisfied.   \nIf not, we get a  stationary  particle current (to the right) equal to $\\langle J_i\\rangle = -C = \\alpha -\\gamma z_1 = \\beta z_N - \\delta$ and thermodynamic \ndriving force $(\\mu_\\ell - \\mu_r)\/T = \\log \\alpha\/\\gamma - \\log \\delta\/\\beta$. Note however that for $C\\neq 0$ the (then nonequilibrium) stationary \ndistribution $\\rho$ also depends on purely kinetic (and not only on thermodynamic) aspects; they will again enter the \nresponse in terms of the dynamical activity.  \nFor example, fixing $\\alpha\/\\gamma$ and $\\delta\/\\beta$ does not determine $C$, trivially but importantly.\\\\\n\n\n\\subsection{Time-reversal}\nTo make explicit use of the formula \\eqref{rrr}, we need to know the time-reversed process, which is interesting in itself.\\\\ \nIn general for a Markov process as we had it described in Section \\ref{nonr} the time-reversed process is again a Markov jump \nprocess with generator $L^*$ in \\eqref{ster} and with rates\n\\[\nk^{\\text rev}(x,y)  = k(y,x)\\,\\frac{\\rho(y)}{\\rho(x)}\n\\]\nfor the stationary distribution $\\rho$.  Because we know the stationary distribution $\\rho$ of the zero range process as the\nproduct distribution \\eqref{ste}, it is actually easy to determine explicitly the time-reversed process.  This is interesting \nalso because, \nby time-reversing, the particle current will be reversed\/change sign but the stationary density profile, as given in terms of the \nfugacities $z_i$, will remain the same.  As can be guessed, that only works because by time-reversing one actually generates \nan external field.  Let us see the details.\\\\\n\nFirst we take a bulk transition in which a particle hops to a neighboring site.  \nTake $y=x-e_i+e_{i+1}$ where $e_i$ stands for the particle configuration with exactly one particle at site $i$.  Then,\n\\[\n\\frac{\\rho(y)}{\\rho(x)} = \\frac{z_{i+1}}{z_i}\\frac{w(x(i))}{w(x(i+1))},\\quad k(y,x) = w(x(i+1)) \n\\]\nwhich means that in the time-reversed process a particle moves from site $i$ to $i+1$ at rate \n$k^{\\text rev}(x,x-e_i+e_{i+1})  = z_{i+1} \\,w(x(i))\/z_{i}$ while \nsimilarly for a jump from $i$ to $i-1$, $k^{\\text rev}(x,x-e_i+e_{i-1})  =  z_{i-1} \\,w(x(i))\/z_{i}$.  \nWe have therefore for the time-reversed process again a zero range process but now in an inhomogeneous bulk field \n\\[\nE_i := 2\\log \\frac{z_{i+1}}{z_{i}}\n\\]\nover the bond $(i,i+1)$, having the sign of $C$, i.e., pushing the particles towards the boundary {\\it where the chemical potential \nwas largest}. \nAt the boundaries we find the creation and annihilation parameters for the time-reversed process to be\n\\[\n\\alpha^{\\text rev} = \n\\gamma z_1,\\quad \\beta^{\\text rev} = \\frac{\\delta}{z_N},\\quad \\gamma^{\\text rev}= \\frac{\\alpha}{z_1},\\quad \\delta^{\\text rev} =\\beta z_N.\n\\]\nThat means that the chemical potentials for the reversed process have become \n\\begin{eqnarray}\n\\mu_\\ell^{\\text rev} &=& - \\mu_\\ell +2 T\\log (e^{\\mu_\\ell\/T} + C\/\\gamma)\\nonumber\\\\\n\\mu_r^{\\text rev} &=& -\\mu_r + 2T \\log(e^{\\mu_r\/T} - C\/\\beta)\\nonumber\n\\end{eqnarray}\n\n\nNote of course that in the case of detailed balance $E_i\\equiv 0 $ and $\\alpha^{\\text rev} =\\alpha$ etc., \nso that the equilibrium process is unchanged\nby time-reversal.\\\\\n\nWe can now write down the explicit expression for the second term in \\eqref{rrr}:\n\\begin{eqnarray}\n(L - L^*)V\\,(x) &=& \\big(\\gamma-\\frac{\\alpha}{z_1}\\big)\\,w(x(1))\\,[V(x-e_1)-V(x)] + \\big(\\alpha-\\gamma z_1\\big)\\,[V(x+e_1)-V(x)] \\nonumber\\\\\n&+& \\big(\\beta-\\frac{\\delta}{z_N}\\big)\\, w(x(N))\\,[V(x-e_N)-V(x)] + \\big(\\delta -\\beta z_N\\big)\\,[V(x+e_N)-V(x)]\\nonumber\\\\\n&&-C\\sum_{i=1}^{N-1} \\frac{w(x(i))}{z_i} [V(x-e_i+e_{i+1})-V(x)]\\nonumber\\\\\n&&+C\\sum_{i=2}^{N} \\frac{w(x(i))}{z_i} [V(x-e_i+e_{i-1})-V(x)]\\nonumber\n\\end{eqnarray}\nApplying that for $V(x) = {\\cal N}(x):= x(1) + x(2) +\\ldots + x(N)$ the total number of particles in the system, we get\n\\begin{equation}\\label{lminl}\n(L - L^*){\\cal N}\\,(x) =\\big(\\frac{\\alpha}{z_1}-\\gamma\\big)\\,w(x(1)) +  \\big(\\frac{\\delta}{z_N} - \\beta\\big)\\, w(x(N))\n\\end{equation}\nwhere we have also used that $\\gamma z_1 + \\beta z_N = \\alpha + \\delta$.