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+{"text":"\\section{Why quantum metrology ?}\n\nMetrology, as the science of measurements, has had an immense impact on the world we live in today. It has improved the quality of peoples' lives by enabling advances in areas such as navigation, telecommunication, transport and medicine~\\cite{Quinn:2005aa} as well as facilitating trade, commerce and even high finance. It encapsulates a wide range of aspects, from defining the units of measurement and realising them in practice, to understanding phenomena and the fundamental limits that can be achieved in the precise estimation of parameters. These fundamental limits are set by the underlying theory of Nature -- quantum mechanics and therefore provide deep insights into the theory of quantum mechanics and hence Nature itself.\n\n\nMetrology is thus the science -- and art -- of devising schemes that extract as precise as possible an estimate of the parameters associated with a system. A typical estimation process can be divided into three stages: probe preparation, interaction with the system and probe readout. For a given interaction with the system, the choice of probe states and measurements determines the precision with which one can measure the parameters of interest.  Appropriately chosen probe states ensure that the maximum amount of information about the parameters is encoded onto the probe, and appropriately chosen measurements maximise the amount of information that can be then extracted from the probe after acquiring this information. Even the most astutely designed and meticulously implemented scheme however, is affected by errors in the estimation process. The errors can either be systematic or statistical. Statistical errors of a stochastic nature can be reduced through repeated interactions between the probe and the system (corresponding to $M$ independent measurements), resulting in the typical statistical error (in standard deviation) scaling of $M^{-\\frac{1}{2}}$. The origins of this scaling lies in the central limit theorem from probability theory, and is possible classically without the invocation of quantum mechanics. Given a probe of size (such as the number of particles or modes, energy) $N,$ the best classical possible scaling is the so-called standard quantum limit (SQL)\\footnote{Although the SQL and the central limit theorem have the same quadratic dependence, they are of entirely different origins.}, whose error also scales as $N^{-\\frac{1}{2}}$~\\cite{Caves:1980aa}.\n\nOnce the stochastic noise is suppressed, quantum mechanics is the ultimate -- and most fundamental -- barrier to the precision of an estimation scheme. This inevitable limit is set by the quantum vacuum fluctuations and can only be overcome by invoking uniquely quantum mechanical techniques.  Quantum probes endowed with such non-classical correlations can attain the so-called Heisenberg limit, identified by a  $N^{-1}$ scaling in the standard deviation~\\cite{Braunstein:1992aa,BRAUNSTEIN:1994aa}. This enhanced scaling, leading to a more precise estimation, is at the root of the appeal of quantum metrology. Quantum metrology can find application in scientific areas from astronomy -- detection of gravitational waves, to biology -- imaging of biological samples sensitive to the total illumination~\\cite{Caves:1981aa,Taylor:2014aa}. It could be relevant for magnetic, electric, and gravitational field sensing, and more precise clock synchronisation protocols~\\cite{Preskill2000,Chuang2000,Giovannetti:2004aa}.\n\nQuantum metrology thus seeks scenarios where non-classical resources can provide improvements in the parameter estimation over the classical strategies and tries to identify the measurements that achieve quantum enhanced precisions. It must be understood that the quantum improvement can be availed only after all classical sources of stochastic noise have been suppressed. The most prominent example of this endeavour is the quest for the detection of gravitational waves using laser interferometry~\\cite{Adhikari2014,Abbott2016}.\n\n\\subsection{Classical metrology}\n\nEstimation theories can be categorised into global and local theories. In the global case, the parameter can be completely unknown and the estimation protocol enables finding the parameter of interest to some precision. Schemes based on the Bayesian theory, where the parameter is a random variable distributed according to some prior probability distribution, can be considered global~\\cite{Demkowicz-Dobrzanski:2015aa,Toth:2014aa}. Examples of Bayesian precision bounds include the Ziv-Zakai and Weiss-Weinstein bounds~\\cite{Ziv1969,Bell1997,Weinstein1988}. On the other hand, in some circumstances, we may already have a good knowledge of the interval where the true parameter value lies. In such cases, local approaches could be beneficial to further improve the precision and accuracy of the parameters of interest. Examples of local precision bounds include the Barankin~\\cite{Barankin1949}, Holevo~\\cite{Holevo:1982aa} and Cram{\\'e}r-Rao bound. The last mentioned bound is the main topic of this article.\n\nA statistical quantity capturing the performance of an estimation process is the variance of the estimator. A crucial result from probability theory, the Cram{\\'e}r-Rao bound (CRB), states that the variance of an (unbiased) estimator is lower bounded by the inverse of the Fisher information (FI). The FI is a function of the probability distribution obtained at the end of an estimation process, and is of independent interest in probability theory, information theory, and information geometry~\\cite{Amari2000}. More precisely, it is a distinguishability metric which provides a statistical distance on the space of probability distributions. It tells us how easily we can distinguish neighbouring probability distributions when separated by an infinitesimally small amount of the parameter value characterising the distributions. Therefore, the FI\\footnote{The FI means different things in different contexts and the relationship between the FI and the entropy is well understood in the classical case. While the entropy is related to the volume of the typical set, the FI is related to the surface area of the typical set~\\cite{Cover2006}.} captures the amount of ``information\" about a given parameter in a probability distribution.\n\nEstimators saturating the CRB are referred to as efficient. One of the difficulties of saturating the CRB is related to finding such efficient estimator whose existence is not guaranteed as explained in Theorem~(\\ref{th:CRBsaturation}) at the beginning of Sec. (\\ref{sec:2}), where we also discuss its asymptotic saturability. In principle, single as well as multi-parameter CRB can always be saturated.\n\n\\subsection{Back to quantum metrology}\nIn the quantum setting of the problem, the probability distribution depends on the input probe state (described by a density matrix) and the measurement (described by the positive operator valued measure). In this framework, maximising the FI over all valid quantum measurements leads to the quantum Fisher information (QFI) -- a distinguishability metric on the space of quantum states which quantifies the maximum amount of ``information\" about a parameter attainable by a given probe state~\\cite{Holevo:1982aa,Helstrom:1976aa,BRAUNSTEIN:1994aa}. Lower bounding the variance of an unbiased estimator with the inverse of the QFI is the so called quantum Cram{\\'e}r-Rao bound (QCRB). Once the assumptions underlying local estimation theory are clearly stated, the QCRB provides the first step in understanding the fundamental limits of quantum enhanced metrology.\n\nSingle parameter QCRB can in principle be always saturated. However, an additional saturability problem arises in the quantum multi-parameter estimation due to the possible non-commutativity of quantum measurements. This additional aspect of quantum multi-parameter estimation is what makes quantum metrology interesting. The lower bound on the precision set by the QCRB can not be always attained when we try to simultaneously gain knowledge about multiple parameters. This is discussed further in Sec. (\\ref{sec:sat}).\n\n\n\nTo illustrate the principle of quantum enhanced metrology, we consider single phase estimation in a noise-free environment using N00N states. For phase estimation protocols in the QCRB setting, states with high photon number variance are preferred since they maximise the QFI\\footnote{QFI for a unitary parameter is proportional to the variance of the generator $\\hat{G}$ and is given by $4M\\left\\langle\\Delta{\\hat{G}}^2\\right\\rangle$, where $M$ is the number of independent measurements. For the phase shift parameter, $\\hat{G}$ is the number operator $\\hat{n}$ and therefore as far as the QFI is concerned states with high photon number variance are preferred for phase estimation protocols. It is worth noting that the definition of the QFI assumes that the input state has a bounded variance and therefore it is not applicable for all probe states for instance $|\\psi\\rangle =\\frac{\\sqrt{3}}{2}\\sum_m 2^{-m} |2^m\\rangle$. In such circumastances, other bounds, such as the Ziv-Zakai bounds, are more suitable and find applications in particular for waveform estimation~\\cite{Tsang:2014aa}.}. N00N states, $|\\psi\\rangle=(|N0\\rangle+|0N\\rangle)\/\\sqrt2$, exhibit this property with photon number variance of $N^2\/4$ which gives the QFI of $MN^2$ and QCRB of $1\/MN^2$ ($M$ is the number of independent measurements), and therefore they attain the desirable Heisenberg scaling. It is important to emphasise that N00N states achieve the QCRB only when one considers unbiased estimators and in particular if sufficient prior knowledge about the parameter value is available. The latter is due to the fact that this state is periodic in phase with period of $2\\pi\/N$ and therefore the phase needs to be known within this interval~\\cite{Hall2012}. Further, N00N states are very fragile to loss. Loss of a photon in any of the two superposition elements quickly collapses the state onto the remaining element which completely destroys the superposition state. In this sense, states with more superposition elements, although smaller photon number variance, such as the Holland Burnett states~\\cite{Holland1993} are more suitable. Additionally, N00N states are very hard to prepare and all the experimental realizations require post selection for $N>1$. Another class of states important for quantum metrology, which are more feasible to prepare experimentally, are squeezed coherent states. Squeezed coherent states exhibit reduced variance in one of the field quadratures in the expense of increased variance in the other quadrature so that to satisfy the Heisenberg uncertainty relation. This also offers enhancements in phase estimation protocols and one of the important applications includes detection of gravitational waves~\\cite{Blair2016}. The topic of quantum enhanced estimation of a single parameter, typically phase, has been studied in great detail, and we direct the reader to the several extensive and excellent reviews~\\cite{Giovannetti:2004aa,Giovannetti:2011aa,Demkowicz-Dobrzanski:2015aa,Toth:2014aa, Paris2009}. \n\nBefore we progress on to the topic of multiple parameters, we must note that the promise of quantum enhancements in precision metrology is limited by the presence of noise, such as dephasing and dissipation in any experiment. Loss is an important limiting factor in photonic experiments, whereas phase diffusion typically plays a crucial role in experiments involving spins in atoms, ions, and vacancy centres. Although it is known that the Heisenberg scaling would eventually vanish in the presence of noise to match the SQL~\\cite{Dobrzanski2013}, it is still possible to gain advantage over the classical schemes. The attainable precision in such realistic cases is an area of active investigation and some progress has been made in obtaining general upper bounds for the QFI corresponding to a single phase estimation in the presence of noise~\\cite{Escher:2011aa,Escher:2012aa, Demkowicz-Dobrzanski2012, Huelga1997, Matsuzaki2011, Macieszczak2015, Smirne2016}.\n\n\\subsection{Why multiple parameters?}\n\nFor equivalent resources, simultaneous quantum estimation of multiple phases or in general, parameters corresponding to non-commuting unitary generators, provides better precision than estimating them individually~\\cite{Humphreys:2013aa, Baumgratz:2015aa}. This has generated interest in multi-parameter quantum metrology in a variety of scenarios and contexts~\\cite{Spagnolo:2012aa,Vaneph:2013aa,Yue:2014aa,Zhang:2014aa,Gao:2014aa,Tsang:2014aa,Zhang:2014ab,Yao:2014aa,Zuppardo:2015aa,Kok:2015aa,Klimov2005}. However, the myriad motivations for studying multi-parameter quantum metrology are deeply interleaved and intertwined. Nevertheless, the following presents a broad delineation of at least three broad seams of interest:\n\n\\begin{enumerate}\n\\item \\textit{Physics}: The measurements that maximise the QFI corresponding to multiple parameters need not necessarily commute. Thus, the enhancements in precision metrology promised by quantum mechanics might eventually be thwarted by quantum mechanics. It is of principal interest in quantum information theory as it explores the information extracting capabilities of quantum measurements, and provides a rich new testing bed for understanding the nature of quantum measurements in great generality. High precision estimation is also beginning to play a role in the detection of novel phenomena such as gravitational waves~\\cite{Abbott2016} and should lead to discoveries yet unknown in other areas of fundamental physics. \n\n\\item \\textit{Mathematics}: The quantum Fisher information matrix -- the multi-parameter extension of the QFI -- is a `metric' on the space of quantum states. Although it is not unique as in the classical case, it is minimal among all monotone metrics~\\cite{Petz:1996aa}. This makes it not only a quantity of inherent interest in quantum metrology, but also capable of unlocking novel features of the space of quantum states whose study underlies all of quantum information theory, non-commutative geometry and quantum information geometry~\\cite{Ercolessi2012, Ercolessi2013, Contreras2016}. \n\n\\item \\textit{Technology}: Numerous high level applications intrinsically involve multiple parameters. Quantum enhanced schemes for imaging~\\cite{Tsang2016}, microscopy, spectroscopy to high precision sensors for classical electric, magnetic, gravitational fields cannot be developed without a clear understanding of multi-parameter quantum metrology. Eventually, highly precise characterisation of components for fault-tolerant quantum technologies~\\cite{Martinis:2015aa} and quantum information science~\\cite{Childs:2000aa} might also benefit from multi-parameter quantum metrology.\n\n\\end{enumerate}\nTo make some of these motivations more concrete and delve into the status of the field in greater details, we first describe the problem mathematically. As we point out along the way, the inception of multi-parameter quantum metrology as a field is just as rich as its future prospects. \n\n\n\n\\section{\\label{sec:2}Multi-parameter estimation}\n\nThe central task here is of estimating a set of parameters $\\bm{\\theta}=(\\theta_1,\\cdots,\\theta_d)^T \\in \\mathbb{R}^d.$ The precision of any estimator $\\bm{\\hat{\\theta}}$ of $\\bm{\\theta}$ is given by the mean square error $\\mathbb{E}\\left[(\\bm{\\theta}-\\bm{\\hat{\\theta}})(\\bm{\\theta}-\\bm{\\hat{\\theta}})^T\\right],$  the expectation value of squared difference. For unbiased estimators, the mean squared error is equal to the covariance matrix $\\mathrm{Cov}(\\bm{\\hat{\\theta}}).$ One of the central results of the classical probability theory, the Cram\\'er-Rao inequality, places a lower bound on the covariance matrix\n\\begin{equation}\n\\mathrm{Cov}(\\bm{\\hat{\\theta}}) \\geq C\\left(\\bm\\theta\\right)^{-1}\n\\end{equation} \nwhere $C\\left(\\bm\\theta\\right)$ is the Fisher information matrix with elements \n\\begin{equation}\n\\label{eq:FI}\n\\left[C\\left(\\bm\\theta\\right)\\right]_{ij}=\\mathbb{E}\\left[\\left(\\frac{\\partial}{\\partial \\theta_i}\\log p(x,\\bm{\\theta})\\right)\\left(\\frac{\\partial}{\\partial \\theta_j}\\log p(x,\\bm{\\theta})\\right)^T\\right],\n\\end{equation}\n which depend on the probability distribution $p(x,\\bm{\\theta})$ of the outcomes $x$. The Cram\\'er-Rao inequality applies only to well behaved probability distributions which satisfy the following regularity condition,\n\n\\begin{equation}\n\\label{eq:RegularityCondition}\n\\mathbb{E}\\left[\\frac{\\partial}{\\partial \\theta_i}{\\log{ p(x,\\bm{\\theta})}}\\right]=0\\hspace{10mm}\\text{for}\\hspace{2mm}\\text{all}\\hspace{2mm}\\bm{\\theta},\n\\end{equation}\n \nwhere the expectation value is taken with respect to $p(x,\\bm\\theta)$. Additionally, the estimator $\\bm{\\hat\\theta}$ saturating the CRB is locally unbiased and therefore must satisfy\n\\begin{equation}\n\\label{eq:UnbiasedEstimator}\n\\mathbb{E}\\left[\\bm{\\hat{\\theta}}\\right]=\\bm\\theta\n\\end{equation}\nin the neighbourhood of the true value of the parameters\\footnote{~A CRB with biased estimators can also be defined \\cite{Cover2006}.}.\nThe CRB is proven using the Cauchy-Schwarz inequality~\\cite{Helstrom:1976aa}. The condition for existence of a locally unbiased estimator saturating the bound is stated in Theorem~(\\ref{th:CRBsaturation}). If such a locally unbiased estimator exists, the bound can always be saturated using the maximum likelihood estimator. Although this saturation by maximum likelihood estimator is in principle asymptotic in the number of experiments, it has found widespread practical use since a finite number of data points usually provides satisfactory performance~\\cite{Braunstein1992}. However, identifying the conditions for the saturation of the CRB is an intricate topic with a variety of technicalities. Most deal with the differentiability of the probability distribution function $p(x,\\bm{\\theta}),$ the most common being the notion of the differentiability of the quadratic mean~\\cite{LeCamYang2000}. The issue of finding the estimator saturating the CRB and the associated saturability is not of quantum origin and cannot be resolved using quantum mechanics.\n\n \n\\begin{theorem}\n\\label{th:CRBsaturation}\nGiven that the probability distribution satisfies Eq. (\\ref{eq:RegularityCondition}), a local unbiased estimator $\\bm{\\hat\\theta}$ saturating the CRB exists iff~\\cite{Kay1993},\n\\begin{equation}\n\\label{eq:SaturationCondition}\n\\frac{\\partial}{\\partial\\bm\\theta}\\log{ p(x,\\bm{\\theta})}=C\\left(\\bm\\theta\\right)\\left({\\bm{\\hat{\\theta}}}-\\bm{\\theta}\\right).\n\\end{equation}\n\\end{theorem}\n\nThe quantum version of estimation theory begins with a quantum state $\\rho_0$ which undergoes an evolution depending on $\\bm{\\theta}$. The resulting state $\\rho_{\\bm{\\theta}}$ is measured by a set of positive operator valued measures (POVMs) $\\{\\Pi_x\\},$ leading to probabilities given by $p(x,\\bm{\\theta}) = \\tr{\\rho_{\\bm{\\theta}}\\Pi_x}.$ All the information about the parameters $\\bm{\\theta}$ is now encapsulated in the probability distribution $p(x,\\bm{\\theta}),$ and can be used to estimate the parameters with a precision given by the classical Fisher information in Eq.