\n\n\\section{Responses in the zero range process}\\label{rzrp}\nLet us consider the perturbation\n\\begin{equation}\\label{mape}\n\\alpha \\rightarrow q\\,\\alpha,\\quad \\beta\\rightarrow p'\\,\\beta,\\quad \\gamma \\rightarrow p\\,\\gamma,\\quad \\delta\\rightarrow q'\\,\\delta\n\\end{equation}\nto the parameters governing the entrance and exit rates at the boundaries of the system.\nTheir thermodynamic meaning is to shift the chemical potentials by $h_\\ell = T\\log q\/p$ for the left and by \n$h_r = T\\log q'\/p'$ for the right reservoir.   Depending on the remaining freedom how to choose the $p, p'$ we can distinguish still several ``kinetic'' possibilities.\n\n\\subsection{``Potential'' perturbation}\\label{potp}\nA first possible perturbation that we consider is that\n\\begin{equation}\\label{shi}\n\\frac{q}{p} = \\frac{q'}{p'} = e^{ h\/T}\n\\end{equation}\nwith $h$ the small (equal) shift in left and right chemical potential.\nEven while the zero range process is not formulated directly in terms of a potential, even at detailed balance, \nit is still easy to fit \\eqref{shi} into the scheme of \\eqref{general}, in particular by choosing \n$h_t\\equiv h$ (time-independent),  $a=b=1\/(2T)$, potential $V={\\cal N}$ equal to the particle number, and\n\\begin{equation}\\label{potper}\ne^{h\/(2T)}=q=q', \\quad e^{-h\/(2T)}=p=p'.\n\\end{equation}\n  We can thus apply \\eqref{rrr} with  formula \\eqref{lminl} to give the correct modification of the Kubo formula as \n \\begin{eqnarray}\\label{rrrtotc}\n\\frac{\\langle O(x_t)\\rangle^{h}- \\langle O \\rangle}{h} &=& \\frac 1{T} \\langle {\\cal N}\\, O\\rangle - \\frac 1{T}\\,\\langle {\\cal N}(x_0)\\, O(x_t)\\rangle \\nonumber\\\\\n&& +\\frac 1{2T}\\int_0^t\\id s\\{\\big(\\frac{\\alpha}{z_1}-\\gamma\\big)\\,\\langle w(x_0(1))\\,O(x_s)\\rangle + \\big(\\frac{\\delta}{z_N}-\\beta\\big)\\, \\langle w(x_0(N))\\,O(x_s)\\rangle\\}  \n  \\end{eqnarray}\nOf course we could also have used \\eqref{stationary} with $L{\\cal N}(x) = \\alpha + \\delta - \\gamma w(x(1)) -\\beta w(x(N))$ to obtain\n\\begin{eqnarray}\\label{rpo}\n\\frac{\\langle O(x_t)\\rangle^{h}- \\langle O \\rangle}{h} &=& \\frac 1{2T} \\langle {\\cal N}(x_t) - {\\cal N}(x_0); O(x_t)\\rangle \\nonumber\\\\\n&& +\\frac{1}{2T}\n\\int_0^t\\id s\\{\\gamma\\,\\langle w(x_0(1));O(x_s)\\rangle + \\beta\\, \\langle w(x_0(N));O(x_s)\\rangle\\}  \n  \\end{eqnarray}\nwhere we have used connected correlation functions $\\langle A;B\\rangle := \\langle A\\,B\\rangle  - \\langle A\\rangle\\,\\langle B\\rangle$. The first term in the right-hand side \n is the entropic or dissipative part of the response, since in that correlation\none sees the observable $O$ correlated with the particle loss; the last term may be called the frenetic part of the response, since one meets there the correlation with the time-integrated escape rates.\\\\\n\n\nFinally one finds place for the Agarwal-Kubo formula \\eqref{ak}, which here is explicit because the stationary density \n$\\rho$ is given in \\eqref{ste}.  For the ``potential'' perturbation \\eqref{shi}--\\eqref{potper} given \nby $\\alpha \\rightarrow (1+ h\/(2T)) \\alpha, \\beta \\rightarrow (1- h\/(2T)) \\beta, \\gamma \\rightarrow (1- h\/(2T)) \\gamma, \\delta \\rightarrow (1+ h\/(2T)) \\delta$ and under discussion so far, that gives\n\\begin{eqnarray}\\label{ak12}\n\\frac{{L^h}^+\\rho - L^+\\rho}{\\rho}(x) \n=  \\alpha \\frac{h}{2T}\\, [\\frac{\\rho(x-e_1)}{\\rho(x)} - 1] + \\delta \\frac{h}{2T}\\,[\\frac{\\rho(x-e_N)}{\\rho(x)} -1]\n+ \\gamma\\frac{h}{2T}\\,w(x(1)) \\nonumber\\\\\n-  \\gamma\\frac{h}{2T}w(x(1)+1)\\frac{\\rho(x+e_1)}{\\rho(x)} +\\beta\\frac{h}{2T}\\,w(x(N)) \n-  \\beta\\frac{h}{2T}w(x(N)+1)\\frac{\\rho(x+e_N)}{\\rho(x)}\\nonumber\\\\\n= \\frac{h}{2T}\\{\\frac{\\alpha}{z_1}(w(x(1)) - z_1) + \\frac{\\delta}{z_N}(w(x(N)) - z_N)\n+\\gamma( w(x(1)) - z_1)  + \\beta (w(x(N)) - z_N) \\}\n\\end{eqnarray}\nThis calculation results in the linear response formula\n\\begin{equation}\\label{lak}\n\\frac{\\langle O(x_t)\\rangle^{h}- \\langle O \\rangle}{h} = \\frac 1{2T}\n\\int_0^t\\id s\\{\\big(\\frac{\\alpha}{z_1}+ \\gamma\\big)\\,\\langle w(x_0(1));O(x_s)\\rangle \n+ \\big(\\frac{\\delta}{z_N}+ \\beta\\big)\\, \\langle w(x_0(N));O(x_s)\\rangle\\}  \n\\end{equation}\n\n\\subsection{General perturbation}\n\nWe emphasize that the three response formul{\\ae} \\eqref{rrrtotc}--\\eqref{rpo}--\\eqref{lak} are mathematically identical. \nThey all start from the ``potential perturbation'' \\eqref{general} as realized in \\eqref{shi}--\\eqref{potper}. They are \nhowever not to be applied for other perturbations even consistent with \\eqref{shi}, except in equilibrium where the \nresponse does not pick up the detailed kinetics.