~(\\ref{eq:FI}). However, the Fisher information now also depends on the POVMs $\\{\\Pi_x\\},$ which stands in the way of obtaining a fundamental quantum limit to the covariance $\\mathrm{Cov}(\\bm{\\theta}).$ The mathematical task is thus that of maximising the classical Fisher information over all POVMs giving rise to the QFI, and the conceptual challenge lies in extending the notion of a derivative to the space of quantum states. \n\n\\subsection{Zeitgeist }\n\\label{sec:zeit}\n\nIn 1967, Helstrom defined a family of operators  $L_i$ called symmetric logarithmic derivatives (SLDs) to capture the notion of the differential of a quantum state as\n\\begin{equation}\n L_i \\rho_{\\bm{\\theta}} + \\rho_{\\bm{\\theta}} L_i = 2 \\frac{\\partial \\rho_{\\bm{\\theta}}}{\\partial \\theta_i}\n\\end{equation}\nleading to the multi-parameter quantum Cram\\'er-Rao bound (QCRB)~\\cite{Helstrom:1967aa,Helstrom:1968aa}\n\\begin{equation}\n\\label{eq:QCRB}\n\\mathrm{Cov}(\\bm{\\theta}) \\geq Q^{-1},\n\\end{equation}\nwhere $ Q_{ij} = \\tr{\\rho_{\\bm{\\theta}}(L_iL_j + L_jL_i)}\/2$ is called the quantum Fisher information matrix (QFIM)\\footnote{This matrix is sometimes referred to as the (symmetric) quantum Fisher information, Helstrom information, Helstrom matrix.}. He showed that the individual parameter $\\theta_i$ can be estimated with a variance lower bounded by the inverse of $\\tr{\\rho_{\\bm{\\theta}} L_i^2},$ and a POVM attaining this precision is given by the eigenvectors of the SLD $L_i.$ He did not consider the collective saturation of the bound for all the parameters simultaneously, but identified the central problem in multi-parameter quantum estimation theory -- that the optimal POVMs corresponding to different $L_i$ need not necessarily commute. \n\nIn 1972, Belavkin in the Soviet Union first exploited the Cram\\'er-Rao bound to formulate generalised Heisenberg uncertainty principle for quantities such as time and energy~\\cite{Belavkin:1976aa}, extending the early work of Mandelstam and Tamm~\\cite{Mandelstam1945}. To that end, he defined the right logarithmic derivative (RLD) $R_i$ as~\\cite{Belavkin:1976aa}\n\\begin{equation}\n\\rho_{\\bm{\\theta}} R_i = \\frac{\\partial \\rho_{\\bm{\\theta}}}{\\partial \\theta_i}.\n\\end{equation}\nIn 1973, Yuen and Lax defined the same quantity to study the attainment of the multi-parameter QCRB with the family of coherent states in thermal noise~\\cite{YUEN:1973aa}. They showed that  to saturate the multi-parameter quantum Cram\\'er-Rao inequality, it may sometimes be necessary to include the possibility of non-Hermitian operators (this is done by considering measurements on a larger system). This is a consequence of the fact that while the SLDs are guaranteed to be Hermitian, the RLD need not be so. Allowing for a non-Hermitian RLD, the estimation of two real parameters can be recast as the estimation of one complex parameter. Furthermore, the Cram\\'er-Rao inequality based on the RLD may not be attainable, even in the single-parameter scenario as the optimal measurements may not be measurable. In 1974, Helstorm and Kennedy studied non-commuting observables in the multi-parameter setting and developed the notion of the most informative bound~\\cite{HELSTROM:1974aa}. Holevo later expanded the results of Belavkin, and Yuen and Lax to the estimation problem of the expectation parameter of family of quantum Gaussian states~\\cite{Holevo:1982aa}, including those involving the RLD. He also obtained a lower bound on the mean square error which can be applied in great generality, and is now called the Holevo bound~\\cite{Holevo:1982aa}.\n\nAlso in 1968, Braginskii realized that the expected amplitude of gravitational wave-induced oscillations of a bar detector signal mode would be on the order of the zero point oscillations of this mode, as predicted by quantum mechanics. Thus, in order to observe gravitational waves, one has to treat a detector quantum-mechanically and consequently, there will be a quantum back action, setting a limitation on the achievable sensitivity, the SQL~\\cite{Braginskii1968}. By the late 1970s, Braginskii and coworkers were seeking different detectors such as ground-based optical interferometers to circumvent the SQL in gravitational wave detectors~\\cite{Braginsky1978}. By 1980, Caves had shown that the limits to the precision of phase estimation in an interferometer is set by the vacuum fluctuations entering its empty port~\\cite{Caves1980}. \n\nHelstrom, Kennedy, Lax and Yuen had been interested in the limits of optical communication engineering and radar systems. The eventual quantum nature of the electromagnetic radiation had led them to quantum estimation theory. Belavkin and Holevo were largely driven towards a deeper understanding of quantum mechanics and quantum measurements. The designers of gravitational wave detectors, and later interferometers in the late 1970s and early 1980s headed into quantum estimation theory to better understand the ultimate limits of their instruments. In Japan, information geometry was being developed in the 1980s. Information geometry applies the methods of differential geometry to probability theory by considering probability distributions as points on a Riemannian manifold, with the Riemannian metric being given by the Fisher information in Eq.~(\\ref{eq:FI})~\\cite{Amari2000}. And the methods were ready to attack the probability distributions arising from quantum systems by the middle of the 1990s. \n\nMulti-parameter quantum metrology has thus emerged as the conflux of several disparate streams of scientific objectives and aspirations. \n\n\\section{Multi-parameter quantum metrology}\n\n\nIn 1994, Braunstein and Caves brought the methods of quantum estimation theory for a single parameter to quantum physics~\\cite{Braunstein:1992aa,BRAUNSTEIN:1994aa}. The main contribution of this work lies in the separation of the classical and quantum optimisation necessary for the saturation of the quantum Cram\\'er-Rao bound.  In the same year, Fujiwara addressed also the same problem, but limited to pure states~\\cite{Fujiwara:1994ab,Fujiwara:1995aa}. He also addressed the theory of multi-parameter estimation for pure states based on the RLD~\\cite{FUJIWARA:1994aa,Fujiwara:1999aa}. In 1995, Massar and Popescu constructed an optimal measurement to determine two parameters that identify a specific pure state, and also showed that the optimal measurement is an entangled measurement over all $N$ probes~\\cite{Massar:1995aa} in an answer to a question by Peres and Wootters~\\cite{Peres:1991aa}. \n\nIn 1996, the mathematical agitation between the SLD and RLD bounds that afflicted the multi-parameter quantum metrology was dramatically resolved by Petz and coworkers~\\cite{Petz1996,Petz:1996aa}.  They show that all stochastically monotone Riemannian metrics are characterized by means of operator monotone functions and prove that there exist a maximal and a minimal among them. In particular, the minimal one is none other than that given by the QFIM. Invoking methods from operator theory, these results endowed the QFIM with a new fundamental character. In 1997, another important result was obtained -- Matsumoto showed that a multi-parameter quantum Cram\\'er-Rao inequality can always be saturated for pure states~\\cite{Matsumoto:1997aa,Matsumoto:2002aa}\\footnote{Parts of the Matsumoto's thesis appear in chapter 20 of Ref.~\\cite{Hayashi:2005aa}.}.\n\nA natural context in which multi-parameter quantum metrology has an operational interpretation is quantum state estimation. Since the multi-parameter QCRB is attainable only when the SLDs commute in the expectation value (See Sec.~\\ref{sec:sat}), Holevo~\\cite{Holevo:1982aa} obtained a lower bound on the MSE but it remained an open problem whether this bound is achievable for mixed states. In the last few years, Gu\\c{t}\\u{a} and coworkers have developed the quantum local asymptotic normality theory for quantum state estimation, and one of its consequences (up to some technicalities) is the achievability of the Holevo bound~\\cite{Kahn2008,Guta2008,Kahn2009}. \n\n\\subsection{Unitary parameters}\n\nThe estimation of a single phase has always been the most ubiquitous form of the problem from a physicists' perspective. This has been driven by the central role that interferometry, which measures a relative phase, plays in numerous areas of physics, and spurred on by the impetus to improve the sensitivity of gravitational wave interferometers~\\cite{Danilishin2012}. More generally,  estimating a unitary operator with fidelity as the figure of merit has been studied~\\cite{Acin:2001aa}. Similarly, simultaneous estimation of multiple phases was considered in~\\cite{Macchiavello2003}. Further, the strategy requires a reference system with entanglement between the system and a reference system. This highlights an important issue in the field -- that for different cost functions different measurements will be optimal~\\cite{Barndorff-Nielsen:2000aa}. This work also discussed the non-attainability of the multi-parameter Cram\\'er-Rao bound because the optimal measurements might not commute. In the estimation of commuting unitary operators $U,$ Ballester showed that no advantage is afforded by using entangled input states~\\cite{Ballester:2004ab}. Note that in this setup, the quantum system was divided into two parts, one of which sensed the unitary, while the other half remained untouched. In the same setup, entangled measurements enabled an improvement of the precision by a factor $2(d+1)\/d$, where $d$ is the dimension of the Hilbert space on which the unitary acts~\\cite{Ballester:2004aa}. For non-commuting unitaries, the transmission of a reference frame through a quantum channel made out of $N$ spins has been studied as a $SU(2)$ estimation problem, leading to an average error that obeys a Heisenberg scaling \\cite{Bagan:2001aa,Fujiwara:2001ab,Kolenderski08}. \n\nThe $SU(2)$ estimation problem has also been studied using methods from group theory such as equivalent representations and multiplicity spaces, showing the requirement of entanglement between spaces, where the action of the group is irreducible and spaces, where the action of the group is trivial~\\cite{Chiribella:2004aa}.  A similar result holds for the optimal Bayesian estimation of an unknown transformation with a quantum-enhanced Heisenberg scaling~\\cite{Chiribella:2005aa}.  $SU(d)$ estimation has also been studied, but the $d$ dependence of the variance was not explored~\\cite{IMAI:2007aa,Kahn:2007aa}. In \\cite{Chiribella:2006aa}, the authors discussed the joint estimation of real squeezing and displacement in phase space. They found optimal measurements for a joint estimation that maximise the likelihood function.  They also highlighted the nonunimodularity of the group as playing a vital role in the estimation process, and once again noted the value of quantum entanglement in precision estimation. The same was noted in the estimation of displacements, a complex parameter, in phase space~\\cite{Genoni:2013aa}. A recent experiment demonstrating a quantum-enhanced tomography of an unknown unitary process is outlined in~\\cite{Zhou2015}.\n\n\\begin{wrapfigure}{c}{25mm}\n  \\vspace{-1.5cm}\n    \\includegraphics[scale=0.2,viewport= 0 0 400 600,clip]{pixels.png}\n  \\caption{Phases $\\phi_i$ label the pixels in an image}\n  \\label{fig:pixels}\n\\end{wrapfigure}\n\n\n\\subsubsection{Recent advances}\n\nIn 2013, Humphreys and co-workers showed that for a fixed number of photons, the precision in estimating a certain number of phases across independent modes is better if they are estimated simultaneously rather than individually~\\cite{Humphreys:2013aa}. They also showed that the total variance decreases linearly with the number of parameters. This makes multi-parameter very attractive from a technological point of view. Of course, the investment needed to harness this advantage is the generation of entangled quantum states of an increasing number of modes. However, this could be worthwhile in imaging applications where an object can be considered as a collection of independent pixels as shown in Fig.~(\\ref{fig:pixels}). Experimental efforts have been made to estimate phases in multi-arm interferometers as first steps towards such a realisation~\\cite{Spagnolo:2012aa,Ciampini15}. The initial proposal has also been studied in realistic circumstances and although the enhancement of multi-phase estimation eventually reduces to the SQL in the presence of loss, an advantage still remains if the loss figure is not too high and robust states are employed~\\cite{Yue:2014aa}.\n\nA similar `multi-parameter' advantage, proportional to the number of parameters, was shown in Ref.~\\cite{Baumgratz:2015aa} in the estimation of fields in 3 dimensions. The technique presented applies to any number of dimensions and works in spite of the non-commutativity of the generators. This covers scenarios of interest such as magnetic, electric, or gravitational field imaging in 3 dimensions simultaneously. It is mathematically identical to the estimation of hamiltonians as in Ref.~\\cite{Skotiniotis:2015aa}, although this work does not exploit its multi-parameter aspects. This aspect was studied in the physical context of a non-demolition measurement of a Bose-Einstein condensate in a double-well optical cavity~\\cite{Zuppardo:2015aa}. The work in~\\cite{Liu2016} also investigates the multi-parameter aspect of phase estimation, but with entangled coherent states. It finds that the simultaneous estimation can provide a better precision than the independent estimation and that the entangled coherent states outperform the generalised N00N states in the equivalent estimation scenario. A case of quantum-enhanced multi-phase estimation using Gaussian inputs has been studied in~\\cite{Gagatsos2016}. The work shows that assuming equally squeezed input states and an orthogonal interferometer, the simultaneous phase estimation strategy is always better than the individual phase estimation with the figure of merit being trace of the QFI matrix.\n\n\n\\subsection{Non-unitary parameters}\n\nWhile pure states and unitary transformations have occupied most of the attention in the realm of quantum metrology, the full characterisation of a system would also require the estimation of decoherence parameters. Simultaneous estimation of all the parameters yield a better understanding of the underlying system, and include parameters such as diffusion and loss. This is an improvement on the typical strategy of estimating the decoherence in independent experiments and using that value to optimise phase estimation in the presence of decoherence~\\cite{Dorner2009,Knysh2011,Knysh:2013aa}. \n\nThe estimation of decoherence parameters is intrinsically linked to mixed states since they are the end result of a decoherent evolution. The quantum estimation theory of mixed states has been covered in the early work of Helstrom and others. However, their emphasis on coherent states avoided explicit investigation of mixed states. Braunstein and Caves also made  explicit the distinction between pure and mixed states. One of the early works within the information geometry framework was by Fujiwara~\\cite{Fujiwara:1995aa,Fujiwara:1999aa}. Optimal estimation of qubit mixed states (all the components of a Bloch vector) was studied in Refs.~\\cite{Fujiwara:2001aa,Fujiwara:2003aa,Bagan:2006aa,Hayashi:2008aa}, and  information geometry was employed to study the estimation of multiple parameters from Markovian dynamics~\\cite{Guta16} and Gaussian states~\\cite{Monras:2010aa,Monras:2011aa}. Gu\\c{t}\\u{a} and others employed local asymptotic normality for the estimation of mixed quantum states~\\cite{Kahn2009}. \n\nOne of the interesting aspects of multi-parameter estimation is the tradeoff in the attainable precisions that arises due to the possible non-commutativity of optimal measurements, for instance in the simultaneous estimation of phase and loss in optical interferometry~\\cite{Crowley:2014aa}. Such a tradeoff can also arise if the dimension of the Hilbert space is not enough to accommodate all the parameters. This was studied by Gill and Massar in 2000 in the problem of estimating parameters related to quantum state estimation~\\cite{Gill:2000aa}. This was later identified in the simultaneous estimation of phase and dephasing for qubits~\\cite{Vidrighin:2014aa}. A specific class of measurements, called Fisher-symmetric informationally complete measurements, that can saturate these tradeoffs have also been studied recently~\\cite{Li15}. \n\n\\subsection{Saturating the multi-parameter QCRB}\n\\label{sec:sat}\n\nThe multi-parameter QCRB is an inequality, and identifying the conditions of its saturation is salient to its understanding. Since, as stated in Sec. (\\ref{sec:zeit}), the SLDs corresponding to the different parameters need not commute, the saturation of the multi-parameter QCRB is not assured. This is an issue that does not arise in single parameter estimation theory, where saturation is guaranteed. For multiple parameters, a sufficient condition for the saturation is $[L_i,L_j]=0,$ the commutation of the SLDs. However, this is not the necessary condition and it is not obvious what this condition is in general. It is known that the Holevo bound can be asymptotically attained within the framework of local asymptotic normality (LAN), where the model converges to a Gaussian shift model~\\cite{Gill2013}. The precision associated with the QCRB is always smaller or equal to the precision associated with the Holevo bound. When the Holevo bound coincides with the QCRB based on SLDs, the necessary and sufficient condition is\n\\begin{equation}\n\\label{eq:lan}\n \\tr{ \\rho_{\\bm{\\theta}}[L_i, L_j]}=0.\n\\end{equation}\nIn the framework of LAN, $N$ copies of the state $\\rho_{\\bm{\\theta}}^{\\otimes N}$ tend to a locally continuous variable system -- the product of a Gaussian probability density function and a tensor product of uncorrelated single mode quantum Gaussian states. The commutation relation of the collective modes of these latter Gaussian states is given by~\\cite{Gill2013} $ \\tr{ \\rho_{\\bm{\\theta}}[X_i, X_j]}$, where $X_i$ is the collective variable. In the instances when the Holevo bound is the same as the SLD quantum Cram\\'er-Rao bound this becomes $\\tr{ \\rho_{\\bm{\\theta}}[L_i, L_j]}$. The parameters of a single mode Gaussian state can be measured simultaneously if the commutation relation vanishes. This leads to the necessity of Eq.~(\\ref{eq:lan}) for saturating the multi-parameter QCRB at least in these special circumstances. Note that the convergence to the saturation is asymptotic, and can be attained by the maximum likelihood estimator~\\cite{Geyer2012}. To the knowledge of the authors, an exact and general relationship between the Holevo bound and the QCRB is not established. However, there has been studies connecting the Holevo bound to the QCRB in special cases. Ref.~\\cite{Suzuki2015} investigates such connection for a two-parameter qubit model and gives a condition for when these two bounds are equivalent.\n\nFor quantum estimation using pure states, the multi-parameter QCRB can however be always saturated asymptotically~\\cite{Matsumoto:2002aa}. The underlying necessary and sufficient condition is still that of commuting SLDs in the expectation value, and for a general hamiltonian estimation the optimal measurement can be constructed explicitly~\\cite{Baumgratz:2015aa}.\n\n\\section{Conclusions and Outlook}\n\nThe task of quantum metrology is to obtain as precise as possible an estimate of a set of parameters using quantum probes. The choice of the measurement is a vital ingredient in this process. This is brought to the fore in multi-parameter quantum metrology. This is precisely why multi-parameter quantum metrology provides a fertile ground for understanding quantum measurements. \n\nThe potential applications of multi-parameter quantum metrology are wide beyond its appeal as a domain for a deeper understanding of quantum mechanics. It has prospective appeal in the development of quantum technology itself. In a fault-tolerant quantum computer, the qubits and their logic interactions must have errors below a threshold of $10^{-18}.$ Characterising such a system will require multi-parameter quantum metrology at the level of 1- and 2-qubit gates~\\cite{Martinis:2015aa}. Another area could be the understanding of the hamiltonians driving quantum phase transitions. Since these are zero temperature phenomena, their direct probing will necessarily require probes with minimal energy and disturbance, the kind provided by quantum metrology. Other scenarios for multi-parameter quantum metrology could include the imaging of electric, magnetic, gravitational and other fields in 3 or more dimensions, as well as multimode quantum imaging~\\cite{Kolobov1999}. All of these have fundamental scientific as well as technological applications. \n\nThe above applications, allied with the intrinsic richness and variety of the topics that touch upon multi-parameter quantum metrology, make it a topic worth pursuing. Open questions abound. One of the first should be a systematic study of multi-parameter quantum metrology in the mould of Ref.~\\cite{Giovannetti:2006aa}, where the authors discuss all four possible scenarios with respect to probe states and measurements -- classical and classical, classical and quantum, quantum and classical, and quantum and quantum. This should clarify the role of quantum correlations in circumventing the tradeoffs that arise in multi-parameter quantum metrology. Another open question is the relation of the Holevo bound to the Cram\\'er-Rao bound, and there have been some recent results for special cases such as the two-parameter qubit estimation problem~\\cite{Suzuki2015}. One very fruitful area could be the use of information geometry methods to identify the tradeoff relations in multi-parameter quantum metrology and optimal measurements necessary to saturate them. A final open challenge could be to go possibly beyond the differential approach of Helstrom and information geometry, and identify measurements that minimise the mean square error over all the parameters. \n\n\n\\section*{Acknowledgements}\n\nAD thanks Mihai Vidrighin for bringing Ref.~\\cite{Gill:2000aa} to his attention and M\\u{a}d\\u{a}lin Gu\\c{t}\\u{a}, Rafa\\l{} Demkowicz-Dobrza\\'{n}ski and Michael Hall for several interesting discussions. This work was supported, in part, by the UK EPSRC (EP\/K04057X\/2), the National Quantum Technologies Programme (EP\/M01326X\/1, EP\/M013243\/1), and the DSTL (DSTLX-1000091903). \n\n\n\n\n\n\\bibliographystyle{tADP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{SURVEY DETAILS AND CANDIDATE SELECTION PROCEDURE}\n\nOur survey consists of 4 WFI fields (deep fields) in the central part\nof the cluster observed in four medium band filters, namely 770\/19,\n815\/20, 856\/14 and 914\/27, where the filter name notation is central\nwavelength \/ FWHM in nm, and the broad band filter $R_{\\rm c}$, 26 WFI\nfields (radial fields, were chosen to extend predominantly in the\ndirection of constant Galactic latitude) with the broad bands $R_{\\rm\n  c}$ and $J$ and the bands 815\/20 and 914\/27, and 5 WFI fields\n(outward fields) with $J$, 815\/20 and 914\/27, for a total coverage of\n$\\sim$10.9 sq.  deg.  The survey extends to a detection limit down to\n0.02\\,M$_\\odot$ (10$\\sigma$) and is centred on RA=08:35:45.1\nDEC=-52:35:58.0.  Candidates were first selected based on\ncolour-magnitude diagrammes and a second selection was performed using\ncolour-colour diagrammes.  Third, astrometry was used to remove\nobjects with high proper motion.  Finally, non-candidates were\nrejected based on a discrepancy between the observed magnitude in\n815\/20 and the magnitude in this band computed with the NextGen model\nand our estimation of $T_{\\rm eff}$.\n\n\\section{RESULTS : MASS FUNCTION OF THE DEEP AND OUTWARD FIELDS}\n\nIn both cases we see an apparent rise in the number of objects below\n0.05\\,M$_\\odot$ (logM=-1.3, Fig. \\ref{fig:mf-deep-89j-fit}, but we\nconcluded that this is an artefact of residual contamination by field\nM dwarfs. This was also observed by \\cite{barrado2004}. The fact that\nwe do not see this rise in the radial fields is because they were\nobserved with both the $J$ and $R_{\\rm c}$ filters in addition to the\nmedium band filters. This provides a longer spectral baseline (better\ndetermination of the energy distributions and helps rejection of field\nM dwarfs based on observed magnitude vs. predicted magnitude from\nmodels). Another apparent rise in the mass function (MF) over the\n0.5--1.0\\,M$_\\odot$ interval (also observed for NGC~2547\n\\cite{jeffries2004}) is due to background giants. Red giant\ncontamination may be reduced by using medium bands such as 770\/19,\n815\/20, 856\/14 and 914\/27, and theoretical colours of red giants\n\\cite{hauschildt1999b}. Our spectroscopic follow-up has confirmed that\nselection based on these filters resulted in no red giant contaminants\namong a sample.\n\n\\begin{figure}\n  \\includegraphics[height=0.3\\textheight]{fig5}\n  \\caption{Filled dots represent the MF based on the four deep fields.\n    Open dots represent the MF based on the outward fields. Also, we\n    present again the 10$\\sigma$ detection limit, the MF of all fields\n    observed in $R_{\\rm c}$, 815\/20, 914\/27 and $J$ within 2.1 from\n    the cluster centre (histogram) and its log normal fit. The\n    vertical thin dotted and thin dashed line lines are the mass for\n    which saturation start to occur in the short exposures for outward\n    and deep field respectively.\\label{fig:mf-deep-89j-fit}}\n\\end{figure}\n\n\\section{RESULTS : RADIAL VARIATION OF THE STELLAR AND SUBSTELLAR\n  POPULATION}\n\nRadial variation is observed in the MF from 0.15 to 0.5\\,M$_\\odot$\n(Fig. \\ref{fig:mf-r89j}) and we argue that this is a signature of mass\nsegregation, presumably via dynamical evolution. This is consistent\nwith theoretical predictions since the age of IC~2391 is half of its\nrelaxation time (estimated at $\\sim$105\\,Myr). We do not observe a\nsignificant radial variation in the MF bellow 0.15\\,M$_\\odot$.\nAlthough this absence of radial variation of the brown dwarf (BD)\npopulation would be in agreement with the ejection scenario of BD\nformation, the fact that we do not observe a discontinuity in the MF\nacross the stellar\/substellar boundary (0.072\\,M$_\\odot$) implies that\nthe ejection formation scenario is not a significant BD formation\nmechanism in this cluster if this formation mechanism results in a\nhigher distribution velocity of BDs compared to stars\n\\cite{kroupa2003}. On the other hand, if ejection mechanism is the\nunique BD formation path, then both BDs and stars should have the same\nvelocity dispersion \\cite{bate2003}.\n\n\\begin{figure}\n  \\includegraphics[height=0.3\\textheight]{fig6}\n  \\caption{MF based on photometry for all radial fields. The\n    10$\\sigma$ detection limit is shown as a vertical dashed line. The\n    MF fit for IC~2391 from \\cite{barrado2004} and for the galactic\n    field \\cite{chabrier2003} stars are shown as thick dashed thin\n    solid lines respectively. The thick solid line is the fitted\n    lognormal function of the MF of IC~2391 (within 2.1). Dots in each\n    panel represent the MF of (left) fields within 1.5 of the cluster\n    centre, fields within the annulus from 1.5 to 2.1 and (right) the\n    MF of fields outside of 2.1. Error bars are Poissonian arising\n    from the number of objects observed in each bin. The histogram is\n    the MF for all fields within 2.1 of the cluster centre. The\n    vertical thin dotted line is the mass for which saturation start\n    to occur in the short exposures.\\label{fig:mf-r89j}}\n\\end{figure}\n\n\\section{RESULTS : SPECTROSCOPIC FOLLOW-UP}\n\nFrom our preliminary spectroscopic follow up, all 77 spectra analysed\nare M-dwarfs (field or in the cluster), demonstrating the efficiency\nof our method to avoid red giant contaminants in our photometric\nselection using medium filter. We also observed that H$\\alpha$\nemission line cannot be used as a membership criterion in IC~2391 when\ntaking fibre spectroscopy because of background contamination, if\nbackground is assumed to be uniform. About 31\\% of our photometric\ncandidates are true physical members of the cluster, where 7 are new\nspectroscopically-confirmed BD members of IC~2391 (Fig.\n\\ref{fig:ic2391-bd}).\n\n\\begin{figure}\n  \\includegraphics[width=1.0\\textwidth]{ic2391-15-20-BD-PROCEEDING}\n  \\caption{Spectra of the seven newly discovered BD members of the\n    IC~2391 cluster found in our survey. Objects are given the\n    notation IC~2391-HYDRA-ZZ-YY where ZZ is the field number and YY a\n    serial identification number (ID). \\label{fig:ic2391-bd}}\n\\end{figure}\n\n\\section{CONCLUSIONS}\n\nWe have performed a photometric survey of the open cluster IC~2391 to\nstudy the radial dependence of the MF. We have 1734 photometric\ncandidates for the outward fields, 499 from the deep fields and 954\nfrom the radial fields, which gives $\\sim$8 candidates per 100 arcmin.\nEjection formation scenario is not a significant BD formation\nmechanism if it results in a higher velocity dispersion of BDs\ncompared to stars. However, if ejection mechanism is the unique BD\nformation path, then both BDs and stars should have the same velocity\ndispersion. Variations in the colours of the main (field star) locus\nin the CMDs are due to spatial extinction-induced variations in\nbackground star contaminants. Selection based on medium filters (such\nas 770\/19, 815\/20, 856\/14 and 914\/27) resulted in no red giant\ncontaminants while a broad spectral baseline (such as the use of\n$R_{\\rm c}$ to $J$) was successful in reducing M-dwarf\nbackground\/foreground contaminants. About 31\\% of our candidates from\nour spectroscopic follow-up are physical members, from which 7 are\nBDs.  H$\\alpha$ emission line cannot be used as a membership criterion\nin IC~2391 when taking fibre spectroscopy if background is assumed to\nbe uniform (we recommend background subtraction to be performed in a\nsimilar way as the one done by \\cite{carpenter1997}, where the same\nfibers for the science targets were also used for sky subtraction but\nshifted 6 arcsec away). More results and informations are presented in\n\\cite{boudreault2008}.\n\n\n\n\n\n\n\n\\bibliographystyle{aipproc}  \n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\vspace{-3.0mm}\nThe presence of non--baryonic dark matter in our Galaxy can \nbe probed by means of different techniques which attempt to\ndetect either directly (through elastic scattering off\nnuclei) or indirectly (through products of annihilation)\nthe dark matter particles which are supposed to be embedded \nin the galactic halo. It has been shown (see, e.g., \nRef. \\cite{ringberg98}) that the sensitivities of the\npresent experiments is currently at the level required \nfor the study of one of the most appealing particle\ncandidates for dark matter, the neutralino. This implies\nthat it is now feasible to start the investigation of possible\nsignatures in the detection rates, originated\nby specific features related to the presence of the dark matter\nparticles. \n\nIn the case of direct detection, a typical signature consists \nin the annual modulation of the detection rate. \nThis effect was first pointed out in the seminal papers of \nRefs. \\cite{ann_mod_th}, where it \nwas observed that, during the orbital motion of the Earth around the Sun,\nthe change of direction of the relic particle velocities\nwith respect to the detector induces a time dependence in the \ndifferential detection rate, i.e.\n$S(E,t) = S_0 (E) + S_m (E) \\cos [\\omega (t-t_0)]$,\nwhere $\\omega = 2\\pi\/365$ days and $t_0 = 153$ days\n(June 2$^{\\rm nd}$). $S_0 (E)$ is the average\n(unmodulated) differential rate and $S_m (E)$ is the\nmodulation amplitude of the rate.\nThe relative importance of $S_m (E)$ with respect to $S_0 (E)$ for a given\ndetector, depends both on the mass of the dark matter particle and on the value of the\nrecoil energy where the effect is looked at. Typical values of $S_m (E)\/S_0 (E)$\nfor a NaI detector range from a few percent up to $\\sim$ 15\\%,\nfor WIMP masses of the order of 20--80 GeV and recoil energies\nbelow 8--10 KeV. \n\nOver the last year, the DAMA\/NaI Collaboration reported on two\ndifferent analyses of the data collected during two periods\nof data taking, obtained with an experimental set--up\nconsisting of nine 9.70 Kg NaI(Tl) detectors \\cite{dama1,dama2}.\nThe data have been analysed by employing a maximum likelihood\nmethod, which allows to test the hypothesis of the presence\nof a yearly modulated signal against a time--independent background,\nby properly considering the energy and time behaviour expected \nfor a recoil of massive WIMPs. The remarkable result of \nRefs. \\cite{dama1,dama2} is that the data support the possible presence of an \nannual modulation effect induced by WIMPS in the counting rate of the detector. \nBy combining the data of the two periods of data taking, a total\nexposure of 19,511 kg $\\times$ day has been collected. The maximum\nlikelihood analysis indicates that the hypothesis of\npresence of modulation against the hypothesis of absence of  modulation\nis statistically favoured at 99.6\\% C.L., and pins\ndown a 2--$\\sigma$ C.L. region in the plane \n$\\xi \\sigma^{(\\rm nucleon)}_{\\rm scalar}$ -- $m_\\chi$, \nwhere $m_\\chi$ is the WIMP mass, \n$\\sigma^{(\\rm nucleon)}_{\\rm scalar}$ is the WIMP--nucleon  scalar elastic\ncross section  and $\\xi = \\rho_\\chi \/ \\rho_l$ \nis the fractional amount of local \nWIMP density $\\rho_\\chi$ with respect to the total local \ndark matter density $\\rho_l$. This region is plotted in\nFig.1 as a closed dashed curve, for $\\rho_l=0.3$ GeV cm$^{-3}$.\nThe ensuing 1--$\\sigma$ ranges for the two quantities \nare: $m_{\\chi} = 59_{- 14}^{+ 22}$ GeV and \n$\\xi \\sigma^{(\\rm nucleon)}_{\\rm scalar} = 7.0_{-1.7}^{+0.4} \\times 10^{-9}$ nb \\cite{dama2}. \nThese results refer to $v_{\\rm rms}$ = 270 Km s$^{-1}$ \nfor the velocity dispersion of the WIMP Maxwellian \nvelocity distribution in the halo, $v_{\\rm esc} = 650$ Km s$^{-1}$\nfor the WIMP escape velocity and $v_\\odot = 232$ Km s$^{-1}$\nfor the velocity of the Sun around the galactic centre. We\nnotice that a variation of these velocity parameters inside their\n2--$\\sigma$ ranges of uncertainty affects the maximum likelihood\nvalue of $m_\\chi$ and $\\xi \\sigma^{(\\rm nucleon)}_{\\rm scalar}$\nby roughly $\\pm$ 30\\% and by $\\sim \\pm$ 12\\%, respectively.\n\nIn this paper, which is based on the results obtained in \nRefs. \\cite{mssm_mod,sugra_mod,ind_mod} we derive the theoretical implications of\nthe experimental data of Refs. \\cite{dama1,dama2}, assuming that the indication of the\npossible annual modulation is due to relic\nneutralinos and we show that these data are fully compatible with an\ninterpretation in terms of a relic neutralino as the major component of dark\nmatter in the Universe. We select the relevant susy\nconfigurations in two different frameworks: the low energy\nminimal supersymmetric extension of the standard model (MSSM) and\nsupergravity--inspired schemes (SUGRA models). In the latter case,\nwe will explicitly consider both a strict unification at the GUT scale \nand the possibility of deviation from universality in the Higgs sector.\nThe present analysis extends a previous analysis of ours \\cite{pinning} \nreferring to the experimental data of Ref. \\cite{dama1}.\nWe then finally also discuss how the susy configurations, selected by \nthe annual modulation data, can be investigated by\nindirect searches for relic WIMPs and at accelerators. \n\n\\vspace{-6.0mm}\n\\section{Selection of susy configurations by the annual modulation data}\n\n\\begin{figure}[t]\n\\hbox{\n\\hspace{-3.5mm}\n\\psfig{figure=idm98_fig1a.ps,width=2.45in,bbllx=60bp,bblly=222bp,bburx=530bp,bbury=675bp,clip=}\n\\hspace{-4.5mm}\n\\psfig{figure=idm98_fig1b.ps,width=2.45in,bbllx=60bp,bblly=222bp,bburx=530bp,bbury=675bp,clip=}\n}\n\\vspace{-3.0mm}\n\\caption{Scatter plot of {\\bf MSSM} configurations compatible with the annual\nmodulation data in the plane \n$m_{\\chi}$--$\\xi \\sigma_{\\rm scalar}^{(\\rm nucleon)}$. \nThe dashed contour line\ndelimits the 2--$\\sigma$ C.L. region, obtained by the DAMA\/NaI Collaboration, \nby combining together the data of\nthe two running periods of the annual modulation experiment.  \nThe solid contour line is obtained from the dashed line, which refers to \n$\\rho_l = 0.3$ GeV cm$^{-3}$, by accounting for \nthe uncertainty range of $\\rho_l$.\n(a) and (b) refer to configurations with \n$0.01 \\leq \\Omega_{\\chi} h^2 \\leq 0.7$ and with $\\Omega_{\\chi} h^2 < 0.01$, \nrespectively.}\n\\vspace{-4.0mm}\n\\end{figure}\n\n\\vspace{-3.0mm}\nIn order to discuss which region in the susy parameter space is selected by the \nDAMA data, we first convert the region delimited by the 2--$\\sigma$ \nC.L. dashed contour line of Fig.1 \ninto an enlarged one, which accounts for the uncertainty in the value of \n$\\rho_l$, due to a possible flattening of the dark matter\nhalo and a possibly sizeable baryonic contribution to the\ngalactic dark matter:\n0.1 GeV cm$^{-3} \\leq \\rho_l \\leq $ 0.7 GeV cm$^{-3}$. One then obtains the\n2--$\\sigma$ C.L. region denoted by a solid contour in Fig.1\n(hereafter denoted as region $R$).\nThe susy configurations are then selected by the requirement that \n($m_{\\chi}, \\xi \\sigma_{\\rm scalar}^{(\\rm nucleon)}) \\in R$, when\n$m_{\\chi}$, $\\sigma_{\\rm scalar}^{(\\rm nucleon)}$ and $\\xi$ are\nevaluated in the MSSM and SUGRA schemes.\nAs for the values to be assigned to the quantity $\\xi = \\rho_{\\chi}\/ \\rho_l$ \nwe adopt the standard rescaling recipe \\cite{mssm_mod}: \n$\\xi = {\\rm min} [1, {\\Omega_\\chi h^2 \/ (\\Omega h^2)_{\\rm min}}]$,\nwhere $\\Omega_\\chi h^2$ denotes the neutralino relic abundance,\ncalculated in the susy model \\cite{omega}, and $(\\Omega h^2)_{\\rm min}$\nis a minimal value compatible with observational data and with \nlarge--scale structure calculations. We use here the value \n$(\\Omega h^2)_{\\rm min} = 0.01$ \\cite{mssm_mod}.\nIn all our analyses, we consider as cosmologically acceptable \nall configurations which provide $\\Omega_\\chi h^2 \\leq 0.7$.\n\n\\vspace{-5.0mm}\n\\section{Analysis in the MSSM}\n\n\\begin{figure}[t]\n\\centerline{\n\\psfig{figure=idm98_fig2.ps,width=4.00in,bbllx=40bp,bblly=190bp,bburx=530bp,bbury=650bp,clip=}\n}\n\\vspace{-5.0mm}\n\\caption{Scatter plot of {\\bf MSSM} configurations compatible with the annual\nmodulation data in the plane \n $\\Omega_{\\chi} h^2$ -- $\\xi \\sigma_{\\rm scalar}^{(\\rm nucleon)}$.