\nLet us therefore do better (more general) and  illustrate the systematic interpretation  with unique formula \\eqref{eq:xsusc} to \nthe perturbation \\eqref{mape}.\\\\\n\n\nWe only need experience with entropy and no calculation to find the first term in \\eqref{eq:xsusc}.  For the perturbation \\eqref{mape} \nthe entropic part in the response follows the usual (irreversible) thermodynamics and we must have the excess in entropy flux given by\n\\begin{equation}\\label{fer}\n \\mbox{Ent}^{\\left[ 0,t \\right]}(\\omega) =  -\\frac{h_r}{T}\\,J_r(\\omega) -\\frac{h_\\ell}{T}\\,J_\\ell(\\omega)\n\\end{equation}\nwhere $J_r$ ($J_\\ell$) is the net number of particles that have exited to the right (left) reservoir (time-integrated current).\nWhen we specify to a perturbation like \\eqref{shi} in which the chemical potentials get shifted together, $h=h_r=h_\\ell$, we can use that $J_\\ell(\\omega) + J_r(\\omega) = {\\cal N}(x_0) - {\\cal N}(x_t)$ so that the excess in entropy flux becomes\n\\begin{equation}\\label{gfer}\n \\mbox{Ent}^{\\left[ 0,t \\right]}(\\omega) =  \\frac{h}{T}\\,({\\cal N}_t - {\\cal N}_0)\n\\end{equation}\n proportional to the change over time in particle number.\\\\\n For the second term in \\eqref{eq:xsusc} we lack the experience and calculation will guide us.  The point is that the \ndynamical activity \\eqref{setd} exactly picks up the time-symmetric part in the action for path-integration. More specifically, \nlet us now call $P^h$ the process started from the unperturbed stationary zero range process \\eqref{ste} but under the \nperturbed dynamics for a time $[0,t]$.  The unperturbed stationary process is denoted by $P$.  We can compute the action \n${\\cal A^h}$ for which\n\\[\nP^h = e^{-{\\cal A}^h}\\,P \\simeq (1-{\\cal A}^h)\\,P\n\\]\nwith\n\\begin{eqnarray}\n{\\cal A}^h &=& - I^\\ell_{\\shortleftarrow}\\,\\log p- I^\\ell_{\\shortrightarrow}\\,\\log q  - I^r_{\\shortrightarrow}\\,\\log p' -  \nI^r_{\\shortleftarrow}\\,\\log q'\\nonumber\\\\\n  &+& \\int_0^t \\id s \\{(p-1)\\,\\gamma\\, w(x_s(1)) + (p'-1)\\,\\beta\\, w(x_s(N)) + (q-1)\\alpha + (q'-1) \\delta \\}\n\\end{eqnarray}\nwhere for example $I^\\ell_{\\shortrightarrow}$ equals the total number of particles that have entered the system from \nthe left, and $I^r_{\\shortrightarrow}$ is the total number of particles that have escaped to the right reservoir.\nWe decompose this action with the time-reversal $\\theta$ which makes $(\\theta x)_s= x_{t-s}$, so that\nthe response (up to higher order in $h$) can be obtained from\n\\begin{eqnarray}\nP^h - P &=& \\frac 1{2}[{\\cal A}^h\\theta - {\\cal A}^h]\\,P - \\frac 1{2}[{\\cal A}^h\\theta + {\\cal A}^h]\\,P\\nonumber\\\\\n&=& \\{\\frac 1{2} \\mbox{Ent}^{\\left[ 0,t \\right]}\n-  \\mbox{Esc}^{\\left[ 0,t\\right] }\\}\\,P\n\\end{eqnarray}\nwhere we indicate the general relation with \\eqref{eq:xsusc}.\\\\\nIn particular, we verify that\n\\[\n{\\cal A}^h\\theta - {\\cal A}^h = \\log\\frac{q}{p} \\,(I^\\ell_{\\shortrightarrow} - I^\\ell_{\\shortleftarrow} ) +\n\\log\\frac{p'}{q'} \\,(I^r_{\\shortrightarrow} - I^r_{\\shortleftarrow} ) \n\\]\nindeed exactly equals \\eqref{fer} (using for example $I^\\ell_{\\shortrightarrow} - I^\\ell_{\\shortleftarrow} = -J_\\ell$). \nOn the other hand, for the time-symmetric part\n\\begin{eqnarray}\n{\\cal A}^h\\theta + {\\cal A}^h &= & -\\log (pq) \\,I^\\ell - \\log (p'q') \\,I^r \n+ 2(p-1) \\gamma \\int_0^t \\id s \\,w(x_s(1))\\nonumber\\\\ &+& 2(p'-1) \n\\beta \\int_0^t \\id s\\, w(x_s(N)) + 2(q-1) \\alpha  t + 2 (q'-1) \\delta \\,t\n\\end{eqnarray}\nwith left activity $I^\\ell := I^\\ell_{\\shortleftarrow} + I^\\ell_{\\shortrightarrow}$ the total number of \ntransitions at the left boundary and similarly for $I^r$ at site $N$.  \nThe excess in dynamical activity $\\mbox{Esc}^{\\left[ 0,t\\right] } = ({\\cal A}^h\\theta + {\\cal A}^h)\/2$ that we need for \nthe general response in \\eqref{eq:xsusc} is thus\n\\begin{eqnarray}\\label{fes}\n\\mbox{Esc}^{\\left[ 0,t\\right] }(\\omega) &= & -\\log \\sqrt{pq} \\,I^\\ell - \\log \\sqrt{p'q'} \\,I^r \n+ (p-1) \\gamma \\int_0^t \\id s \\,w(x_s(1))\\nonumber\\\\ &+& (p'-1) \\beta \\int_0^t \\id s\\, w(x_s(N)) + (q-1) \\alpha  t +  (q'-1) \\delta \\,t\n\\end{eqnarray}\nNote that of course here the separate $p,p'$ and $q,q'$ play a role, and not just their ratio\n$p\/q, p'\/q'$ as for \\eqref{fer} --- that is how the frenetic contribution picks up kinetic information, while \nthe entropic part is purely thermodynamic.\nSubstituting \\eqref{fer} and \\eqref{fes} into \\eqref{eq:xsusc} gives the general response of the zero range process under \\eqref{mape}.\nA natural application is to look at how the current into the left reservoir changes when $h_r=0, h_\\ell = -a$ or $q'=p=p'=1$ but $q=1 -a\/T$, \ndecreasing (for $a>0$) the chemical potential of the left reservoir.  Then, for that choice, \\eqref{fer} and \\eqref{fes} give\n\\begin{equation}\\label{gkr}\n\\langle J_\\ell\\rangle^h - \\langle{J_\\ell}\\rangle =  \\frac{a}{2T}\\langle J_\\ell;J_\\ell\\rangle -\\frac{a}{2T}\\langle J_\\ell;I^\\ell\\rangle\n\\end{equation}\nwhich is the modification to the Green-Kubo relation \\cite{yan10} , for all times $t>0$, for the boundary driven zero range process. \nObserve that it is the correlation between current $J_\\ell$ and dynamical activity $I^\\ell$ that governs the correction. \nWhen $t\\uparrow +\\infty$, the conductivity will of course coincide with the change of $C$ in \\eqref{ste} under $\\alpha$.\nThere is a similar relation for the change in expected dynamical activity, so that in fact\n\\[\n\\langle J_\\ell + I^\\ell \\rangle^h - \\langle{J_\\ell + I^\\ell}\\rangle =  \n\\frac{a}{2T}\\langle J_\\ell;J_\\ell\\rangle -\\frac{a}{2T}\\langle I^\\ell;I^\\ell\\rangle\n\\]\nis given by a difference between variances of the current and dynamical activity,\nwhere still $\\langle J_\\ell\\rangle = C =-\\alpha+\\gamma z_1,\\langle I^\\ell\\rangle = \\alpha + \\gamma z_1$.\\\\\nFormul{\\ae} \\eqref{gfer}--\\eqref{fes} in \\eqref{eq:xsusc} will of course also lead again to a formula equal \nto each of the \\eqref{rrrtotc}--\\eqref{rpo}--\\eqref{lak} when restricting to \\eqref{shi}--\\eqref{potper}.\n\n\\subsection{``External'' perturbation} \nShifting the chemical potentials (from the outside) realistically means to change $\\alpha\\rightarrow q\\,\\alpha$ and \n$\\delta \\rightarrow q'\\, \\delta$ but not the exit rates $\\beta$ and $\\gamma$.  That is thermodynamically the same \n(in the shift of chemical potentials) as for the ``potential'' perturbation in Section \\ref{potp} but it is kinetically different. \nThe response formul{\\ae} \\eqref{rrrtotc}--\\eqref{rpo}--\\eqref{lak} are then invalid except at equilibrium.\n Here we look when we change  only the rates of the incoming particles in \\eqref{mape} but restricting ourselves to \\eqref{shi}:\n\\begin{equation}\n\\label{rep}\np=1=p',\\quad q=q'=1+ h\/T\n\\end{equation}\n  Note that the expected total activity in the unperturbed steady regime equals\n\\[\n\\langle I^\\ell+I^r \\rangle = (\\alpha + \\gamma z_1 + \\beta z_N + \\delta)t = 2(\\alpha + \\delta)t\n\\]\nbecause the stationary current equals $\\alpha-\\gamma z_1 = \\beta z_N - \\delta$.\nThat means that the excess dynamical activity \\eqref{fes} (for perturbation \\eqref{rep}) simply equals\n\\begin{equation}\n\\mbox{Esc}^{\\left[ 0,t\\right] }(\\omega) =  \\frac{h}{2T} \\{ \\langle I^\\ell+I^r \\rangle  - [I^\\ell + I^r]\\} \n\\end{equation}\nwhich is now very visibly related to the dynamical activity.  