\nThe two vertical solid lines \ndelimit the $\\Omega_{\\chi} h^2$--range of cosmological interest. The two\ndashed lines delimit the most appealing interval for \n$\\Omega_{\\chi} h^2$, as suggested by the most recent observational data. \nThe hatched area is excluded by cosmology.}\n\\vspace{-4.0mm}\n\\end{figure}\n\n\\begin{figure}[t]\n\\hbox{\n\\hspace{-3.5mm}\n\\psfig{figure=idm98_fig3a.ps,width=2.45in,bbllx=60bp,bblly=222bp,bburx=530bp,bbury=675bp,clip=}\n\\hspace{-4.5mm}\n\\psfig{figure=idm98_fig3b.ps,width=2.45in,bbllx=60bp,bblly=222bp,bburx=530bp,bbury=675bp,clip=}\n}\n\\vspace{-3.0mm}\n\\caption{Explorability at accelerators of the {\\bf MSSM} configurations compatible with the annual\nmodulation data. (a): Scatter plot in the plane $m_h$ -- $\\tan \\beta$. \nThe hatched region on the right is excluded by theory. \nThe hatched region on the left is \nexcluded by present LEP data at 183 GeV. The dotted and the dashed \ncurves denote the reach of LEP2 at energies 192 GeV and \n200 GeV, respectively. The solid line represents the \n95\\% C.L. bound reachable at LEP2, in case of non discovery of a neutral \nHiggs boson. (b): Scatter plot in the plane \n$m_{\\chi}$ -- $\\tan \\beta$. The hatched region on the left is \nexcluded by present LEP data. The dashed and the \nsolid vertical lines denote the reach of LEP2 and TeV33, \nrespectively.}\n\\vspace{-4.0mm}\n\\end{figure}\n\n\\vspace{-3.0mm}\nIn our first analysis \\cite{mssm_mod} we employ the \nminimal supersymmetric extension of the standard model (MSSM)\nwhich conveniently  describes the \nsusy phenomenology at the electroweak scale, without too strong \ntheoretical assumptions.\nThe MSSM is based on the same gauge group as the Standard Model, \ncontains the susy extension of its particle content and \ntwo Higgs doublets. The free parameters of the\nmodel are: the SU(2) gaugino mass parameter $M_2$, related to \nthe U(1) gaugino mass parameter by the GUT relation\n$M_1= (5\/3) \\tan^2 \\theta_W M_2$; the Higgs--mixing \nparameter $\\mu$; the ratio of the two Higgs\nvev's $\\tan\\beta$; the mass of the pseudoscalar neutral Higgs $m_A$;\na common soft--mass parameter for all the squarks and sfermions $m_0$;\na common trilinear parameter for the third family $A$ (the other trilinear parameters\nare all set to zero).\nThe parameters are varied in the following ranges:\n$10\\;\\mbox{GeV} \\leq M_2 \\leq  500\\;\\mbox{GeV},\\;\n10\\;\\mbox{GeV} \\leq |\\mu| \\leq  500\\;\\mbox{\\rm GeV},\\;\n75\\;\\mbox{GeV} \\leq m_A \\leq  1\\;\\mbox{TeV},\\; \n100\\;\\mbox{GeV} \\leq m_0 \\leq  1\\;\\mbox{TeV},\\;\n-3 \\leq A \\leq +3,\\;\n1 \\leq \\tan \\beta \\leq 50$. \n\nOur susy parameter space is constrained by\nthe latest data from \nLEP2 on Higgs, neutralino, chargino and \nsfermion masses \\cite{LEP}. Moreover, the constraints \ndue to the $b \\rightarrow s + \\gamma$ process \nhas been taken into account, considering the \nlatest results both in the theoretical evaluation \nand in the experimental determination of the\nbranching ratio \\cite{bsgamma}.\n\nBy varying the susy parameters inside the ranges defined above,\nwe find that a large portion of the modulation region $R$ is indeed covered \nby susy configurations, \ncompatible with all present physical constraints. This set of susy states (set $S$)\nis displayed in Fig.1 with \ndifferent symbols, depending on the neutralino composition. \nIn Fig.1(a) we notice that a quite sizeable portion of region $R$ is \npopulated by susy configurations\nwith neutralino relic abundance inside the cosmologically interesting range \n$0.01 \\lsim \\Omega_{\\chi} h^2 \\lsim 0.7$. \nThus we obtain  the first main result of our analysis, i.e. \n{\\it the annual \nmodulation region, singled out by the DAMA\/NaI experiment, is largely \ncompatible with a relic neutralino as the major component of dark matter}. \nThis is certainly the most remarkable possibility. However, we also keep under\nconsideration neutralino configurations with a small contribution to \n$\\Omega_{\\chi} h^2$ (see Fig.1(b)), since also the detection of relic \nparticles with these features would provide in itself a very noticeable \ninformation. \n\nThe neutralino relic abundance $\\Omega_{\\chi} h^2$ is plotted versus \n$\\xi \\sigma_{\\rm scalar}^{(\\rm nucleon)}$ in \nFig.2. We notice that a large fraction of \nthe neutralino relic abundance falls into the restricted range \n$0.02 \\lsim \\Omega_{\\rm CDM} h^2 \\lsim 0.2$, which turns out to be the most\nappealing interval for relic neutralinos as indicated from recent observations\nand analyses on the value of the matter content of the Universe \\cite{mssm_mod}.\n\nThe properties of set $S$ relevant to searches at accelerators\nare displayed in Fig.3. Section (a) of this figure \nshows a scatter plot of set $S$ in term  of $m_h$ and $\\tan \\beta$,\nwhere it is apparent  a correlation between $\\tan \\beta$ \nand $m_h$. This is due to the fact that the rather \nlarge values\n$\\sigma_{\\rm scalar}^{(\\rm nucleon)} \\sim (10^{-9} - 10^{-8}$) nb, as required\n by the annual modulation data, impose that either the couplings are large \n(then large $\\tan \\beta$) and\/or the\nprocess goes through the exchange of  a light particle. \nThus, Higgs--exchange dominance (which turns out to occur here) and \n$\\sigma_{\\rm scalar}^{(\\rm nucleon)} \\sim (10^{-9} - 10^{-8})$ nb require a \nvery light $h$ at small $\\tan \\beta$, \nand even  put {\\it a lower bound on tan $\\beta$: $\\tan \\beta \\gsim 2.5$}. \n{From} Fig.3(a) we notice that a good deal of susy\nconfigurations are explorable at LEP2, while others will require experimental\ninvestigation at a high luminosity Fermilab Tevatron, which \nshould be capable to explore Higgs masses up to $m_h \\sim$ 130 GeV. \nIn Fig.3(b) we display the scatter plot of set $S$ in the plane \n$m_{\\chi}$ -- $\\tan \\beta$. Since the reach of LEP2 extends only up to\nthe dashed vertical line, at $m_\\chi \\simeq 50$ GeV, the exploration of the \nwhole interesting region will require Tevatron upgrades or LHC. Under favorable\nhypothesis, TeV33 could provide exploration up to the vertical solid line. \n\n\\vspace{-5.0mm}\n\\section{Analysis in SUGRA schemes}\n\n\\begin{figure}[t]\n\\hbox{\n\\hspace{-3.5mm}\n\\psfig{figure=idm98_fig4a.ps,width=2.45in,bbllx=60bp,bblly=222bp,bburx=530bp,bbury=675bp,clip=}\n\\hspace{-4.5mm}\n\\psfig{figure=idm98_fig4b.ps,width=2.45in,bbllx=60bp,bblly=222bp,bburx=530bp,bbury=675bp,clip=}\n}\n\\vspace{-3.0mm}\n\\caption{Scatter plot of {\\bf SUGRA} configurations compatible with the annual\nmodulation data in the plane \n $\\Omega_{\\chi} h^2$ -- $\\xi \\sigma_{\\rm scalar}^{(\\rm nucleon)}$. \n(a): universal SUGRA models. (b): SUGRA models with\ndeviations from universality in the Higgs sector.\nNotations are as in Fig.2.}\n\\vspace{-4.0mm}\n\\end{figure}\n\n\\begin{figure}[t]\n\\hbox{\n\\hspace{-3.5mm}\n\\psfig{figure=idm98_fig5a.ps,width=2.45in,bbllx=60bp,bblly=222bp,bburx=530bp,bbury=675bp,clip=}\n\\hspace{-4.5mm}\n\\psfig{figure=idm98_fig5b.ps,width=2.45in,bbllx=60bp,bblly=222bp,bburx=530bp,bbury=675bp,clip=}\n}\n\\vspace{-3.0mm}\n\\caption{Explorability at accelerators of the {\\bf non--universal SUGRA} configurations \ncompatible with the annual modulation data. (a): Scatter plot in the plane \n$m_h$ -- $\\tan \\beta$. (b): Scatter plot in the plane $m_{\\chi}$ -- $\\tan \\beta$. \nNotations are as in Fig.3.}\n\\vspace{-4.0mm}\n\\end{figure}\n\n\\vspace{-3.0mm}\nIn this Section we show that the susy features,\nimplied by the DAMA\/NaI data, are also compatible with more \nambitious supersymmetry \nschemes, where the previous phenomenological model is implemented in \na supergravity (SUGRA) framework, especially if the unification \nconditions, which are frequently imposed at the Grand  \nUnification (GUT) scale, are appropriately relaxed \\cite{sugra_mod}. \n\nThe essential elements of the SUGRA models employed here are:\na Yang--Mills Lagrangian, the\nsuperpotential and  the soft--breaking\nLagrangian. In this class\nof models the electroweak symmetry breaking (EWSB) is induced radiatively.\nThis supergravity  framework  is usually implemented by\nsome restrictive assumptions about unification at\n$M_{GUT}$: (i) unification  of the gaugino masses: $M_i(M_{GUT}) \\equiv m_{1\/2}$, \n(ii) universality of the scalar masses: $m_i(M_{GUT}) \\equiv m_0$,\n(iii) universality of the trilinear scalar couplings:\n$A^{l}(M_{GUT}) = A^{d}(M_{GUT}) = A^{u}(M_{GUT}) \\equiv A_0 m_0$. \nAs extensively discussed in Ref. \\cite{bere}, these conditions have strong \nconsequences for low--energy supersymmetry\nphenomenology, and in particular for the properties of the neutralino.\n\nThe unification conditions represent a theoretically attractive\npossibility, which makes strictly universal SUGRA models very predictive.\nHowever, the above assumptions, particularly (ii) and (iii), are not fully justified, \nsince universality may occur at a scale higher\nthan $M_{GUT}$, i.e. the Planck scale or string scale, \nin which case renormalization above $M_{GUT}$ weakens universality.\nWe therefore discuss the DAMA\/NaI data both in a SUGRA model \nwith strict unification conditions and in a SUGRA framework, where \nwe introduce a departure from universality in the scalar masses at $M_{GUT}$ \nwhich splits the Higgs mass parameters:\n$ M^2_{H_i}(M_{GUT}) = m_0^2(1+\\delta_i)$.\nThe parameters $\\delta_i$ will be varied in the range ($-1$,$+1$), but are taken to be\nindependent of the other susy parameters. \n\nBecause of the requirements of radiative EWSB and\nof the universality conditions, the independent susy parameters are reduced to \n(apart from the $\\delta_i$'s): $m_{1\/2}, m_0, A_0, \\tan \\beta$\nand ${\\rm sign}(\\mu)$. They are varied in the following ranges:\n$10\\;\\mbox{GeV} \\leq m_{1\/2} \\leq  500\\;\\mbox{GeV},\\;\n m_0 \\leq  1\\;\\mbox{TeV},\\;\n-3 \\leq A_0 \\leq +3,\\;\n1 \\leq \\tan \\beta \\leq 50$; the parameter $\\mu$ is taken positive.\nThe values taken as upper limits of the ranges for \n$m_{1\/2}, m_0$ are inspired by the upper \nbounds which may be\nderived for these quantities in SUGRA theories, when one requires that the \nEWSB, radiatively induced by the soft supersymmetry\nbreaking, does not occur with excessive fine tuning. The same argument\nwas also used in the previous Section in setting the upper limits on \nthe dimensional parameters of the MSSM. \n\nThe susy parameter space is constrained by the same experimental\nbounds discussed in the previous Section for the MSSM, with the additional\nconstraint arising from the limits on the bottom--quark mass $m_b$. The \nbottom mass is computed as a function of the susy\nparameters  and required to be compatible with the present \nexperimental bounds \\cite{mb}.\n\nThe susy configurations compatible with the\nDAMA data are shown in the plane \n$\\xi \\sigma^{(\\rm nucleon)}_{\\rm scalar}$ -- $\\Omega_\\chi h^2$\nin Fig.4(a) for universal SUGRA models and in Fig.4(b) for\nmodels with deviation from strict universality.\nWe notice that also in SUGRA theories a fraction of the selected susy\nconfigurations fall into the cosmologically interesting\nrange of $\\Omega_\\chi h^2$.\n\nOther qualifications for the configurations which lie inside the\nregion $R$, which are relevant for\nsearches at accelerators, concern the ranges for the $h$--Higgs boson mass, \nthe neutralino mass and the lightest top--squark mass. In the case\nof universal SUGRA models, we find:\n$m_h \\lsim 115$ GeV, 50 GeV $\\lsim m_{\\chi} \\lsim$ 100 GeV,\n200 GeV $\\lsim m_{\\tilde t_1} \\lsim$ 700 GeV and \n$\\tan\\beta \\gsim 42$. For deviations from universality, the situation\nis shown in Fig.5 where we notice that the sample of representative \npoints covers a slightly wider domain.\nThe ranges of the Higgs and neutralino masses\nare similar to those already found in the universal case,\nbut now $\\tan\\beta$ extends to the interval $10 \\lsim \\tan \\beta \\lsim 50$,\ninstead of being limited only to very large values.\n\n\\vspace{-5.0mm}\n\\section{Indirect detection of neutralino dark matter}\n\n\\begin{figure}[t!]\n\\centerline{\n\\psfig{figure=idm98_fig6.ps,width=3.90in,bbllx=40bp,bblly=180bp,bburx=550bp,bbury=650bp,clip=}\n}\n\\vspace{-3.0mm}\n\\caption{Scatter plots for the up--going muon fluxes from the center of the Earth versus\nthe neutralino mass. The {\\bf MSSM} configurations compatible with the annual modulation data are\nsubdivided into the 4 panels, depending on the corresponding value \nthe local density:\n$\\rho_l\/({\\rm GeV cm}^{-3})$ = $0.1$, $0.3$, \n$0.5$, $0.7$. \nDots denote configurations which could be excluded on the basis of the BESS95 \nantiproton data, circles denote configurations which survive this exclusion\ncriterion. The dashed line denotes the MACRO upper bound. \n}\n\\vspace{-4.0mm}\n\\end{figure}\n\n\\vspace{-3.0mm}\nThe susy configurations which have been proved to\nbe compatible with the annual modulation data can also\nbe searched for by using methods of indirect \nsearch for relic particles \\cite{ind_mod}. The two most promising techniques\nare the measurement of cosmic--ray antiprotons \\cite{pierre} and the\nmeasurement of neutrino fluxes from Earth and Sun \\cite{indiretta}.\n\nThe signals we are going to discuss here\nconsists of the fluxes of up--going muons in a neutrino telescope, generated by\nneutrinos which are produced by pair annihilations\nof neutralinos captured and accumulated inside the Earth and the\nSun. The calculation of the up--going muon signal is performed according to the\nmethod described in Ref. \\cite{indiretta}.\n\nIn Fig. 6 we display the \nscatter plots for the flux of the up--going muons from the center of the \nEarth, for various values of the  local dark matter density \n$\\rho_l$. In this figure is also reported the current $90\\%$ C.L. \nexperimental upper \nbound on $\\Phi_{\\mu}^{\\rm Earth}$, obtained by MACRO \\cite{macro}.\nWe notice the particular enhancement at \n$m_{\\chi} \\sim (50 - 60)$ GeV, due to a mass--matching effect\nbetween the WIMP and the Fe nuclei in the Earth core.\nIt is also noticeable an effect of suppression and spreading \nof the fluxes for $m_{\\chi} \\gsim 70$ GeV at  \n$\\rho_l \\gsim 0.3$ GeV cm$^{-3}$. \nThis is due to the fact that the configurations with these \nlarge values of $\\rho_l$ may imply, because of the annual modulation data,\na neutralino--nucleus cross--section \nwhich is too small to establish an efficient capture rate, necessary for \nthe capture-annihilation \nequilibrium in Earth \\cite{ind_mod}.\nIn Fig. 6  the configurations which would be \nexcluded on the basis of the antiproton data \nare denoted differently from those which would survive this \ncriterion (for details see Refs. \\cite{ind_mod,fiorenza}).\n\nBy comparing our scatter plots with the experimental MACRO upper limit, \none notices that a number of susy configurations provide \na flux in excess of this experimental bound and might then be considered\nas excluded. However, \nit has to be recalled that a possible neutrino oscillation \neffect may \nbe operative here and  affect   the indirect neutralino signal as \nwell as the background consisting in atmospheric neutrinos.  \nTherefore a strict enforcement of the\ncurrent upper bound on $\\Phi_\\mu^{\\rm Earth}$ should be applied with caution\nas long as the neutrino oscillation properties are not fully considered.  \nHowever, it is rewarding that the set $S$ of \nsusy configurations is quite accessible to relic neutralino \nindirect search by measurements of up--going fluxes.\n\n\\vspace{-5.0mm}\n\\section{Conclusions}\n\n\\vspace{-3.0mm}\nIn this paper, we have analysed in terms of relic neutralinos\nthe total sample of new \nand former DAMA\/NaI data \\cite{dama1,dama2}, which provide \nthe indication of a possible annual modulation effect in the \nrate for WIMP direct detection.\nThe remarkable result of our analysis is that {\\em the annual\nmodulation data are widely compatible with a relic neutralino making \nup the major part of dark matter in the Universe}, both in the low\nenergy MSSM and in SUGRA schemes.\n\nWe have also investigated the possibility of exploring at accelerators\nthe same susy configurations which are compatible with the \nannual modulation data. We have shown that an analysis of the main \nfeatures of these susy configurations is within the reach of present \nor planned experimental set--ups. \nIn particular, we have obtained the following results:\n\n\\vspace{-2.0mm}\n\\begin{description}\n\\item[$\\bullet$]\n      {\\underline{MSSM:}}\n      The sizeable neutralino--nucleon elastic cross--sections, implied by the \n      annual modulation data, entail  a rather stringent upper bound for \n      $m_h$ in terms of $\\tan \\beta$. In particular, this property  implies that no \n      susy configuration would be allowed for $\\tan \\beta \\lsim 2.5$. \n      Another property, discussed in Ref. \\cite{mssm_mod}, is that\n      the annual modulation data and the $b \\rightarrow s + \\gamma$ \n      constraint complement each other in providing a correlation \n      between $\\tan \\beta$ and the mass of the lightest top--squark. \n\\item[$\\bullet$]\n      \\vspace{-2.0mm}\n      {\\underline{SUGRA:}}\n      In the universal SUGRA model the constraints imposed by the DAMA\/NaI data \n      imply for the $h$--Higgs boson mass, \n      the neutralino mass and the lightest top--squark mass, the following ranges: \n      $m_h \\lsim 115$ GeV, 50 GeV $\\lsim m_{\\chi} \\lsim$ 100 GeV and \n      200 GeV $\\lsim m_{\\tilde t_1} \\lsim$ 700 GeV, respectively. \n      In universal SUGRA $\\tan \\beta$ is constrained to be large, \n      $\\tan \\beta \\gsim 42$, whereas, \n      with departure from universality in the scalar masses, the range for \n      $\\tan \\beta$ widens to $10 \\lsim \\tan \\beta \\lsim 50$. \n\\end{description}\n\n\\vspace{-2.0mm}\nMany of the above configurations will be explored by LEP2, and almost all\nof them are under reach of the future planned high--energy accelerators, namely\nthe upgrade of the Tevatron and LHC.\n\nThe same configurations can also be probed by indirect dark matter searches.\nWe have shown \\cite{ind_mod} that a sizeable fraction of the \nsusy neutralino configurations singled out by the \nDAMA\/NaI data may provide signals detectable by\nmeasurement of cosmic--ray antiprotons and detection of \nneutrino fluxes from Earth and Sun. \n\nFor the case of cosmic antiprotons, it has been shown \\cite{ind_mod,fiorenza}\nthat present data are well fitted by total spectra which include a $\\bar p$\ncontribution from neutralino--pair annihilation, with neutralino \nconfigurations which are relevant for annual modulation in direct detection. \nThese data can also be used to reduce the total sample of the\nsusy configurations under study, and to narrow the range \nof the local density, by disfavoring its largest values.\nInvestigation by measurements of cosmic $\\bar p$'s looks very promising \nin view of the collections and analyses of more statistically \nsignificant sets of data in the \nlow--energy regime which are currently under way and\nwhich may soon provide further relevant information, like, for instance, \nin the case of BESS and AMS detectors.\n\nMeasurements of neutrino fluxes from Earth and Sun, due to capture and annihilation\nof neutralinos inside these celestial bodies, have been proved to be sensitive\nto neutralino configurations singled out by the annual modulation data. However, an\nappropriate interpretation of these measurements preliminarily requires \nsome clarification of the oscillation neutrino properties, especially in the\nlight of the recent Kamiokande result \\cite{kamioka}.\n\nFinally, we conclude with a few comments. A solid confirmation of the\nannual modulation effect, as singled out by the DAMA\/NaI Collaboration,\nnecessarily requires further accumulation of data with very stable \nset--ups over a few years, a project which is currently\nundertaken by the DAMA\/NaI Collaboration. For the status of the\nexperimental activity in WIMP direct search with other detectors,\nsee the contributions of the various experimental Collaborations to\nthese Proceedings.\n\nIt is really worth noticing that the detection of the effect of annual\nmodulation, if confirmed by further experimental evidence, would \nturn out to be a major breakthrough in establishing the existence of \nparticle dark matter in the Universe, and this would henceforth be a \nmajor breakthrough for astrophysics, cosmology and particle physics as well.\n\n\\vspace{-5.0mm}\n\\section*{References}\n\\vspace{-2.0mm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:introduction}\n\nIn recent years there has been an increasing interest in the question of joint measurability of noncommuting quantum observables, both from a foundational\n\\cite{OQP,FQMEA,AnBaAs05,SoAnBaKi05,Kar-etal06,JaBo06}\nand quantum information theoretical\n\\cite{DArSa00,DArMaSa01,BrAnBa06,FePa07} perspective. The connection of this issue with certain impossible tasks in quantum mechanics, such as universal copier and Bell's telephone, is lucidly explained in \\cite{Werner01}. Since two observables represented as selfadjoint operators do not have a joint observable if they do not commute, it is necessary in such cases to understand joint measurability in a wider sense. \n\nAs intuitively understood by Heisenberg already in 1927 \\cite{Heisenberg27}, one has to allow for a degree of imprecision in order to make room for a notion of joint measurement of noncommuting observables. This idea can be appropriately investigated if the wider class of observables represented as positive operator measures (POMs) is taken into consideration. Projection valued measures among the POMs correspond to the standard observables represented as selfadjoint operators; they are called {\\em sharp observables}.\n\nIn the class of POMs, there are pairs of noncommuting observables that possess a joint observable, which thus has these two observables as its marginals. Commutativity is necessary for joint measurability if at least one of the POMs is a sharp observable, but generally commutativity is not required. A joint measurement of two POMs $\\E^1$ and $\\E^2$ can be regarded as an {\\em approximate joint measurement} of two noncommuting observables $\\A$ and $\\B$ if $\\E^1,\\E^2$ are close (in some suitable sense) to $\\A,\\B$, respectively.\n\nThe problem of approximate joint measurements of position and momentum has been treated comprehensively in related publications \\cite{Werner04b,BuHeLa06,BuPe06}. The case of observables with discrete spectra requires somewhat different concepts and will be treated in the present paper for the case of joint measurements of qubit observables. We will introduce an appropriate measure of the quality of the approximation of one observable by another observable. It will then be shown that the quality of approximations in an approximate joint measurement of two sharp observables is limited if these observables do not commute. This limitation can indeed be formulated rigorously as a form of Heisenberg uncertainty relation.\n\nOne factor limiting the accuracy in an approximate joint measurement of noncommuting sharp observables is the fact that the approximating marginal observables must have a sufficient degree of {\\em intrinsic unsharpness} as a consequence of their joint measurability. Hence there is yet another form of Heisenberg uncertainty relation for appropriately defined degrees of unsharpness in joint measurements. The distinction between the relational feature of inaccuracy (distance between two POMs) and the intrinsic property of unsharpness (of an individual POM) was until now blurred due to the fact that joint measurements were considered in which the marginals were coarse-grained versions of the sharp observables to be approximated; in such cases the intrinsic unsharpness and the inaccuracy are interconnected.\n\nA theory and first models of approximate joint measurements of qubit observables\nwere presented for special cases in \\cite{Busch86,Busch87}. In those works and all subsequent developments, only a restricted class of joint measurements was used to approximate two sharp spin components. Here this restriction will be lifted, thereby allowing one to determine optimal joint measurements and to formulate uncertainty relations that can be used to characterize the optimal cases.\n\nThe paper is organized as follows. In Section \\ref{sec:simple} we\nreview the condition of joint measurability  of two simple\nobservables  (i.e. observables representing yes-no measurements),\nand give a precise definition of the approximate joint measurability\nof two observables. These conditions and concepts are investigated\nin Sections~\\ref{sec:qubit}-\\ref{sec:approx-joint} in the case of\nqubit observables. Section~\\ref{sec:conclusion} gives our\nconclusions and an outlook.\n\n\n\\section{Simple observables and their (approximate) joint measurability}\\label{sec:simple}\n\n\\subsection{Effects and observables}\\label{sec:effects}\n\n\nThe general definition of an observable $\\A$ as a positive operator\nmeasure (POM) reduces, in the case of measurements with finitely\nmany outcomes $\\omega_i$, to the specification of a map\n$\\omega_i\\mapsto\\A_i$, where the $\\A_i$ are {\\em effects}, that is,\npositive operators satisfying the ordering\nrelation\\footnote{Relation $A\\leq B$ for two selfadjoint operators\n$A$ and $B$ means that $\\ip{\\psi}{A\\psi}\\leq\\ip{\\psi}{B\\psi}$ for\nevery vector $\\psi$.} $\\nul \\le \\A_i\\le\\id$. (Here $\\nul,\\id$ are\nthe null and unit operators, respectively.) Together with any state\n(density operator) $T$, $\\A$ determines a probability distribution\nover the outcomes of $\\A$ via the trace formula, $\\omega_i\\mapsto\n\\tr[T\\A_i]$. The additivity and normalization of probability\ndistributions is ensured by the condition $\\sum_i\\A_i=\\id$.\n\nA {\\em simple observable} is one that represents a measurement with\ntwo possible outcomes; it is given as a POM with two values and\nassociated effects,\n\\begin{equation}\n\\mathcal{A}:\\quad \\omega_+\\mapsto \\A_+,\\quad \\omega_-\\mapsto \\A_-.\n\\end{equation}\nNormalization entails that $\\A_+ + \\A_-=\\id$, so that a simple observable\nis commutative. We note that any effect $A$ together with its \ncomplement effect $A':=\\id- A$ defines a class of simple\nobservables, distinguished only by their outcome sets\n$\\{\\omega_+,\\omega_-\\}$.\\footnote{For clarity we denote observables with\nscript capital letters, while effects are denoted with italic letters.}\n\nAs noted in the introduction, an observable (POM) is {\\em sharp} if its effects are\nall projections. Otherwise an observable is called {\\em unsharp}. \nA measure of the intrinsic unsharpness of an effect $A$ and thus  of\nthe associated simple observable $\\A$ that is independent of the\noutcomes of $\\A$ is obtained as follows. Let $\\sa$ denote the\nspectrum of an effect $A$, then the {\\em spectral width} of $A$ is\nthe length of the smallest closed interval containing $\\sa$, that\nis: $\\W(\\sa):=\\max\\sa-\\min\\sa=\\no{A}+\\no{A'}-1$. The {\\em sharpness}\nof $A$ is defined as\n\\begin{equation}\\label{eqn:sharpness-def}\n\\S(A):=\\W(\\sa)-\\W(\\saap).\n\\end{equation}\nThe operator $AA'$ can be written as $AA'=A^{\\half}A'A^{\\half}$.\nThis shows that $AA'$ arises from the sequential L\\\"uders\nmeasurement of the complement effects $A$ and $A'$, and this is a\nmotivation for the formula (\\ref{eqn:sharpness-def}); see \\cite{Busch07} for further discussion.\n\nSince $\\W(\\sa)=\\W(\\sap)$, it follows that $\\S(A')=\\S(A)$. On can prove the following facts about the sharpness\\footnote{See \\cite{Busch07}. These statements are proved for qubit observables in Subsection \\ref{sec:sharpness-distance}.}:\n$\\S(A)\\in[0,1]$; $\\S(A)=0$ exactly when $A$ is a\ntrivial effect (i.e. $A=k\\id$ for some $0\\leq k\\leq 1$);\n$\\S(A)=1$ exactly when $A$ is a nontrivial projection. These are properties one would expect any measure of sharpness to possess: the measure should single out the perfectly sharp effects and the trivial effects.\n\n\nThe {\\em sharpness} and {\\em unsharpness} of a simple observable $\\A$ may now be defined as\n\\begin{eqnarray}\n\\S(\\A) &:=& \\S(\\A_+)=\\S(\\A_-), \\\\\n\\U(\\A) &:=& 1-\\S(\\A)^2. \\label{eqn:u}\n\\end{eqnarray}\n\n\n\n\\subsection{Joint measurability}\\label{sec:joint}\n\nTwo observables are jointly measurable if there is a measurement\nscheme that allows the determination of the values of both\nobservables. This means that the POM representing that joint\nmeasurement contains the two observables as marginals. In this way\nit is ensured that there is a joint probability distribution for\neach state. We spell out this definition\\footnote{For the general\ndefinition of joint measurability and a detailed discussion on this\ntopic, see e.g. \\cite{Lahti03} and references given therein.} for\nthe case of a pair of simple observables.\n\nTwo simple observables $\\E^1$ and $\\E^2$ are \\emph{jointly\nmeasurable} if there is an observable $\\G:\\omega_{ij}\\mapsto \\G_{ij}$,\n$i,j=\\pm$, such that\n\\begin{equation}\\label{simple-jm}\\begin{split}\n\\E^1_+=\\G_{++}+\\G_{+-},&\\qquad \\E^1_-=\\G_{-+}+\\G_{--},\\\\\n\\E^2_+=\\G_{++}+\\G_{-+},&\\qquad \\E^2_-=\\G_{+-}+\\G_{--}.\n\\end{split}\\end{equation}\nIn this case the observables $\\E^1$ and $\\E^2$ are the\n\\emph{marginals} of $\\G$, and we also say that $\\G$ is a \\emph{joint\nobservable} for $\\E^1$ and $\\E^2$. The outcomes $\\omega_{ij}$ of\n$\\G$ could be taken to be (or replaced by) the pairs\n$(\\omega_i,\\omega_j)$.\n\nThe joint measurability of two simple observables $\\E^1$ and $\\E^2$\nis equivalent to the statement \\cite{SEO} that there exists an\noperator $\\G_{++}$ satisfying the  following set of operator\ninequalities:\n\\begin{equation}\\label{simple-coex}\n\\begin{split}\n\\nul\\le\\G_{++},\\quad\n\\G_{++}&\\le\\E^1_+,\\quad\n\\G_{++}\\le\\E^2_+,\\\\\n\\E^1_+ + \\E^2_+ - \\id&\\le\\G_{++}.\n\\end{split}\n\\end{equation}\nIn fact, these inequalities ensure that the following four operators are effects:\n\\begin{equation}\\label{simple-joint}\\begin{split}\n&\\G_{++},\\quad \\G_{+-}\\equiv \\E^1_+ - \\G_{++},\\quad \\G_{-+}\\equiv\\E^2_+ - \\G_{++},\\\\\n&\\G_{--}\\equiv\\id-\\G_{++}-\\G_{+-} - \\G_{-+} = \\id-\\E^1_+ - \\E^2_+ + \\G_{++}.\n\\end{split}\\end{equation}\nIt is straightforward to verify that equations (\\ref{simple-jm}) hold and hence, these effects define a joint observable $\\G$ for $\\E^1$ and $\\E^2$.\n\nThe joint measurability condition for two simple observables can be interpreted\nas the requirement that the intersection of four cones in the set of effects is\nnonempty. The order relation $A\\le B$ for two selfadjoint operators is equivalent to\neither of $B-A\\ge\\nul$ and $A-B\\le\\nul$. The condition $A\\ge\\nul$ defines a convex cone in the real vector space of selfadjoint operators.\\footnote{This means that whenever $A,B\\ge \\nul$, then $tA+(1-t)B\\ge\\nul$ for any $t\\in [0,1]$.} Thus we can define the upward and\ndownward cones of a selfadjoint operator $A$ as\n$\\mathcal{C}^\\lor(A):=\\{B: A\\le B\\}$ and $\\mathcal{C}^\\land(A):=\\{B:A\\ge B\\}$. The joint measurability condition for observables $\\E^1$ and $\\E^2$ now reads:\n\\begin{equation}\\label{eqn:cone-condition}\n\\mathcal{C}^\\lor(\\nul)\\cap\\mathcal{C}^\\land(\\E^1_+)\\cap\\mathcal{C}^\\land(\\E^2_+)\\cap\\mathcal{C}^\\lor(\\E^1_++\\E^2_+-\\id)\\ne\\emptyset.\n\\end{equation}\n\n\\begin{example}[Trivial cases of joint measurability]\\label{ex:trivial-coex}\nIn the following four cases, joint measurability falls out trivially:\\\\\n(a) $\\E^1_+\\ge\\E^2_+$: put $\\G_{++}=\\E^2_+$, $\\G_{+-}=\\E^1_+-\\E^2_+$, $\\G_{-+}=\\nul$, $\\G_{--}=\\id-\\E^1_+$.\\\\\n(b) $\\E^1_+\\le\\E^2_+$: put $\\G_{++}=\\E^1_+$, $\\G_{+-}=\\nul$,\n$\\G_{-+}=\\E^2_+-\\E^1_+$, $\\G_{--}=\\id-\\E^2_+$.\\\\\n(c) $\\E^1_+\\ge \\E^2_-$: put $\\G_{++}=\\E^1_++\\E^2_+-\\id$,  $\\G_{+-}=\\E^2_-$, $\\G_{-+}=\\E^1_-$,\n$\\G_{--}=\\nul$.\\\\\n(d) $\\E^1_+\\le \\E^2_-$:  put $\\G_{++}=\\nul$, $\\G_{+-}=\\E^1_+$, $\\G_{-+}=\\E^2_+$, $\\G_{--}=\\id-\\E^1_+-\\E^2_+$.\n\\end{example}\n\nWe conclude that interesting (i.e. nontrivial) cases arise when\n$\\E^1_+-\\E^2_+$ and $\\E^1_++\\E^2_+-\\id$\nare neither $\\le \\nul$ nor $\\ge \\nul$. In terms of a joint observable $\\G$, nontrivial cases are exactly those in which $\\G_{ij}\\neq\\nul$ for every $i,j=\\pm$.\n\nAnother simple instance of joint measurability arises from commutativity.\n\n\\begin{example}[Mutually commuting observables]\\label{ex:com-coex}\nIf $\\E^1$ and $\\E^2$ commute mutually in the sense that $\\E^1_i\\E^2_j=\\E^2_j\\E^1_i$ for every $i,j=\\pm$, then they are jointly measurable. In this case the formula $\\G_{ij}=\\E^1_i\\E^2_j$ defines a joint observable.\n\\end{example}\n\nIt is instructive to prove the following well-known proposition, which supplements Example \\ref{ex:com-coex}. (Its statement is valid also for observables that are not simple, and then the proof requires only minor changes.)\n\n\\begin{proposition}\\label{prop:com}\nLet $\\E^1$ and $\\E^2$ be simple observables which are jointly measurable. If one of them is a sharp observable, then they commute and the unique joint observable $\\G$ is of the product form $\\G_{ij}=\\E^1_i\\E^2_j$.\n\\end{proposition}\n\n\\begin{proof}\nLet, for instance, $\\E^1$ be a sharp observable and suppose that $\\G$ is a joint observable for $\\E^1$ and $\\E^2$. Since $\\G_{ij}\\le\\E^1_i$, one obtains $\\E^1_i\\G_{ij}=\\G_{ij}\\E^1_i=\\G_{ij}$. This shows also that $\\E^1_i\\G_{-ij}=(\\id-\\E^1_{-i})\\G_{-ij}=\\nul$ and similarly $\\G_{-ij}\\E^1_i=\\nul$. It follows that\n\\begin{equation}\n\\E^1_i\\E^2_j=\\E^1_i(\\G_{+j}+\\G_{-j})=\\G_{ij}\n\\end{equation}\nand\n\\begin{equation}\n\\E^2_j\\E^1_i=(\\G_{+j}+\\G_{-j})\\E^1_i=\\G_{ij}.\n\\end{equation}\nA comparison of these equations proves the claim.\n\\end{proof}\n\nWe note that in general two observables may have many different joint observables; this fact will be demonstrated in Section \\ref{sec:covjoint}.\n\nFor later use we recall the following general fact. The set of observables on a fixed outcome space is convex: for two observables $\\E,\\F$ and any $t\\in[0,1]$, a new observable $t\\E+(1-t)\\F$ is defined as $\\omega_i\\mapsto t\\E_i+(1-t)\\F_i$.\n\n\\begin{proposition}\\label{prop:joint-convex}\nLet $(\\E^1,\\E^2)$ and $(\\F^1,\\F^2)$ be two pairs of jointly measurable observables. Then for any\n$t\\in[0,1]$, the observables $t\\E^1+(1-t)\\F^1$ and $t\\E^2+(1-t)\\F^2$ are jointly measurable.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\G$ be a joint observable of $\\E^1,\\E^2$ and $\\mathcal{H}$ of $\\F^1,\\F^2$. Then $t\\G+(1-t)\\mathcal{H}$ is a joint observable of $t\\E^1+(1-t)\\F^1$ and $t\\E^2+(1-t)\\F^2$.\n\\end{proof}\n\n\\subsection{Approximate joint measurability}\\label{sec:approximate}\n\nAssume that two observables $\\A$ and $\\B$ do not have a joint measurement. We may still ask if it could be possible to obtain some information on both observables in a single measurement scheme. One way of approaching this task is to consider whether there are two jointly measurable observables $\\E^1,\\E^2$ that are close to $\\A,\\B$, respectively, in a sense to be determined. Any joint measurement of $\\E^1$ and $\\E^2$ can then be regarded as an approximate joint measurement of $\\A$ and $\\B$.\n\nA natural characterization of the closeness between two observables (assumed to have the same outcome space) is based on the degree of similarity of their associated probability distributions for all states. Hence we define the distance between observables $\\A$ and $\\B$ in the following way:\n\\begin{equation}\n\\D(\\A,\\B):=\\max_{j} \\sup_{T} \\big| \\tr[T\\A_j]-\\tr[T\\B_j]\\big|=\\max_{j}\\no{\\A_j-\\B_j}.\n\\end{equation}\nClearly, $0\\leq\\D(\\A,\\B)\\leq 1$, and $\\D(\\A,\\B)=0$ if and only if $\\A=\\B$. Moreover, the triangle inequality holds for a triple of observables, so that $\\D$ is indeed a metric.\nIf $\\A$ and $\\B$ are simple observables, then $\\A_--\\B_-=\\B_+-\\A_+$ and therefore \n\\begin{equation}\n\\D(\\A,\\B)=\\no{\\A_+ - \\B_+}=\\no{\\A_- - \\B_-}.\n\\end{equation}\n\nA conventional approach to realizing approximate joint measurements consists of replacing\nthe observables $\\A,\\B$ to be approximated with some coarse-grained versions $\\E^1,\\E^2$. Here we briefly illustrate this approach in the case of simple observables from the perspective of the general framework. We refer to \\cite[Chapter 7]{FQMEA} for a review and examples on this topic.\n\n\nIf $\\A$ and $\\B$ are simple observables, one defines, using $2\\times 2$ stochastic matrices\\footnote{A stochastic matrix is a square matrix whose entries are non-negative real numbers and each column sums to 1.} $(\\lambda_{ik}),(\\mu_{j\\ell})$, the coarse-grainings $\\E^1$ and $\\E^2$ by\n\\begin{equation}\\label{smear}\n\\E^1_i=\\lambda_{i+}\\A_++\\lambda_{i-}\\A_-,\\quad \\E^2_j=\\mu_{j+}\\B_++\\mu_{j-}\\B_-.\n\\end{equation}\nWe expect that $\\E^1$ is a good approximation of $\\A$ if $\\lambda_{++}$ is close to 1 and $\\lambda_{+-}$ is close to 0. Indeed, if, for instance, $\\A$ is a sharp observable then\n\\begin{equation}\\label{distance-of-coarsegraining}\n\\D(\\E^1,\\A)=\\max \\{ 1-\\lambda_{++},\\lambda_{+-}\\}.\n\\end{equation}\n\nTo give an example of jointly measurable observables, assume that $\\E^1$ and $\\E^2$ are defined as in (\\ref{smear}). Since we want to approximate $\\A$ by $\\E^1$ and $\\B$ by $\\E^2$, it is natural to require that\n\\begin{equation}\\label{smatrix}\n\\lambda_{++}\\ge\\lambda_{+-},\\quad \\mu_{++}\\ge\\mu_{+-}.\n\\end{equation}\nIn fact, if these inequalities do not hold, one can choose $\\lambda'_{ik}=\\lambda_{-ik}$,\n$\\mu'_{jl}=\\mu_{-jl}$ to obtain new coarse-grainings which do satisfy the inequalities.\n\nNow, define $\\G_{++}=\\min\\{\\lambda_{+-},\\mu_{+-}\\}\\,\\id$. Condition (\\ref{smatrix}) implies that $\\nul\\le\\G_{++}\\le\\E^1_+$ and $\\G_{++}\\le\\E^2_+$. The remaining inequality\nrequired for joint measurability, $\\E^1_+ + \\E^2_+ - \\id\\le\\G_{++}$, depends on the specific structure of the observables $\\A$ and $\\B$; it is ensured to hold independently of $\\A$ and $\\B$ if\n\\begin{equation}\n\\lambda_{++} + \\mu_{++} \\le 1 + \\min\\{\\lambda_{+-},\\mu_{+-}\\}.