We therefore find the linear response formula  \\eqref{eq:xsusc} to become\n\\begin{equation}\\label{LINR}\n\\frac{\\langle O(\\omega)\\rangle^h - \\langle O(\\omega)\\rangle}{h} = \\frac{1}{2T} \\,\\langle({\\cal N}_t - {\\cal N}_0);O(\\omega)\\rangle \n+\\frac{1}{2T}\\langle (I^\\ell+I^r); O(x)\\rangle\n\\end{equation}\nwhich is another result for the linear response of the boundary driven zero range model when both left and right entrance rates have been \nincreased with the same small amount. Note that from \\eqref{eq:xsusc} it is here also possible to take a general path-observable \n$O(\\omega)$ that depends on the whole trajectory $\\omega$.  The first term is entropic corresponding to the dissipation of particles and \nthe second term is frenetic with the total dynamical activity \n$I := I^\\ell+I^r = I^\\ell_{\\shortleftarrow} + I^\\ell_{\\shortrightarrow} +I^r_{\\shortleftarrow} + I^r_{\\shortrightarrow}$.\\\\  \nLet us check the formula \\eqref{LINR} for the linear response around equilibrium ($C=0$, detailed balance), \nand with $O = I$ the total activity.  Then, since the first term $\\langle({\\cal N}_t - {\\cal N}_0);O(\\omega)\\rangle^{\\text eq} = 0$ \nfor time-symmetric $O$, we have a Green-Kubo type formula for the linear response of the dynamical activity around equilibrium:\n\\begin{equation}\n\\frac{\\langle I \\rangle^h - \\langle I\\rangle^{\\text eq}}{h} =\\frac{1}{2T}\\,\\mbox{Var} I > 0\n\\end{equation}\nwith, in the right-hand side, the unperturbed equilibrium variance of the dynamical activity giving the expected change in \nthat same dynamical activity when the left and right chemical potentials get slightly shifted.  Whether, say for positive $h$, \nthe change in dynamical activity remains positive also for boundary driven zero range processes depends apparently on whether \nthe dynamical activity is positively or negatively correlated with the dissipation of particles.  One could guess that \nfor very small $\\alpha, \\delta \\ll 1$ while keeping $ \\gamma,\\beta w(k) \\simeq 1$ (low temperature reservoirs) there is \na negative correlation between ${\\cal N}_t - {\\cal N}_0$ and $I$ which would make at least the first \nterm in \\eqref{LINR} for $O=I$ negative.\\\\\nIn any event however, be it equilibrium or nonequilibrium, we have the positivity of\n\\begin{equation}\n \\frac{\\langle {\\cal N}(x_t) + I\\rangle^h - \\langle {\\cal N}(x_0)+ I\\rangle}{h} =\\frac{1}{2T} \\,\\,\\mbox{Var} ({\\cal N}_t - {\\cal N}_0 + I)  \n> 0\n\\end{equation}\nby taking the observable $O =  {\\cal N}_t - {\\cal N}_0 + I$ in \\eqref{LINR}.\\\\\n\n \nLet us further simplify and take $O$ in \\eqref{LINR} a state function. It is then relevant to see how the stationary \ndistribution \\eqref{ste} gets modified under \\eqref{rep}.  It is straightforward to check that $C, B \\rightarrow qC, qB$ so that \nthe new ``fugacities'' become equal to $qz_i$.  The stationary distribution thus simply changes by multiplying $\\exp[h{\\cal N}(x)\/T]$ \nto the weights $\\rho(x)$. It is therefore not so surprising that the linear response drastically simplifies.  \nTo check it we take the opportunity to illustrate again the Agarwal-Kubo procedure \\eqref{ak} but now for the perturbation \\eqref{rep}:\n\\begin{eqnarray}\n\\frac{{L^h}^+\\rho - L^+\\rho}{\\rho}(x) &=&   \n\\alpha (q-1)\\, [\\frac{\\rho(x-e_1)}{\\rho(x)} - 1] + \\delta (q'-1) \\,[\\frac{\\rho(x-e_N)}{\\rho(x)} -1]\\nonumber\\\\\n&=& \\alpha \\frac{h}{T}\\, [\\frac{w(x(1))}{z_1} - 1] + \\delta  \\frac{h}{T} \\,[\\frac{w(x(N))}{z_N} -1]\\nonumber\n\\end{eqnarray}\nwhere we substituted the known stationary distribution $\\rho$ from \\eqref{ste}.  On the other hand, the backward generator \nof the time-reversed process equals\n\\[\nL^*{\\cal N}\\,(x) =  -\\frac{\\alpha}{z_1} \\,w(x(1)) + \\gamma z_1 + \\beta z_N  -\\frac{\\delta}{z_N}\\,w(x(N)) \n\\]\nand $\\alpha-\\gamma z_1 + \\delta -\\beta z_N=0$.\nTherefore,\n\\begin{equation}\\label{acc}\n\\frac{{L^h}^+\\rho - L^+\\rho}{\\rho} = -\\frac{h}{T}\\,L^* {\\cal N}\n\\end{equation}\nAs a consequence, using \\eqref{ak} results in the linear response exactly of the same form \\eqref{eqform} as in equilibrium, because\n (with $V = {\\cal N}$ in \\eqref{stra}),\n \\begin{eqnarray}\n\\frac{\\id}{\\id s} \\langle {\\cal N}(x_s)\\,O(x_t)\\rangle &=& - \\frac{\\id}{\\id t} \\langle {\\cal N}(x_0)\\,O(x_{t-s})\\rangle\\nonumber\\\\\n&=& -\\langle {\\cal N}(x_0) LO(x_{t-s})\\rangle = -\\langle L^*{\\cal N}(x_0)\\,O(x_{t-s})\\rangle\n\\label{kuboform}\n\\end{eqnarray} \nIn other words, for state observables the linear response of any boundary driven zero range process \nto ``external'' perturbations \\eqref{rep} \nhas always the same equilibrium Kubo-form \\eqref{eqform}, independent of being close or far from detailed balance.\n\n\n\n\n\\section{Intersections of equilibrium and nonequilibrium evolutions}\\label{eqd}\nThe difference between equilibrium and nonequilibrium processes is not always so crystal clear.  For exampe, if one starts \nwith a dynamics for which the Gibbs distribution $\\sim e^{-\\beta H}$ is invariant, for some Hamiltonian $H$, then that \ndistribution is also obviously unchanged when adding extra transformations or updating that leave the Hamiltonian $H$ invariant.  \nOn a more formal level, suppose we modify the Liouville equation to\n\\begin{equation}\n\\frac{\\partial}{\\partial t} \\rho(x,t) + \\{\\rho,H\\} = \\int\\id x [k(y,x)\\,\\rho(y) - k(x,y)\\, \\rho(x)]\n\\end{equation}\nwhere the right-hand side involves transition rates $k(x,y)$ between states $x\\rightarrow y$.  \nIf these $k(x,y)$ are zero unless $H(x) = H(y)$, then $\\rho\\sim \\exp[-\\beta H]$ remains of course invariant.  \nOn the other hand, the modified dynamics need not at all to satisfy detailed balance and then the resulting \nstationary regime will not be time-reversal invariant.\n\nThe Kubo formula \\eqref{eqform} summarizes equilibrium linear response in terms of a fluctuation-dissipation formula.  As we have seen in the previous \nsection with the combination \n\\eqref{acc}--\\eqref{kuboform}, \nthe Kubo formula extends to the zero range process and for \nexternal perturbations \\eqref{rep} to the nonequilibrium case.  \nIn the present section we look at that from a more general perspective.\n\n\n\n\n\\subsection{Special perturbations}\\label{spp}\nA special case arises when $b=0$ and $a=1\/T$ in (\\ref{general}),\nbecause then the response is of the equilibrium form \\eqref{eqform}.\\\\\nSuppose we have (quite arbitrary) a Markov jump process with rates $k(x,y)$ that we perturb\nby adding a time-dependent potential into\n\\begin{equation}\\label{suft}\nk_t(x,y) = k(x,y)\\, e^{-h_t V(x)\/T}\n\\end{equation}\nwhere $h_t$ is the small parameter.  \nThe linear response formula is obtained by putting $b=0$ in \\eqref{stationary} \nwhich gives the Kubo-equilibrium formula.\\\\\nThat can also be seen\nfrom the following consideration. Take $h$ to be constant; the law\n$\\rho^h$ defined by $\\rho^h(x) \\propto \\rho(x)e^{ h V(x)\/T}$ is\nstationary for the new dynamics (to all orders in $h$).  In other\nwords, here the resulting behavior under this perturbation is like\nin equilibrium, even though the unperturbed dynamics can be far\nfrom equilibrium.\\\\\n\nThe case of perturbation \\eqref{rep} for zero range is just slightly different and is summarized in \\eqref{acc}, which \nis the condition that there exists a function $V$ for which\n\\[\n({L^h}^+ - L^+)\\rho = h\\,\\rho\\,L^*V = h\\, L^+(V\\rho)\n\\]\n for the stationary density $\\rho$.  That\nis equivalent with  finding a potential $V$ so that for all functions $f$\n\\begin{equation}\\label{suf}\n\\sum_x ((L^h-L)f )(x)\\,\\rho(x) = h\\sum_x (Lf)(x)\\, V(x)\\, \\rho(x)\n\\end{equation}\nIt is easily seen that \\eqref{suf} exactly follows when $L^h = (1+ hV)\\,L$ which (basically) is \n\\eqref{suft}.  Therefore, \\eqref{acc} or \\eqref{suf} is only slightly weaker than \\eqref{suft}.\n\n\n\n\n\n\n\n\\subsection{Density response in the boundary driven Lorentz gas}\\label{lg}\n\\begin{figure}\\begin{center}\\includegraphics[scale=0.9]{lorentzfig.eps}\\caption{The boundary driven Lorentz gas. \nA flat rectangular slab is placed between two thermo-chemical reservoirs and contains \nan array of fixed discs, which scatter particles (red dots) via elastic collisions. \nThe centers of the scatterers of radius $R$ are placed in a regular triangular lattice with {\\it finite horizon}; that is, \nthe distance among the centers\nof contiguous disks ($4R\/\\sqrt 3$) ensures that a particle cannot cross the distance of a unit \ncell without \ncolliding at least once with a scatterer. There is a uniform temperature in the \nreservoirs $T$, which determines the velocities of all gas particles. \nIn the molecualr dynamics simulation, when a particle hits a boundary wall \nit disappears from the system, while other particles are injected \nto the system at given rates, proportional to each reservoir density.