\n\\end{equation}\nThis shows that one can always construct (nontrivial) jointly measurable coarse-grainings; a possible choice is, for instance, $\\lambda_{++}=\\mu_{++}=\\frac{2}{3}$ and $\\lambda_{+-}=\\mu_{+-}=\\frac{1}{3}$. \n\n\n\\section{Qubit observables}\\label{sec:qubit}\n\n\\subsection{Effects and observables}\\label{sec:qubit-effects}\n\nIn the 2-dimensional Hilbert space of a qubit one can take the unit\noperator $\\id$ together with the Pauli operators\n$\\sigma_1,\\sigma_2,\\sigma_3$ as a basis of the real vector space of\nselfadjoint linear operators. The latter can be defined with respect\nto any fixed basis of orthogonal unit vectors $\\varphi_+,\\varphi_-$ so\nthat the usual relations are satisfied:\n$\\sigma_3\\varphi_\\pm=\\pm\\varphi_\\pm$, $\\sigma_1\\varphi_\\pm=\\varphi_\\mp$,\n$\\sigma_2\\varphi_\\pm=\\pm i\\varphi_\\mp$. We will write $\\vsigma$ for the\noperator triple $(\\sigma_1,\\sigma_2,\\sigma_3)$. States of a qubit\ncan be written in the form $T_{\\vr}=\\half \\left( \\id +\n\\vr\\cdot\\vsigma\\right)$, where $\\vr\\in\\R^3$ and $\\no{\\vr}\\leq 1$.\nThe pure states are characterized by the condition $\\no{\\vr}=1$.\n\nFor each $(\\alpha,\\va)\\in\\R^4$, we denote\n\\begin{equation}\n\\Aaa:=\\frac{1}{2}\\left( \\alpha \\id + \\va\\cdot\\vsigma \\right).\n\\end{equation}\nThe eigenvalues of the operator $\\Aaa$ are $\\frac{1}{2}(\\alpha \\pm\n\\no{\\va})$. Hence, $\\Aaa$ is an effect if\n\\begin{equation}\n\\no{\\va}\\leq \\alpha \\leq 2-\\no{\\va},\n\\end{equation}\nwhich implies, in particular, that $\\no{\\va}\\leq 1$. The operator $\\Aaa$ is a nontrivial projection if\n\\begin{equation}\n\\alpha=\\no{\\va}=1.\n\\end{equation}\nThe spectral decomposition of the effect $\\Aaa$, $\\va\\neq 0$, is\n(putting $\\hat{\\va}:=\\no{\\va}^{-1}\\va$)\n\\begin{equation}\\label{eqn:spectraldecom}\n\\Aaa=\\frac{1}{2}(\\alpha+\\no{\\va})A(1,\\hat\\va) +\n\\frac{1}{2}(\\alpha-\\no{\\va})A(1,-\\hat\\va).\n\\end{equation}\nFor later use we note the commutator of two effects $A(\\alpha,\\va)$\nand $A(\\beta,\\vb)$:\n\\begin{equation}\n\\left[ A(\\alpha,\\va),A(\\beta,\\vb) \\right]=\\tfrac{1}{2}\n\\left(\\va\\times\\vb \\right) \\cdot \\vsigma.\n\\end{equation}\n\n\nSince the Hilbert space of a qubit is 2-dimensional, any sharp qubit\nobservable is (effectively) simple. It is clear that this\nrestriction does not apply for a qubit observable in general; one\ncan write the identity operator $\\id$ as a sum of arbitrarily many\ndifferent effects. It is also known that for some quantum\ninformational tasks, such as unambiguous state discrimination, one\nneeds other qubit observables than the simple ones; see e.g. \\cite{Chefles00}. Here we shall,\nhowever, concentrate on simple qubit observables as our aim is to\nstudy approximate joint measurements of sharp qubit observables. \nTo clarify further the nature of sharp qubit observables, we note that again due to the low dimensionality, \nthe projections that constitute such an observable are of rank one, which implies that their\nrepeatable measurements are von Neumann measurements \\cite{OQP}.\n\nWe denote by $\\Eaa$ the simple qubit observable defined as\n\\begin{equation*}\\begin{split}\n\\omega_+\\mapsto\\Eaa_{+} &:= \\Aaa,\\\\ \\omega_-\\mapsto\\Eaa_{-}&:=\\id-\\Aaa=A(2-\\alpha,-\\va).\n\\end{split}\n\\end{equation*}\nA special case is given by the sharp observables $\\E^{1,\\hat\\va}$.\nThe spectral decomposition (\\ref{eqn:spectraldecom}) of $A(\\alpha,\\va)$ shows that $\\Eaa$ is a\ncoarse-graining of $\\Esa$.\n\nFrom the above commutator formula we recover the well known fact\nthat the observables $\\Eaa$ and $\\Ebb$ commute exactly when the\nvectors $\\va$ and $\\vb$ are collinear. Together with Proposition\n\\ref{prop:com}, this shows that an observable $\\Ebb$ is jointly\nmeasurable with a sharp observable $\\Esa$ if and only if $\\Ebb$ is a\ncoarse-graining of $\\Esa$. A joint measurement of that kind is of\nlittle value; one can simply measure $\\Esa$ alone to get the same\ninformation.\n\n\\subsection{Covariance}\\label{sec:covariance}\n\nLet $U$ be a unitary operator describing some symmetry\ntransformation of the system. We assume that $U^2=\\id$, so that\n$\\{\\id,U\\}$ form a two-element group. In other words, $U$ is a\nselfadjoint unitary operator. We say that an observable $\\Eaa$ is\n\\emph{covariant with respect to} $U$, or $U$-\\emph{covariant} for\nshort, if\n\\begin{equation}\nU\\Eaa_+U=\\Eaa_-.\n\\end{equation}\nThis covariance condition means that the symmetry transformation described by $U$ swaps the outcomes of the observable $\\Eaa$ but has no other effect on its measurement outcome distributions.\n\nEffects $\\Eaa_+$ and $\\Eaa_-$ can be unitarily equivalent only if they have the same eigenvalues, which is the case exactly when $\\alpha=1$. Hence, $\\Eaa$ can be covariant only if $\\alpha=1$. Assume that $\\alpha=1$ and fix a unit vector $\\vu\\in\\R^3$ orthogonal to $\\va$. The operator $U=\\vu\\cdot\\vsigma$ is a selfadjoint unitary operator and\n\\begin{equation}\\label{eqn:Eacov}\nU\\Ea_+U =\\Ea_-.\n\\end{equation}\nMoreover, any selfadjoint unitary operator $U$ satisfying (\\ref{eqn:Eacov}) is of the form $U=\\vu\\cdot\\vsigma$ for some unit vector $\\vu$ orthogonal to $\\va$.\n\nIn \\cite{AnBaAs05}, an observable $\\Eaa$ was selected in relation\nto a sharp observable $\\En$ by the requirement that the expectation\nvalues of $\\Eaa$ are proportional to those of $\\En$. This\nrequirement, called there {\\em unbiasedness}, is equivalent with the\nfact that $\\alpha=1$ and the vectors $\\va$ and $\\vn$ are parallel.\nHence, the unbiasedness requirement means that $\\Eaa$ is covariant\nwith respect to the same unitary operators as $\\En$, i.e., the\nobservables $\\Eaa$ and $\\En$ have the same symmetry properties.\n\n\n\n\\subsection{Sharpness and distance}\\label{sec:sharpness-distance}\n\n\nThe spectral width of an operator acting on two dimensional Hilbert\nspace is simply the difference of its greater and lower eigenvalues.\nThe sharpness of an observable $\\Eaa$ is thus found to be\n\\begin{equation}\\label{eqn:sharpness-qubit}\n\\S(\\Eaa)=\\no{\\va}(1-|1-\\alpha|)=\\no{\\va}\\min\\{\\alpha,2-\\alpha\\}.\n\\end{equation}\nWith this expression one can easily confirm the statements of\nSection \\ref{sec:effects} for simple qubit observables: $\\S(\\Eaa)=1$\nexactly when $\\Eaa$ is a sharp observable and $\\S(\\Eaa)=0$ exactly\nwhen $\\Eaa$ is a trivial observable. We also note the following useful observation:\n\\begin{equation}\\label{eqn:sharpness-ineq}\n\\S(\\Eaa)\\le\\S(\\Ea).\n\\end{equation}\n\nThe distance between two qubit\nobservables $\\Eaa$ and $\\Ebb$ is given by the formula\n\\begin{equation}\\label{eqn:distance-ab}\n\\D(\\Eaa,\\Ebb)=\\half\\no{\\va-\\vb}+\\half |\\alpha-\\beta|.\n\\end{equation}\nThis shows, in particular, that the distance of a given observable $\\Eaa$ from any sharp observable $\\En$ is minimal when $\\vn=\\hat\\va$, or in other words, when $\\Eaa$ is a coarse-graining of $\\En$. In this case we have\n\\begin{equation}\\label{eqn:distance-aa}\n\\D(\\Eaa,\\E^{1,\\hat\\va})=\\half\\left( 1-\\no{\\va} \\right) + \\half |1-\\alpha|.\n\\end{equation}\nWe also note the following:\n\\begin{equation}\n\\D(\\Eaa,\\Esa)\\ge\\D(\\Ea,\\Esa).\n\\end{equation}\n\nFinally, from equations (\\ref{eqn:sharpness-qubit}) and (\\ref{eqn:distance-aa}) we get the following relations:\n\\begin{equation}\\label{eqn:dist-sharp}\\begin{split}\n\\D(\\Eaa,\\En)+\\half\\S(\\Eaa)&\\ge\n\\D(\\Eaa,\\E^{1,\\hat\\va})+\\half\\S(\\Eaa)\\\\\n&\\ge\\D(\\Ea,\\E^{1,\\hat\\va})+\\half\\S(\\Ea) =\\half.\n\\end{split}\n\\end{equation}\nThe last equation shows that the distance between $\\Ea$ and $\\E^{1,\\hat\\va}$ is directly related to the sharpness of $\\Ea$. This is not surprising when we recall that $\\Ea$ is a coarse-graining of $\\E^{1,\\hat\\va}$.\n\n\\section{Joint measurability of qubit observables}\\label{sec:joint-qubit}\n\n\\subsection{General criterion for joint measurability}\\label{sec:genjoint}\n\nThe joint measurability conditions (\\ref{eqn:cone-condition}) applied to two qubit\nobservables $\\Eaa,\\Ebb$ takes the following form: there exists an\noperator $\\G_{++}=\\half(\\gamma\\id+\\vg\\cdot\\vsigma)$ such that\n\\begin{eqnarray}\n\\no{\\vg}&\\le&\\gamma;\\\\\n\\no{\\va-\\vg}&\\le&\\alpha-\\gamma;\\\\\n\\no{\\vb-\\vg}&\\le&\\beta-\\gamma;\\\\\n\\no{\\va+\\vb-\\vg}&\\le&2+\\gamma-\\alpha-\\beta.\n\\end{eqnarray}\nLet $\\mathbf{B}(\\vx,r)$ denote the closed ball with center $\\vx$ and radius\n$r$. Then it is seen that the joint measurability of $\\Eaa,\\Ebb$ is\nequivalent to the statement that there exists a number $\\gamma\\ge 0$ such\nthat the intersection of four balls is non-empty:\n\\begin{equation}\\label{eqn:ball-coex}\n\\mathbf{B}(\\vnull,\\gamma)\\,\\cap\\, \\mathbf{B}(\\va,\\alpha-\\gamma)\\,\\cap\\,\\mathbf{B}(\\vb,\\beta-\\gamma)\\,\\cap\\,\\mathbf{B}(\\va+\\vb,2+\\gamma-\\alpha-\\beta)\\ne\\emptyset.\n\\end{equation}\n\nThe criterion (\\ref{eqn:ball-coex}) immediately gives the\nfollowing as a necessary condition for joint measurability: the two pairs of balls diagonally opposite to each\nother must have separations which are no greater than the sum of\ntheir radii; thus, there must be a $\\gamma\\ge 0$ such that\n\\begin{eqnarray}\n\\no{\\va-\\vb}&\\le&\\alpha+\\beta-2\\gamma,\\\\\n\\no{\\va+\\vb}&\\le&2-\\alpha-\\beta+2\\gamma,\n\\end{eqnarray}\nor equivalently,\n\\begin{equation}\\label{eqn:gamma12}\n\\gamma_1:=\\half\\no{\\va+\\vb}+\\half[\\alpha+\\beta-2]\\le\n\\gamma\\le\\half[\\alpha+\\beta]-\\half\\no{\\va-\\vb}=:\\gamma_2.\n\\end{equation}\nThis gives an interval for $\\gamma$ to lie in which has to be\nnonempty. Therefore the following is a necessary joint measurability\ncondition:\n\\begin{equation}\n\\gamma_2-\\gamma_1=1-[\\half\\no{\\va+\\vb}+\\half\\no{\\va-\\vb}]\\ge0.\n\\end{equation}\n\n\\begin{proposition}\\label{prop:generalineq}\nIf observables $\\Eaa$ and $\\Ebb$ are jointly measurable,\nthen\\footnote{This condition has the following geometric meaning:\nfor a observable $\\Ea$, a jointly measurable observable $\\Eb$ is\nsuch that the vector $\\vb$ is inside a prolate spheroid. The center\nof the spheroid is in the origin and its major axis is in the\ndirection of $\\va$. The polar radius of the spheroid is 1 and the\nequatorial radius is $(1-\\no{\\va}^2)^{1\/2}$. In fact, in coordinates\nfor which $\\va$ is in the $z$-direction, the inequality  becomes\n$b_x^2+b_y^2+(1-a^2)b_z^2\\le1-a^2$, to be read as a condition for\n$\\vb$.}\n\\begin{equation}\\label{eqn:generalineq}\n\\no{\\va + \\vb} + \\no{\\va - \\vb} \\leq 2.\n\\end{equation}\n\\end{proposition}\n\nIn the case of covariant qubit observables (for which $\\alpha=\\beta=1$) the condition (\\ref{eqn:generalineq}) is found to be also sufficient for joint measurability, as was\nshown in \\cite{Busch86}. A new proof of this fact, stated below, will arise as a corollary of our investigation in Subsection \\ref{sec:covjoint}.\n\n\\begin{proposition}\\label{prop:Paulsineq}\nObservables $\\Ea$ and $\\Eb$ are jointly measurable if and only if\ninequality $(\\ref{eqn:generalineq})$ holds.\n\\end{proposition}\n\n\nIn the following example we demonstrate that (\\ref{eqn:generalineq})\nis not sufficient in general to guarantee the joint measurability of\nobservables $\\Eaa$ and $\\Ebb$.\n\n\\begin{example}\nLet us consider the case where the vectors $\\va$ and $\\vb$ are\northogonal and  equality holds in (\\ref{eqn:generalineq}), or in\nother words, $\\no{\\va+\\vb}=\\no{\\va-\\vb}=1$. Assume that $\\Eaa$ and\n$\\Ebb$ are jointly measurable observables. We have\n$\\gamma=\\gamma_1=\\gamma_2$ and therefore, there is only one point\n$\\vg$ in the intersection\n$\\mathbf{B}(\\va,\\alpha-\\gamma)\\cap\\mathbf{B}(\\vb,\\beta-\\gamma)$, and similarly in\nthe intersection\n$\\mathbf{B}(\\vnull,\\gamma)\\cap\\mathbf{B}(\\va+\\vb,\\gamma+2-\\alpha-\\beta)$.\nThus, $\\vg$ is in the boundary of $\\mathbf{B}(\\va,\\alpha-\\gamma)$ and it must satisfy the equation\n\\begin{equation}\n\\vg = \\va+(\\alpha-\\gamma)(\\vb-\\va)\n\\end{equation}\nand three similar equations corresponding to the other balls. These\nequations taken together imply that $\\alpha=\\beta=1$. As condition\n(\\ref{eqn:generalineq}) does not restrict $\\alpha$ and $\\beta$, we\nconclude that (\\ref{eqn:generalineq}) is not sufficient to ensure\nthe joint measurability of $\\Eaa$ and $\\Ebb$. In fact, we could have\nchosen $\\alpha=\\beta=\\no{\\va}=\\no{\\vb}=1\/\\surd 2$, in which case the\nobservables $\\Eaa$ and $\\Ebb$ are not jointly measurable although\n(\\ref{eqn:generalineq}) is satisfied.\n\\end{example}\n\n\n\nPropositions \\ref{prop:generalineq} and \\ref{prop:Paulsineq} lead to the following observation, which we will need later.\n\n\\begin{proposition}\\label{prop:alsojoint}\nIf $\\Eaa$ and $\\Ebb$ are jointly measurable, then also $\\Ea$ and\n$\\Eb$ are jointly measurable.\n\\end{proposition}\n\n\\subsection{Sufficient conditions for joint measurability}\\label{sec:sufficient}\n\nThe problem of finding necessary {\\em and} sufficient conditions for the joint measurability of a pair of qubit observables $\\Eaa$ and $\\Ebb$ beyond the above case of $\\Ea,\\Eb$ has only recently been solved by the present authors in different collaborations. In \\cite{StReHe08}, this is achieved by analyzing the sphere intersection condition (\\ref{eqn:ball-coex}), whereas in \\cite{BuSch08} the cone intersection condition (\\ref{eqn:cone-condition}) is elucidated. The sets of inequalities found for $\\alpha,\\va,\\beta,\\vb$ are rather involved and not easily comparable, hence we refrain from reproducing them here.\nInstead we give a  sufficient condition for the joint measurability of $\\Eaa$ and\n$\\Ebb$ which is an obvious strengthening of (\\ref{eqn:generalineq}). The fact that this stronger\ncondition may appear quite natural at first sight but is actually not necessary highlights the intricate nature of the general problem solved in \\cite{StReHe08} and \\cite{BuSch08}. \n\nFirst we identify two distinguished effects $A_1:=A(\\gamma_1,\\vg_1)$ and\n$A_2:=A(\\gamma_2,\\vg_2)$, where $\\gamma_1,\\gamma_2$ are the parameters from\nEq.~(\\ref{eqn:gamma12}) and\n\\begin{equation}\\begin{split}\n\\vg_1&=\n\\frac 12\\left[ 1-\\frac{2-\\alpha-\\beta}{\\no{\\va+\\vb }} \\right]\\,( \\va+\\vb ),\\\\\n\\vg_2&\n=\\frac 12(\\va+\\vb )-\\frac{\\alpha-\\beta}{\\no{\\va-\\vb }}\\frac 12\n(\\va-\\vb)  .\n\\end{split}\n\\end{equation}\nThe effect $A_1$ is in $\\mathcal{C}^\\lor(\\nul)\\cap\\mathcal{C}^\\lor(\\E^1_++\\E^2_+-\\id)$ and it is the unique\nelement of all effects $A(\\gamma,\\vg)$ in that intersection with the lowest possible\n$\\gamma$. The effect $A_2$ is in $\\mathcal{C}^\\land(\\E^1_+)\\cap\\mathcal{C}^\\land(\\E^2_+)$\nand it is the unique element of all effects $A(\\gamma,\\vg)$ in that intersection with the\ngreatest possible $\\gamma$. Now, joint measurability is guaranteed if $A_1\\le A_2$,\nwhich is equivalent to the condition:\n\\begin{equation}\\label{eqn:sufficient}\n\\no{\\va+\\vb}+\\no{\\va-\\vb}\n+\\no{\\frac{2-\\alpha-\\beta}{\\no{\\va+\\vb}}(\\va+\\vb)-\n\\frac{\\alpha-\\beta}{\\no{\\va-\\vb}}(\\va-\\vb)}\n\\le 2.\n\\end{equation}\nIt is not hard to verify that  this condition is automatically satisfied in all trivial and commutative cases,\nwhere joint measurability is given. Furthermore it follows from the stronger condition\n\\begin{equation}\\label{eqn:strong-suff}\n\\no{\\va+\\vb}+\\no{\\va-\\vb}+|2-\\alpha-\\beta|+ |\\alpha-\\beta| \\le 2,\n\\end{equation}\nwhich can also be written in operator terms as\n\\begin{equation}\n\\no{\\Eaa_+-\\Ebb_+}+\\no{\\Eaa_+-\\Ebb_-}\\le 1.\n\\end{equation}\nThis sufficient condition for joint measurability is satisfied in all cases with $\\alpha=\\beta=1$ but is generally not necessary, as can be seen from the example $\\Eaa_+=\\id$, $\\Ebb_+=A(1,\\vn)$ (where\n$\\vn$ is any unit vector).\n\nIn nontrivial cases the above sufficient joint measurability conditions can be further strengthened and simplified. For two qubit observables $\\Eaa$ and $\\Ebb$, the nontriviality requirement in the sense of Example \\ref{ex:trivial-coex} amounts to the following:\n\\begin{equation}\\label{eqn:nontrivial}\\begin{split}\n|\\alpha-\\beta|&< \\no{\\va-\\vb}\\quad (\\mathrm{not\\ (a),(b)}); \\\\\n|2-\\alpha-\\beta|&< \\no{\\va+\\vb}\\quad (\\mathrm{not\\ (c),(d)}).\n\\end{split}\n\\end{equation}\nUnder these nontriviality assumptions, the conditions (\\ref{eqn:sufficient}) and (\\ref{eqn:strong-suff}) are\nseen to be satisfied if\n\\begin{equation}\n\\no{\\va+\\vb}+\\no{\\va-\\vb}\\le 1.\n\\end{equation}\n\n\nThe next example shows that the sufficient condition\n(\\ref{eqn:sufficient}) is not a necessary condition.\n\\begin{example}\nWe consider the case where $\\va\\perp\\vb$. Furthermore, let $\\va=\\hat\\va$ be a unit vector, so that\n$A:=\\Eaa_+=\\alpha A(1,\\hat\\va)$ is a multiple of a projection. Note that $A$ being an effect entails that\n$\\alpha\\le 1$. Next we denote $B:=\\Ebb_+=A(1,\\vb)$, where we assume that $b:=\\no{\\vb}\\ne 0$. \n\nJoint measurability of $A,B$ is given if and only if there is an operator $G$ which is bounded above by $A,\\ B$ and bounded below by $\\nul,\\ A+B-\\id$. The inequality $G\\le A$ is satisfied if and only if $G$ is a multiple of the projection $A(1,\\hat\\va)$, hence: $G=\\gamma A(1,\\hat\\va)$, and $\\gamma\\le\\alpha$. Further, $\\gamma$\nmust be chosen such that $\\gamma A(1,\\hat\\va)\\le B$; thus:\n\\[\n1-\\gamma\\ge\\sqrt{\\gamma^2+b^2}.\n\\]\nThis is equivalent to $\\gamma\\le\\gamma_0:=\\half(1-b^2)$.\nThe inequality $\\id-A-B+G\\ge\\nul$ is equivalent to\n\\[\n1-\\alpha+\\gamma\\ge \\sqrt{(\\alpha-\\gamma)^2+b^2}.\n\\]\nThis is solved by $\\gamma\\ge \\alpha-\\half(1-b^2)=\\alpha-\\gamma_0$. \n\nTo summarize: the given effects $A,B$ are jointly measurable if and only if\n\\[\n\\alpha-\\gamma_0\\le\\min\\{\\gamma_0,\\alpha\\},\\quad \\gamma_0\\equiv\\half(1-b^2).\n\\]\nThe nontriviality conditions assume here the form\n\\[\n|\\alpha-\\beta|=|2-\\alpha-\\beta|=1-\\alpha < \\no{\\va-\\vb}=\\no{\\va+\\vb}=\\sqrt{\\alpha^2+b^2},\n\\]\nwhich is equivalent to $\\gamma_0<\\alpha$. In this case the joint measurability condition reduces to\n$\\alpha\/2\\le\\gamma_0$.\n\nWe are now ready to show that condition (\\ref{eqn:sufficient}) can be violated in nontrivial cases. In the given constellation, this inequality assumes the form\n\\[\n\\sqrt{\\alpha^2+b^2}+\\frac{\\alpha(1-\\alpha)}{\\sqrt{\\alpha^2+b^2}}\\le 1.\n\\]\nFor the choice $\\alpha=\\half=1-b^2=2\\gamma_0$, the left hand side becomes $2\/\\sqrt 3$, which is greater than 1. However, this choice fulfills the joint measurability and nontriviality conditions.\n\\end{example}\n\n\n\\subsection{Covariant joint observables}\\label{sec:covjoint}\n\nIn what follows we will investigate implications of covariance. In\nthis way we establish a far-reaching analogy to similar studies made\non approximate joint measurements of position and momentum where\ncovariance (under translations on phase space) was found to be\nparamount (cf. the review \\cite{BuHeLa06}).\n\nLet us first note that there is a unitary operator $U$ such that\nboth $\\Ea$ and $\\Eb$ are covariant with respect to $U$. Namely, fix\na unit vector $\\mathbf{u}$ orthogonal to both $\\mathbf{a}$ and\n$\\mathbf{b}$ and choose $U=\\vu\\cdot\\vsigma$.\n\nWe say that a joint observable $\\G$ of $\\Ea$ and $\\Eb$ is covariant\nwith respect to $U$, or $U$-covariant, if\n\\begin{equation}\\label{covG}\n\\begin{array}{ll}\nU\\G_{++}U=\\G_{--}, & \\\\\nU\\G_{+-}U=\\G_{-+}. &\n\\end{array}\n\\end{equation}\nSince $\\Ea$ and $\\Eb$ are $U$-covariant, the two equations in\n(\\ref{covG}) are equivalent and thus, already one of them implies\nthat $\\G$ is $U$-covariant.\n\n\\begin{proposition}\\label{prop:covjoint}\nIf $\\Ea$ and $\\Eb$ are jointly measurable, then they have a\n$U$-covariant joint observable.\n\\end{proposition}\n\n\\begin{proof}\nLet $\\G$ be a joint observable of $\\Ea$ and $\\Eb$. Define\n\\begin{eqnarray*}\n\\widetilde{\\G}_{++} &=& \\half \\left( \\G_{++} + U\\G_{--}U \\right), \\\\\n\\widetilde{\\G}_{+-} &=& \\half \\left( \\G_{+-} + U\\G_{-+}U \\right), \\\\\n\\widetilde{\\G}_{-+} &=& \\half \\left( \\G_{-+} + U\\G_{+-}U \\right), \\\\\n\\widetilde{\\G}_{--} &=& \\half \\left( \\G_{--} + U\\G_{++}U \\right).\n\\end{eqnarray*}\nEach operator $\\widetilde{\\G}_{\\pm\\pm}$ is a convex combination of\ntwo effects, hence an effect. Moreover, the sum of these effects is\n$\\id$ and thus, $\\widetilde{\\G}$ is an observable.\n\nWe have\n\\begin{eqnarray*}\n\\widetilde{\\G}_{++} + \\widetilde{\\G}_{+-} &=& \\Ea_+,\\\\\n\\widetilde{\\G}_{++} + \\widetilde{\\G}_{-+} &=&\\Eb_+,\n\\end{eqnarray*}\nshowing that $\\widetilde{\\G}$ is a joint observable of $\\Ea$ and $\\Eb$.\nUsing the fact that $U^2=\\id$ we immediately see that $U\\widetilde{\\G}_{++}U=\\widetilde{\\G}_{--}$, meaning that $\\widetilde{\\G}$ is $U$-covariant.\n\\end{proof}\n\nWe proceed by characterizing all $U$-covariant joint observables of\n$\\Ea$ and $\\Eb$. Denoting $\\G_{++}=\\half \\left( \\gamma\\id +\n\\vg\\cdot\\vsigma \\right)$ the covariance\ncondition (\\ref{covG}) can be written in the form\n\\begin{equation}\n\\vg-(\\vu\\cdot\\vg)\\vu=\\half(\\va+\\vb),\n\\end{equation}\nwhich means that $\\vg=\\half(\\va+\\vb)+p\\vu$ for some $p\\in\\R$.  The joint measurability condition (\\ref{simple-coex}) reduces to the requirement that\n\\begin{equation}\\label{gammap}\n\\sqrt{\\tfrac{1}{4}\\no{\\va+\\vb}^2+p^2} \\leq \\gamma \\leq 1 -\\sqrt{\\tfrac{1}{4}\\no{\\va-\\vb}^2+p^2}.\n\\end{equation}\nWe conclude that $U$-covariant joint observables of $\\Ea$ and $\\Eb$\nare characterized by the pairs $(\\gamma,p)$ satisfying\n(\\ref{gammap}). The covariant joint observable $\\G$ corresponding to $(\\gamma,p)$ is\n\\begin{eqnarray*}\n\\G_{++} &=& \\frac{\\gamma}{2} \\id + \\frac{1}{4}(\\va+\\vb)\\cdot\\vsigma+\\frac{p}{2}\\vu\\cdot\\vsigma, \\\\\n\\G_{+-} &=& \\frac{1-\\gamma}{2} \\id + \\frac{1}{4}(\\va-\\vb)\\cdot\\vsigma - \\frac{p}{2}\\vu\\cdot\\vsigma, \\\\\n\\G_{-+} &=& \\frac{1-\\gamma}{2} \\id - \\frac{1}{4}(\\va-\\vb)\\cdot\\vsigma - \\frac{p}{2}\\vu\\cdot\\vsigma, \\\\\n\\G_{--} &=& \\frac{\\gamma}{2} \\id -\\frac{1}{4}(\\va+\\vb)\\cdot\\vsigma + \\frac{p}{2}\\vu\\cdot\\vsigma.\n\\end{eqnarray*}\n\n\nIf a pair $(\\gamma,p)$ satisfies condition (\\ref{gammap}), then so\ndoes $(\\gamma,0)$. Hence, $\\Ea$ and $\\Eb$ have a $U$-covariant joint\nobservable if and only if there is a $\\gamma$ such that\n\\begin{equation}\\label{gamma}\n\\half\\no{\\mathbf{a}+\\mathbf{b}} \\leq \\gamma \\leq 1 -\n\\half\\no{\\mathbf{a}-\\mathbf{b}},\n\\end{equation}\nor equivalently, if and only if inequality (\\ref{eqn:generalineq})\nholds. This together with Proposition \\ref{prop:covjoint} gives the result\ncited in Proposition \\ref{prop:Paulsineq}. Inequality (\\ref{gamma}) implies that\n\\begin{equation}\\label{eqn:covariantineq}\n\\half\\no{\\mathbf{a}+\\mathbf{b}} \\leq \\half( 1+\\va\\cdot\\vb )\n\\leq 1- \\half\\no{\\mathbf{a}-\\mathbf{b}}.\n\\end{equation}\nThus, if $\\Ea$ and $\\Eb$ are jointly measurable, then they have a joint observable $\\G^0$ corresponding to the choice $\\gamma=\\gamma_0:=\\half( 1+\\va\\cdot\\vb )$ and $p=0$. The effects of $\\G^0$ can be written in the form\n\\begin{equation}\n\\G^0_{ij}=\\half \\left( \\Ea_i\\Eb_j + \\Eb_j\\Ea_i \\right),\\quad i,j=\\pm.\n\\end{equation}\n\nIf we have the limiting case of condition (\\ref{gamma}), i.e. \n\\begin{equation}\n\\half\\no{\\mathbf{a}+\\mathbf{b}}=1 -\n\\half\\no{\\mathbf{a}-\\mathbf{b}},\n\\end{equation}\nthen $(\\gamma_0,0)$ is the only possible pair\nand hence, in this case $\\G^0$ is the unique $U$-covariant joint observable of $\\Ea$ and $\\Eb$.\nIn all other situations of covariant joint measurements except this limiting case, there is a\ncontinuum of possible pairs $(\\gamma,p)$. The joint observable $\\G$\ncorresponding to $(\\gamma,p)$ is informationally complete if and\nonly if $p\\neq 0$; this follows directly from \\cite[Theorem\n4.7]{Busch86}.\n\nFinally, we note that the covariance of $\\Ea$ and $\\Eb$ does not imply that they have only covariant joint observables. To give an example, assume that the vectors $\\va$ and $\\vb$ satisfy $\\va\\cdot\\vb\\geq 0$ and $\\no{\\va+\\vb} < 1$, so that $\\Ea$ and $\\Eb$ are jointly measurable. Fix a number $t$ such that $0<t\\leq \\no{\\va+\\vb}^{-1} -1$, and define\n\\begin{eqnarray*}\n\\G_{++} &=& \\frac{1}{4} \\id + \\frac{1}{4}(1+t)(\\va+\\vb)\\cdot\\vsigma, \\\\\n\\G_{+-} &=& \\frac{1}{4} \\id + \\frac{1}{4}((1-t)\\va-(1+t)\\vb)\\cdot\\vsigma, \\\\\n\\G_{-+} &=& \\frac{1}{4} \\id + \\frac{1}{4}((1-t)\\vb-(1+t)\\va)\\cdot\\vsigma \\\\\n\\G_{--} &=& \\frac{1}{4} \\id -\\frac{1}{4}(1-t)(\\va+\\vb)\\cdot\\vsigma.\n\\end{eqnarray*}\nThen $\\G$ is a joint observable for $\\Ea$ and $\\Eb$ but it is not covariant.\nIndeed, the above condition on $t$ guarantees that inequalities (\\ref{simple-coex}) are satisfied. The eigenvalues of $\\G_{++}$ and $\\G_{--}$ are different and thus, the covariance condition (\\ref{covG}) cannot be satisfied with any unitary operator $U$.\n\n\n\\subsection{Joint measurability vs. sharpness}\\label{sec:jm-sharpness}\n\nIt is instructive to write down the joint measurement condition for two covariant observables $\\Ea$ and $\\Eb$, assuming that the vectors $\\va$ and $\\vb$ are orthogonal. The inequality (\\ref{eqn:generalineq}) takes now the form\n\\begin{equation}\n\\U(\\Ea)+\\U(\\Eb)\\geq 1,\n\\end{equation}\nshowing that the joint measurability is achieved exactly when the observables are made unsharp enough. \n\nIn the general case, we can transform (\\ref{eqn:generalineq}) by repeated squaring into the equivalent inequality\n\\begin{equation}\\label{eqn:generalineq2}\n\\no{\\va}^2 + \\no{\\vb}^2 \\leq 1+ (\\va\\cdot\\vb)^2,\n\\end{equation}\nwhich can be written in the form\n\\begin{equation}\n\\no{\\va\\times\\vb}^2\\le(1-\\no{\\va}^2)(1-\\no{\\vb}^2).\n\\end{equation}\nThe term on the left hand side is equal to $4\\no{[\\Eaa_+,\\Ebb_+]}^2$.\nThe term on the right hand side turns out to give a bound for the degrees\nof sharpness of $\\Eaa$ and $\\Ebb$. Considering the formulas (\\ref{eqn:u}) and (\\ref{eqn:sharpness-ineq})\nwe obtain the following.\n\\begin{proposition}\nIf two qubit observables $\\Eaa,\\Ebb$ are jointly measurable,\nthe degrees of their unsharpness satisfy the inequality\n\\begin{equation}\\label{eqn:jm-unsharp}\n\\U(\\Eaa)\\,\\U(\\Ebb)\\,\\ge\\,\\U(\\Ea)\\,\\U(\\Eb)\\,\\ge \\,4\\no{[\\Eaa_+,\\Ebb_+]}^2.\n\\end{equation}\nIf $\\alpha=\\beta=1$, this inequality is in fact equivalent to the joint measurability condition.\n\\end{proposition}\nThis shows that the intrinsic sharpness of two jointly measurable simple qubit observables\n$\\Eaa$ and $\\Ebb$ is limited by the noncommutativity of the generating effects.\n\n\n\n\\section{Approximate joint measurement for two sharp qubit observables}\\label{sec:approx-joint}\n\n\nTwo sharp observables $\\En$ and $\\Em$ are jointly measurable exactly\nwhen they commute, and this happens if and only if $\\vn=\\pm\\vm$. In\nthis section we consider the case $\\vn\\neq\\pm\\vm$, so that only\napproximate joint measurements are possible. The idea is to choose a\njointly measurable pair $(\\Eaa,\\Ebb)$ to approximate the sharp pair\n$(\\En,\\Em)$. To be specific, and without loss of generality, we assume\n$\\cos\\theta:=\\vn\\cdot\\vm > 0$.\n\nWe call a point $\\left(\\D_1,\\D_2\\right)\\in[0,1]\\times[0,1]$\n\\emph{admissible} if  $\\D_1=\\D(\\Eaa,\\En)$ and $\\D_2=\\D(\\Ebb,\\Em)$ for some jointly measurable observables $\\Eaa$ and $\\Ebb$. Not all points in the square $[0,1]\\times[0,1]$ are admissible; for instance the point $(0,0)$ is not an admissible point since this would mean that $\\Eaa=\\En$ and $\\Ebb=\\Em$. We show in the following that there are also other points which are not admissible. The set of admissible points gives us a characterization on the quality of possible approximate joint measurements. \n\nThe search for admissible points $\\left( \\D_1,\\D_2\\right)$\nis narrowed down by the following simple observation:\n\\begin{example}\nLet $\\alpha\\in[0,2]$. \nThen $\\D(\\E^{\\alpha,\\vnull},\\En)=\\half\\max\\{\\alpha,2-\\alpha\\}$ and therefore\n\\begin{eqnarray}\n\\left\\{\\D(\\E^{\\alpha,\\vnull},\\En)\\,:\\,\\alpha\\in[0,2]\\right\\}\n=[\\half,1].\n\\end{eqnarray}\n\\end{example}\n\nThus, approximations by means of trivial observables\nwill never give distances below $\\half$. Furthermore, since\n$\\E^{\\alpha,\\vnull}$ is jointly measurable with any observable\n$\\Ebb$, and since $\\D(\\Ebb,\\Em)$ can assume any value in $[0,1]$, it\nfollows that all points in the set\n$[0,1]\\times[0,1]\\setminus[0,\\half]\\times[0,\\half]$ are trivially admissible.\nWe will therefore concentrate on admissible points $\\left(\\D_1,\\D_2\\right)$ in the region $[0,\\half]\\times[0,\\half]$.\n\nThe next two results are not complicated but require some preparation and will be proven in the Appendix.\n\\begin{proposition}\\label{prop:admiss-realization}\nAny admissible point\n$\\left(\\D_1,\\D_2\\right)\\in[0,\\half]\\times[0,\\half]$ has a\nrealization of the type $\\D_1=\\D(\\Ea,\\En)$, $\\D_2= \\D(\\Eb,\\Em)$,\nwhere $\\va$ and $\\vb$ are in the plane spanned by $\\vn$ and $\\vm$.\n\\end{proposition}\n\\begin{proposition}\\label{prop:convex}\nThe set of admissible points is a closed convex set which is reflection\nsymmetric with respect to the axis $\\D_1=\\D_2$; that is, with every\nadmissible point $\\left(\\D_1,\\D_2\\right)$ the point $\\left(\\D_2,\\D_1\\right)$\nis also admissible. Thus the segment of the boundary curve defined as\nthe graph of the function\n\\begin{equation}\n\\D_1\\mapsto\\inf\\{\\D_2\\,:\\,\\left(\\D_1,\\D_2\\right)\\ \\mathrm{is\\ admissible}\\}\n\\end{equation}\nis convex, symmetric and belongs to the set of admissible points.\n\\end{proposition}\n\n\n\\begin{example}\\label{ex:coex0}\nIf $\\D_1=\\D(\\Ea,\\En)=0$ (i.e. $\\va=\\vn$), then the joint measurability\nrequirement implies that $\\va||\\vb$ and thus,\n$$\n\\D(\\Eb,\\Em)=\\half\\no{\\vb-\\vm}\\geq \\half \\sqrt{1-(\\vn\\cdot\\vm)^2}=\\half\\sin\\theta.\n$$\nThe lower bound is attained when $\\vb=\\cos\\theta\\ \\vn=(\\vn\\cdot\\vm)\\vn$.\nWe conclude that\n$\\left( 0,\\half\\sin\\theta \\right)$ and $\\left( \\half\\sin\\theta,0 \\right)$\nare points in the boundary of the admissible region.\n\\end{example}\n\n\n\\begin{figure}\\label{fig:admissible}\n\\begin{center}\n\\includegraphics[width=7cm]{admissible.eps}\n\\caption{The admissible region (dotted area) and the line $\\D_1+\\D_2=2\\D_0$ (thick line). The dashed line is the symmetry axis $\\D_1=\\D_2$.}\n\\end{center}\n\\end{figure}\n\nWe next determine the boundary point with $\\D_1=\\D_2=:\\D_0$. Due to the convexity\nof the admissible region and its reflection symmetry  with respect to the line $\\D_1=\\D_2$,\nit follows immediately that the admissible region is bounded below tightly by the straight line\n$\\D_1+\\D_2=2\\D_0$. This situation is sketched in Figure \\ref{fig:admissible}. Determination of the value of $\\D_0$ yields the following result.\n\n\n\\begin{proposition}\\label{prop:D0}\n Any admissible point $\\left(\\D_1,\\D_2\\right)=\\left(\\D(\\Eaa,\\En),\\D(\\Ebb,\\Em)\\right)$ satisfies\n the inaccuracy trade-off relation\n\\begin{equation}\\label{eqn:dist-approx}\n\\D(\\Eaa,\\En)+\\D(\\Ebb,\\Em)\\ge 2\\D_0,\n\\end{equation}\nwhere\n\\begin{equation}\n2\\D_0=\\tfrac1{\\sqrt2}\\left[\\half\\no{\\vn+\\vm}+\\half\\no{\\vn-\\vm}-1\\right]=\n\\tfrac1{\\sqrt2}\\left(\\cos\\tfrac\\theta2+\\sin\\tfrac\\theta2-1\\right).\n\\end{equation}\nThe point $\\left(\\D_0,\\D_0\\right)$ is admissible.\n\\end{proposition}\n\n\\begin{proof}\nConsider the set of all jointly measurable covariant observables $\\Ea,\\Eb$\nsuch that $\\va,\\vb$ have equal fixed distance from $\\vn,\\vm$,\nrespectively: $\\no{\\va-\\vn}=\\no{\\vb-\\vm}\\equiv d$ (so that\n$\\D(\\Ea,\\En)=\\D(\\Eb,\\Em)=d\/2$). If $(\\va,\\vb)$ is not symmetric\nunder reflection with respect to the line parallel to $\\vn+\\vm$, denote by\n$\\bar{\\va}$ and $\\bar{\\vb}$ the mirror images of $\\vb$ and $\\va$,\nrespectively. Then, if $\\Ea,\\Eb$ are jointly measurable, so are\n$\\E^{1,\\bar{\\va}},\\E^{1,\\bar{\\vb}}$ as the condition (\\ref{eqn:generalineq}) is invariant under reflections. Due to Proposition \\ref{prop:joint-convex}, the observables\n$\\half\\Ea+\\half\\E^{1,\\bar{\\va}}=\\E^{1,\\half(\\va+\\bar{\\va})}$ and\n$\\half\\E^{1,\\vb}+\\half\\E^{1,\\bar{\\vb}}=\\E^{1,\\half(\\vb+\\bar{\\vb})}$ are jointly measurable. It is clear from their definitions that the vectors $\\half(\\va+\\bar{\\va})$ and $\\half(\\vb+\\bar{\\vb})$ are mirror images of each other. As $\\va,\\vb$ have equal distance $d$ from $\\vn,\\vm$, respectively, this means that $\\va$ and $\\bar{\\va}$ have equal distance $d$ from $\\vn$. It follows that the distance from $\\vn$ to $\\half(\\va+\\bar{\\va})$ is less than $d$ (or $d$ if $\\va=\\bar{\\va}$). We conclude that if $\\va,\\vb$ are not mirror images of each other, there is a pair of jointly measurable covariant observables with smaller (equal) distances from $\\En,\\Em$ and mirror symmetric vectors. This shows that the minimal equal distance approximations of $\\En,\\Em$ by means of jointly measurable\nobservables occur among the covariant pairs with $\\va,\\vb$ mirror symmetric with respect to $\\vn+\\vm$.\n\nIf coordinates are chosen such that\n$\\vn=(\\sin\\frac\\theta2,\\cos\\frac\\theta2)$,\n$\\vm=(-\\sin\\frac\\theta2,\\cos\\frac\\theta2)$, then let a symmetric\npair $\\va,\\vb$ be given by $\\va=(u,v)$ and $\\vb=(-u,v)$, with\n$u,v>0$. For such pairs, the joint measurability condition for\n$\\Ea,\\Eb$ assumes the form $u+v\\le1$. It follows that the shortest\n(equal) distances $d$ of $\\va,\\vb$ from $\\vn,\\vm$ are assumed when\n$u+v=1$ and $\\vn-\\va$ is perpendicular to the line $u+v=1$. But this\ndistance $d$ is equal to the distance of the lines $u+v=1$ and\n$u+v=\\cos(\\frac\\theta2)+\\sin(\\frac\\theta2)$, hence\n\\[\nd=\\tfrac1{\\sqrt2}\\left(\\cos\\tfrac\\theta2+\\sin\\tfrac\\theta2-1\\right).\n\\]\n\\end{proof}\n\n\\begin{figure}\\label{fig:arrows}\n\\begin{center}\n\\includegraphics[width=8cm]{arrows.eps}\n\\caption{The vectors corresponding to the optimal approximations $\\Ea,\\Eb$ and of optimal coarse-grainings $\\Ead,\\Ebd$.}\n\\end{center}\n\\end{figure}\n\nThe result of Proposition \\ref{prop:D0} shows in which way the quality of the approximations is limited by the separation of the sharp observables to be  approximated in a simultaneous measurement. This relation becomes perhaps even more transparent when we write the number $\\D_0$ in the form\n\\begin{equation}\\label{eqn:D0}\n\\D_0=\\tfrac1{2\\sqrt 2}\\left[\\D(\\En,\\Em)+\\D(\\En,\\Emn)-1\\right].\n\\end{equation}\nThe appearance of $\\D(\\En,\\Emn)$ in (\\ref{eqn:D0}) is explained by the fact that the joint measurability criterion is blind to the labeling of outcomes. \n\nNote that \n$\\cos\\frac\\theta2+\\sin\\frac\\theta2=(1+\\sin\\theta)^{1\/2}$ and $\\sin\\theta=\\no{\\vn\\times\\vm}=2\\no{[\\En,\\Em]}$. Thus $\\D_0$ is an increasing function of the degree of noncommutativity of the sharp observables to be estimated.\n\nThe approximations $\\Ea$ and $\\Eb$ leading to the boundary point $(\\D_0,\\D_0)$ are generally not among the  coarse-grainings of $\\En$ and $\\Em$ (in the sense of Section \\ref{sec:approximate}). Indeed, let us denote by $\\D_0^c$ the smallest number achieved under the assumptions that $\\D_0^c=\\D(\\Ead,\\En)=\\D(\\Ebd,\\Em)$ and that $\\Ead,\\Ebd$ are jointly measurable and coarse-grainings of $\\En,\\Em$, respectively. If the vectors $\\vn$ and $\\vm$ are orthogonal, then $\\D_0^c=\\D_0$. However, if $0<\\theta<\\frac{\\pi}{2}$, then\n\\begin{equation}\n\\D_0^c=\\half \\left( 1- \\frac{\\sqrt{1-\\sin\\theta}}{\\cos\\theta} \\right) > \\D_0.\n\\end{equation}\nThe vectors $\\va,\\vb$ and $\\va',\\vb'$ are illustrated in Figure \\ref{fig:arrows}.\nWe conclude that to attain the best jointly measurable approximations of two sharp qubit observables, we are forced to seek approximating observables beyond their coarse-grainings.  \n\nFinally, we note that it would be interesting to determine the full convex boundary curve of the region of admissible points $\\left(\\D_1,\\D_2\\right)$. Some numerically calculated boundary curves are drawn in \\cite{HeReSt08}, but their analytic form is not yet known.\n\n\n\\section{Conclusion and outlook}\\label{sec:conclusion}\n\nIn this paper we have quantified the necessary inaccuracies in approximating noncommuting sharp qubit observables by means of a pair of jointly measurable pair of observables (Eq.~(\\ref{eqn:dist-approx})). We also exhibited the necessary unsharpness that observables $\\Eaa,\\Ebb$ must have in order to be jointly measurable (Eq.~(\\ref{eqn:jm-unsharp})).\nIf a sharp observable $\\E^{1,\\hat \\va}$ is approximated by one of its coarse-grainings $\\Eaa$, the distance is related to the sharpness of $\\Eaa$ via the relation (\\ref{eqn:dist-sharp}).\n\nTrough the case study of qubit observables we have demonstrated the conceptual difference of measurement inaccuracy and intrinsic unsharpness. This sheds some new light on the joint measurement problem raised by Uffink in \\cite{Uffink94}, so we shortly recall his argumentation. Uffink analyzed a definition of ``non-ideal\" or ``unsharp\" joint measurement of two noncommuting observables that had previously been sketched out more or less informally by various authors. This definition\ncaptures the idea that smearings of two noncommuting sharp observables may have a\njoint observable. As it was formulated, this definition allowed\nany smeared or coarse-grained version of an observable to be an approximation of\nthat observable, without further stipulations on the quality of the approximation. This entails\nthat even trivial observables (which are always among the coarse-grainings of any observable)\ncan be taken to represent a sort of non-ideal measurement of a given observable.\n\nUffink presented an example that makes this definition look absurdly comprehensive and indeed counter-intuitive: he considered two pairs of observables, $(\\sigma_x,Q)$ and $(\\sigma_z,P)$ and took\n$\\sigma_x$ as a coarse-graining of the first pair and $P$ as a coarse-graining of the second. Then\n$(\\sigma_x,P)$ is a joint observable for these two, and according to the letter of the\ndefinition, it would have to be considered as representing a non-ideal or unsharp joint measurement of the original pairs.\n\nNow, Uffink argued that while the final joint observable had $\\sigma_x$ and $P$ as coarse-grainings (namely, marginals), the original observables were in no way coarse-grainings of it. Hence there was no plausible sense in which $(\\sigma_x,P)$ could be regarded as representing a non-ideal joint measurement of the original pairs of observables. He thus pointed out rightly that a universal definition or criterion of approximate joint measurability was missing. But then he jumped to the conclusion  that POMs  do not contribute to solving the joint measurement problem.