\n}\\label{fig:lgs}\\end{center}\\end{figure}\nThe Lorentz gas is a well known mechanical model of particle scattering that reproduces \nelectron transport in metals \\cite{lor05, dru00}.  Concerning our present focus and subject what becomes \nimportant is the fact that in the appropriate scales of time and energy the \nLorentz gas is diffusive, see \\cite{sza00, kla07} and references therein. \nMoreover, when the system is connected to reservoirs, \nthe ``external'' perturbations \\eqref{rep} become very natural. \nThus, one can expect that the response for the density profile follows the zero range process \nas studied in the previous sections.  \nWe have performed \nextensive numerical experiments in such model to corroborate our expectations. \\\\\n\nTo be more precise, consider the two-dimensional slab containing a Lorentz gas illustrated in Fig \\ref{fig:lgs}. \nThere is a cloud of point particles which \nmove freely in the space between the array of scatterers and collide elastically with them. \nThe vertical coordinate is periodic and in the horizontal direction \nthere are left and right boundary walls, which connect the system to thermo-chemical reservoirs, \ncharacterized by chemical potentials $\\mu_\\ell, \\mu_r$ with uniform \ntemperature $T$. In terms of the mean reservoir density $\\rho $, the reservoir chemical potential \n$\\mu \\propto T\\ln (\\rho\/T)$. During time evolution, as a particle hits the boundaries, it moves into a reservoir; \nadditionally, other particles are emitted to the system at given rates \n$\\pi _{\\ell,r}\\sim \\rho _{\\ell,r}\\sqrt{T}$ and incoming velocities taken \nfrom Maxwellians at temperature $T$. \nThe complete model of stochastic thermal and particle reservoirs connected to \nthe Lorentz slab is \nborrowed from a similar work on a modified Lorentz gas; a detailed description about the choice \nof emission rates and chemical potential, temperature and incoming particle velocities from the reservoirs \ncan be found there \\cite{splg03}. In our present case we are interested in independent particles \nwith constant temperature $T$; with this setting in mind the planar Lorentz gas slab of \nFig. \\ref{fig:lgs} evolves to a nonequilibrium stationary state with diffusive transport of particles, \nwhenever $\\Delta \\mu \\equiv \\mu_\\ell - \\mu_r\\neq 0$.\\\\\n\nWe now wish to connect this model with the zero range model. \nThe rates at which particles enter (like $\\alpha$ and $\\delta$ in \nthe zero range process) are controlled externally by the nominal reservoir \ntemperature and the chemical potentials. \nFor the rates at which individual particles leave, that is only controlled by the temperature and the local \n(boundary) density. Thus, one is under perturbation \\eqref{rep}.  We have therefore proceeded to test \nwhether our boundary driven Lorentz gas satisfies the response as predicted by the Kubo-formula \\eqref{eqform} \nindependent of the distance to equilibrium.  The simulation result is indeed positive. \n\n\\begin{center}\\begin{figure}\\includegraphics[scale=1.2]{neqfdtsol.eps}\n\\caption{The response in the number of particles \n$\\mathcal{N}$ of the driven Lorentz gas when both reservoir chemical potentials are shifted, \n$\\mu_{\\ell,r}\\rightarrow \\mu_{\\ell,r} + h$. \nThe full curve is the Kubo-equilibrium formula, calculated with $\\Delta \\mu \/T=0.2$, $T=150$. \nThe dotted curve \ncorresponds to direct measurements of $\\mathcal{N}$ while performing the shift at $t=0$. These curves are \nobtained from averages over $1.5\\times 10^6$ initial conditions. Also in the plot, \nthe crosses (blue) show the response obtained by solving \nthe diffusion equation $\\partial _t \\rho(x,t)= \\lambda \\partial _{xx}\\rho(x,t)$, with $\\lambda $ diffusivity, \ntaking the stationary unperturbed particle density profile as initial condition, and perturbed densities \nas boundary conditions. \n}\\label{fig:responselgs}\n\\end{figure}\\end{center}\n\nWe have carried out nonequilibrium molecular dynamics simulations of the system in Fig. \\ref{fig:lgs} and have taken \nas observable the total number ${\\cal N}$ of particles in the system. \nThe perturbation simply consists of modifying the reservoir densities, so that the entrance rates $\\pi _{\\ell,r}$ \nare shifted by the same small amount $\\pi _{\\ell,r} \\rightarrow \\pi _{\\ell,r}e^{h\/T}$ (depending also on the constant \ntemperature). The response of $\\mathcal{N}_t$ to this perturbation is shown in Fig \\ref{fig:responselgs} \nfor a nonequilibrium stationary regime with moderate driving of $\\Delta \\mu\/T=0.2$ and $T=150$, which relaxes to \na new stationary regime with different chemical potentials. The perturbation \nis applied at time $t=0$ and the system is then observed in transient states which evolve to the new stationary state. \nEach response curve consists of averages over an ensemble of $1.5\\times 10^3$ initial conditions from the steady regime; \nrelaxation to the final (stationary) state takes about $9.15\\times 10^4$ collisions in the gas. \nThe response for a similar setting with a higher driving $\\Delta \\mu \/T = 2.0$, and using either \nof the terms in \\eqref{kuboform}, gives similar outcomes: indeed we see that \nthe Kubo-relation \\eqref{eqform} follows no matter how far from equilibrium we are. That is not surprising because of the \nindependence of the particles; actually we can predict all density \nresponses simply from solving the linear diffusion equation. This is also shown in \nFig. \\ref{fig:responselgs} with the curve in crosses. Yet, one must note that this interesting example \nis just a special case of what happens more generally \nin the zero range model (possibly showing non-linear hydrodynamics).\n\n\\section{Conclusions}\nOne of the less understood facts of nonequilibrium physics is that the regime of linear response around equilibrium appears \nto extend sometimes quite beyond its theoretical boundaries.  Depending on the situation, that is the case for certain \ntransport equations like the Fourier or even sometimes Ohm's  law, but also for the more general regime of hydrodynamics where local \nequilibrium often appears to be a very good approximation.  In nonequilibrium and irreversible thermodynamics, \nGreen-Kubo relations and general principles like the minimum\/maximum \nentropy production principle often continue to work \nand are used beyond their theoretical limits of validity.  \n\nIn fact, one of the reasons for not having yet an established nonequilibrium statistical mechanics may well be the lack of \nurgent questions as irreversible thermodynamics continues to work surprisingly well in a large range of transport and \nrate processes in physical or chemical systems.  Much of standard thermodynamics can even be mimicked for relatively \nsmall systems without feeling the urge for new concepts beyond those available in close-to-equilibrium regimes.  \nOnly with turbulence and very-far-from-equilibrium processes where new phenomena such as pattern formation and \nself-organization appear, do we really see major modifications with respect to the traditional approach. \\\\\n\nIn this paper we have studied response in the nonequilibrium zero range process, \ngiving explicit expressions of the entropic and frenetic terms in which such response is formally decomposed.\nThat was done for various types of perturbations to the boundary rates.  We have found systematic contributions\nof correlation functions with the dynamical activity to correct in general the Kubo-equilibrium formula.\nThere are in particular modified Green-Kubo relations where the current and the dynamical activity complement their responses.\nThere is however also an important case of ``external'' perturbations where the response retains the equilibrium form; \nthat can also be checked for the driven Lorentz gas, which is a microscopic mechanical model.\nWe may expect similar behavior for other boundary driven systems with diffusive transport \nfor which the analogy with certain aspects of the zero range process can be argued. \\\\\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}