\n\nWe think that the present paper and many preceding it demonstrate that POMs do provide an appropriate language to clarify the definition and quantification of approximate measurements, and to determine any limitations to the accuracy of joint approximations of noncommuting pairs of observables. It is obvious that any measurement can be considered as an ``approximate\" joint\nmeasurement of an arbitrary collection of observables. Even doing nothing and\nrandomly picking outcomes constitutes a trivial ``non-ideal\" joint\nmeasurement of any given set of observables. There is no problem in\nallowing a definition of non-ideal or approximate joint\nmeasurements to include trivial cases; what makes any such\ndefinition useful is whether it allows one to give quantifications\nof how well each of the observables in question is being\napproximated by a given scheme. As we have shown in this paper and its companion\n\\cite{BuHeLa06}, such quantifications\ncan indeed be formulated and yield a nontrivial notion of approximate measurement, leading\nto the conclusion that there are universal limitations to the accuracies with which noncommuting pairs of observables can be approximately measured together. \n\nIf the quality of the approximation is to be  optimized, the approximating observables being measured jointly must be unsharp; and the required\ndegree of unsharpness is linked with the quality of the approximations specified.\nUsing the definition of approximation introduced here, and keeping in mind the conceptual difference between the relation of approximation and the property of intrinsic unsharpness, it is clear that the above ``absurd\" example considered by Uffink is simply not based on good approximations and would therefore not be regarded as a useful joint measurement.\n\nThe quantifications of inaccuracy and intrinsic unsharpness presented here for the case of qubit observables complements analogous investigations carried out in the case of continuous observables in \\cite{BuHeLa06, BuPe06, CaHeTo07}. A unified approach and associated trade-off relations for the approximate joint measurements of general pairs of noncommuting quantities is still outstanding.\n\n\\section*{Appendix: Proofs of Propositions \\ref{prop:admiss-realization} and \\ref{prop:convex}}\n\n\n\\noindent\n(a) If $(\\D_1,\\D_2)$ is an admissible point, then also $(\\D_2,\\D_1)$\nis an admissible point.\\\\\n{\\em Proof.}\nIf $(\\alpha,\\va)$ and $(\\beta,\\vb)$ realize the distances $\\D_1$ and $\\D_2$,\nrespectively, then choose $(\\alpha',\\va')$ and $(\\beta',\\vb')$ as follows: $\\alpha'=\\beta$,\n$\\va'$ has the length of $\\vb$ and its angle relative to $\\vn$ is equal to the angle between\n$\\vb$ and $\\vm$; similarly, $\\beta'=\\alpha$, $\\vb'$ has the length of $\\va$ and its angle relative to\n$\\vm$ is the same as the angle between $\\va$ and $\\vn$. This ensures that $(\\D_1',\\D_2')=\n(\\D_2,\\D_1)$.\n\\qed\\\\\n\n\\noindent (b) Assume that $\\left( \\D_1,\\D_2\n\\right)=\\left(\\D(\\Eaa,\\En),\\D(\\Ebb,\\Em) \\right)$ is an admissible\npoint. As shown in Proposition \\ref{prop:alsojoint}, the joint\nmeasurability of $\\Eaa$ and $\\Ebb$ implies that $\\Ea$ and $\\Eb$ are\njointly measurable.\nDefine $\\va_0$ and $\\vb_0$ to be the projections of the vectors\n$\\va$ and $\\vb$, respectively, onto the plane spanned by $\\vn$ and\n$\\vm$. Then\n\\begin{equation*}\n\\no{\\va+\\vb}\\geq\\no{\\va_0+\\vb_0},\\qquad\n\\no{\\va-\\vb}\\geq\\no{\\va_0-\\vb_0},\n\\end{equation*}\nand hence, $\\E^{1,\\va_0}$ and $\\E^{1,\\vb_0}$ are jointly measurable.\nUsing (\\ref{eqn:distance-ab}) one finds that\n\\begin{equation}\\begin{split}\n\\D(\\E^{1,\\va_0},\\En) &\\leq  \\D(\\Ea,\\En) \\leq \\D_1,\\\\\n\\D(\\E^{1,\\vb_0},\\Em) &\\leq  \\D(\\Eb,\\Em) \\leq \\D_2 .\n\\end{split}\\end{equation}\nWe conclude that the best approximations are to be found from the\nsubset of covariant qubit observables, with vectors $\\va$ and $\\vb$\nin the plane spanned by $\\vn$ and $\\vm$.\\\\\n\n\\noindent\n(c)  If $(\\D_1,\\D_2)$ is an admissible point, then also $(\\D'_1,\\D'_2)$ is an admissible point whenever\n$\\D_i\\leq\\D'_i\\leq\\half$.\\\\\n{\\em Proof.}\nIn view of (b) it is sufficient to show the result for admissible points which have realizations\n$\\left(\\D(\\Ea,\\En),\\D(\\Eb,\\Em)\\right)$. Thus let $\\Ea$, $\\Eb$ be two jointly measurable observables. Using Proposition \\ref{prop:Paulsineq}, we note that also $\\Ea$ and $\\E^{1,r\\vb}$ are jointly measurable for any $0\\leq r\\leq 1$. Since the function\n\\[\nr\\mapsto\\D(\\E^{1,r\\vb},\\Em)=\\half\\no{\\vm-r\\vb}\n\\]\nis continuous, it takes all values between $\\D(\\Eb,\\Em)$ and $\\half$. We can similarly realize all values between $\\D(\\Ea,\\En)$ and $\\half$.\n\\qed\\\\\n\n\\noindent\n(d) Observations (b) and (c) taken together entail\nProposition \\ref{prop:admiss-realization}.\\qed\\\\\n\n\\noindent\n(e) The admissible region is a convex set.\\\\\n{\\em Proof.} Let $\\left(\\D_1,\\D_2\\right)$ and $\\left(\\D_1',\\D_2'\\right)$ be realized by\n$(\\alpha,\\va), (\\beta,\\vb)$ and $(\\alpha',\\va'), (\\beta',\\vb')$ respectively. Let $t\\in[0,1]$.\nThen for\n$(\\alpha_t,\\va_t):=(t\\alpha+(1-t)\\alpha',t\\va+(1-t)\\va')$ and\n$(\\beta_t,\\vb_t):=(t\\beta+(1-t)\\beta',t\\vb+(1-t)\\vb')$, we obtain associated distances\n$\\D_{1,t}$ and $\\D_{2,t}$ which satisfy\n\\[\n\\D_{k,t}\\le t\\D_k+(1-t)\\D_k',\\quad k=1,2.\n\\]\nThis together with (c) proves the claim.\n\\qed\\\\\n\n\\noindent\n(f) The set of admissible points is closed.\\\\\n{\\em Proof.} The mapping\n\\begin{equation}\n(\\va,\\vb)\\mapsto (\\D(\\Ea,\\En), \\D(\\Eb,\\Em))=\\half(\\no{\\va-\\vn},\\no{\\vb-\\vm})\n\\end{equation}\nfrom $\\R^3\\times\\R^3$ to $\\R\\times\\R$ is continuous. The set of admissible points is the image of the compact set\n\\begin{equation}\n\\{ (\\va,\\vb)\\in\\R^3\\times\\R^3 \\mid \\no{\\va}\\leq 1, \\no{\\vb}\\leq 1, \\no{\\va-\\vb}+\\no{\\va+\\vb}\\leq 2 \\},\n\\end{equation}\nhence it is itself closed and  contains its boundary. This and (b) shows that for given\n$\\D_1\\in[0,\\half]$, there is a minimal number $\\D_2^{\\min}(\\D_1)$ such that\nall $\\left(\\D_1,\\D_2\\right)$ with $\\D_2^{\\min}(\\D_1)\\leq\\D_2\\leq\\half$ are admissible pairs\nwhile pairs with $\\D_2<\\D_2^{\\min}(\\D_1)$ are not admissible.\\qed\\\\\n\n\\noindent\n(g)  Since the admissible region is a convex set, the function $\\D_1\\mapsto\\D_2^{\\min}(\\D_1)$ is convex and therefore continuous. Due to (a), the curve is symmetric under reflection with respect to the line $\\D_1=\\D_2$. We conclude that this function gives the lower boundary curve of the set of  admissible points, and that the points on this curve are admissible. Together with (e) and (f), this completes the proof of Proposition \\ref{prop:convex}.\n\n\\vspace{10pt}\n\n\\noindent\n{\\bf Acknowledgement.} This work was initiated during T.H.'s visit at Perimeter Institute. Hospitality and support to both authors during this visit and to P.B. during the completion phase are gratefully acknowledged. T.H. acknowledges the support of the European Union project CONQUEST during the final phase of this work.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nThe bar and cobar constructions are an adjoint pair of functors\nbetween the categories of augmented differential graded (dg) algebras and \ncoaugmented dg coalgebras, both defined over a fixed commutative\nring $k$,\n\\begin{align*}\n&\\xymatrix{\\augdgAlgk \\ar@<-1.2ex>[rr]_{\\Bar {}} &&\n \\coaugdgCoalgk. \\ar@<-1.2ex>[ll]_{\\cobar{}}}\\\\\n\\intertext{This adjoint pair is the algebraic analogue of the\nclassifying space and Moore loop space adjoint pair between topological monoids\nand based topological spaces.\nAnalogous to the situation in topology, the bar and cobar functors\n                                                              are\n                                                              decidedly\n                                                              nontrivial:\n                                                              the unit\n                                                              of the\n                                                              adjunction\n                                                              is a\n                                                              homotopy\n                                                              equivalence. This\n                                                              non-triviality\n                                                              is\n                                                              at the\n                                                              root of their\n                                                                                                                            usefulness\n                                                                                                                           in\n                                                              algebra. \n                                                              In\nparticular, the bar construction gives\ncanonical resolutions, of both modules and bimodules, and plays a\nlarge role in infinitesimal deformation theory.\n\\endgraf\nStasheff, in \\cite{MR0158400}, relaxed the assumption of\nassociativity in a topological monoid to define an $\\text{A}_{\\infty}$-space, using a generalized notion of\nclassifying space. He then showed that a connected topological space has the homotopy type of a\nloop space exactly when it is an $\\text{A}_{\\infty}$-space (see also \\cite[Chapter\n2]{MR505692}). The algebraic analogue of a connected\n$\\text{A}_{\\infty}$-space is an augmented $\\text{A}_{\\infty}$-algebra \\cite{MR0158400part2},\n                                                            generalizing\n                                                            an augmented dg-algebra.\nAn augmented $\\text{A}_{\\infty}$-algebra is an augmented complex $(A, m^{1})$ and a sequence\n                                                            of\n                                                            augmented\n                                                            maps\n                                                            $m^{n}:\n                                                            A^{\\otimes\n                                                            n} \\to A,$ \nwhere $m^{2}$ satisfies the Leibniz rule with respect to $m^{1},$ $m^{3}$ is a nullhomotopy for the associator of $m^{2}$, and\nmore generally, the $m^{n}$ satisfy the\n                                                            quadratic equations necessary for a bar construction,\n                                                           \n                                                           \n                                                            giving the following diagram:}\n   &\\xymatrix{\\augaiAlgk \\ar@<-1.2ex>[rr]_{\\Bar {}} &&\n \\coaugdgCoalgk. \\ar@<-1.2ex>[ll]_{\\cobar{}}}\n\\end{align*}\nEvery $\\text{A}_{\\infty}$-algebra is homotopy equivalent to a dg-algebra, so nothing is gained at\nthe homotopy level by enlarging the category of dg-algebras, but often\nthere are dramatically smaller $\\text{A}_{\\infty}$-versions \nsmaller than the dg-models, e.g., the cochains of the classifying space of a finite\ngroup; see \\cite[\\S 6]{MR2844537}.\n\nThe augmentation assumption plays a vital but subtle role in the\nnontriviality of the above functors. Indeed, by bar construction of an augmented\ndg-algebra\n$\\epsilon: A \\to k,$ we mean the bar construction\napplied to the nonunital algebra $\\ker \\epsilon$. The bar\nconstruction of\na unital dg-algebra is homotopy equivalent to the trivial coalgebra, destroying the ``homotopy type'' of the unital\ndg-algebra. In particular, the resolutions traditionally constructed\nusing the bar construction, will not necessarily be resolutions if one\ndoesn't kill the unit.  Augmented\nalgebras are exactly those we can do this to, without\nlosing information. All of this remains true for augmented versus strictly unital\n$\\text{A}_{\\infty}$-algebras, summarized in the following diagram:\n\\begin{displaymath}\n  \\begin{tikzpicture}\n  \\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]{\n    \\suaiAlgk & {} \\\\\n    \\augaiAlgk & \\augdgCoalgk. \\\\\n  };\n \\path[-stealth, auto, scale = 2] (m-2-1) edge [subseteq]                  (m-1-1);\n  \\path[-stealth, auto]     (m-2-1) edge node[swap] {$\\bar$} (m-2-2)\n                        (m-1-1) edge node[auto = false, draw, minimum size = 20pt,\n                        dashed, cross out, red, thick]{} (m-2-2);\n\\end{tikzpicture}\n\\end{displaymath}\nPositselski had the insight \nthat the right side of the diagram can be extended to \\emph{curved\ndg-coalgebras}. He showed how to construct, for a strictly unital, but not\nnecessarily augmented, $\\text{A}_{\\infty}$-algebra, a curved bar construction, killing\nthe unit and transferring the potentially lost information\nto a curvature term \\cite{MR1250981,\n  MR2830562}, giving the following diagram:\n\\begin{displaymath}\n  \\begin{tikzpicture}\n  \\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]{\n    \\suaiAlgk & \\curvdgCoalgk \\\\\n    \\augaiAlgk & \\augdgCoalgk. \\\\\n  };\n  \\path[-stealth, auto, scale = 2] (m-2-1) edge [subseteq]                  (m-1-1)\n  (m-2-2) edge [subseteq]                  (m-1-2);\n    \\path[-stealth, auto]          (m-2-1) edge node[swap] {$\\bar$} (m-2-2)\n                        (m-1-1) edge node {$\\bar$} (m-1-2);\n\\end{tikzpicture}\n\\end{displaymath}\nHe proved analogous results for $\\text{A}_{\\infty}$-morphisms and $\\text{A}_{\\infty}$-modules, and also stated a strong converse: the curved\nbar construction characterizes strictly unital $\\text{A}_{\\infty}$-algebras\n(and morphisms, and representations).\n\nThe fundamental idea that a curvature term compensates for lack of\naugmentation is not particularly emphasized in the long paper\n\\cite{MR2830562} (a paper that contains many new and powerful ideas),\nfull details of\nthe proofs are not given, and, most importantly for us, the ground ring is assumed to be a\nfield. In this paper, we give careful proofs of Positselski's results,\nvalid for an arbitrary commutative ground ring (in fact with a few small\nadjustments, noted in remarks, the results hold when\nreplacing modules over a commutative ring with any symmetric monoidal\ncategory with countable coproducts, where finite coproducts are also\nfinite products). Positselski also showed that the bar construction is\nhomotopically non-trivial, and this opens the door to using it for the\nconstruction of resolutions. We do not pursue the generalization from\na field to arbitrary base ring here, but hope to return to it in the future.\n\nThe proofs in this paper use a characterization of\nstrictly unital $\\text{A}_{\\infty}$-algebra structures as Maurer-Cartan elements of a\ncertain dg-Lie algebra (the coassociative analogue of a construction used by Schlessinger and Stasheff for Lie coalgebras\n\\cite{SchlStash}). We also use this characterization to show the dg-Lie algebra of reduced Hochschild\ncochains controls the\nstrictly unital infinitesimal deformations of the corresponding\n$\\text{A}_{\\infty}$-algebra. As motivation and context for using Maurer-Cartan\nelements of a dg-Lie algebra, we include a detailed discussion of the\ndeformation theory of nonunital $\\text{A}_{\\infty}$-algebras via the dg-Lie algebra\nof Hochschild cochains. We also prove linear analogues of all\nof the above results. In particular, we recover, and generalize to arbitrary commutative base\nring, Positselski's result that strictly unital modules correspond\nfunctorially to cofree curved dg-comodules over the curved bar\nconstruction.\n\nThere has been considerable further work developing Positselski's\nideas, especially for operads\n\\cite{MR2993002,DerKosDuality,1403.3644,1612.02254}, see also \\cite{MR3597150}. There has also been much work\non homotopy, or weak, units in an $\\text{A}_{\\infty}$-algebra; see\n\\cite{MR2596638,MR2769322,MR3176644} and the references contained\nthere. An $\\text{A}_{\\infty}$-automorphism does not necessarily preserve a strict\nunit, but it does preserve a homotopy unit (one can take for the\ndefinition of homotopy unit that there is an automorphism that takes it to\na strict unit). Positselski's idea on curvature gives a way of\nmaintaining a strict unit through certain processes, e.g., transfer of\n$\\text{A}_{\\infty}$-structure, rather than working in the larger category of homotopy unital\n$\\text{A}_{\\infty}$-algebras.\n\nFinally, let us mention one motivation for this paper. In\n\\cite{1508.03782} we study projective resolutions of modules over a\ncommutative ring $R = Q\/I$ by putting $Q$-linear strictly unital $\\text{A}_{\\infty}$-structures\non $Q$-projective resolutions of $R$ and its modules. (This example emphasizes the importance of working with\nan arbitrary commutative base ring.) In particular, we show that minimality of $\\text{A}_{\\infty}$-structures\ncharacterizes Golod singuliarities, and the bar construction can then\nbe used to construct the minimal free resolution of every module over\na Golod ring. To work effectively with different classes of singularities, e.g., complete intersections, a\nrelative Koszul duality (relative to $Q$) is needed. We hope to develop this in future work. Throughout this paper we give a sequence of running\nexamples illustrating the elementary, but\ninteresting, example of the Koszul complex on a single element $f$ of\nthe ground ring $k,$ where e.g., if\n$k = \\overbar{C}[x_{1}, \\ldots, x_n]$, then we are studying the zero set of\n$f$ relative to $\\overbar{C}^{n}.$\n\nI would like to thank the referee for his or her careful reading and very\nhelpful comments that improved the exposition of the paper.\n\n\\section{Notation and conventions}\\label{sect:notation}\n\\begin{enumerate}\n\\item Throughout, $k$ is a fixed commutative ring. By module, complex, map,\n  etc.\\ we mean $k$-module, complex of $k$-modules, $k$-linear map, etc.  We place no boundedness or\nconnectedness assumptions on complexes. For graded modules\n  $M,N,$ define graded modules $\\Hom {} M N$ and $M \\otimes N$ by\n  \\begin{displaymath}\n    \\Hom {} M N_n = \\prod_{i \\in \\mathbb{Z}} \\Hom {} {M_i} {N_{i + n}} \\quad (M\n    \\otimes N)_n = {\\bigoplus_{i \\in \\mathbb{Z}} M_i \\otimes N_{n - i}}.\n  \\end{displaymath}\n  If $(M, \\delta_{M})$ and $(N, \\delta_{N})$ are complexes, then\n  $\\Hom {} M N$ and $M \\otimes N$ are complexes with differentials\n  \\begin{displaymath}\n    \\delta_{\\operatorname{Hom}}(f) = \\delta_{N}f - (-1)^{|f|}f \\delta_{M} \\quad \\delta_{\\otimes} = \\delta_M \\otimes 1 + 1 \\otimes \\delta_N.\n  \\end{displaymath}\n  A morphism of complexes is a degree 0 cycle of the complex\n  $(\\Hom {} M N, \\delta_{\\operatorname{Hom}}).$\n\n\\item All elements of graded objects are assumed to be homogeneous. We\n  write $|x|$ for the degree of an element $x$. If $M$ is a graded\n  module, $\\Pi M$ is the graded module with $(\\Pi M)_n = M_{n -1}$. Set\n  $s \\in \\Hom {} M {\\Pi M}_{1}$ to be the identity map. For $x\\in M,$\n  we set $[x] = s(x) \\in \\Pi M$ and more generally\n  $[x_1|\\ldots|x_n] = sx_{1}\\otimes \\ldots\\otimes s x_n.$ If\n  $(M, \\delta_{M})$ is a complex, set\n  $\\delta_{\\Pi M} = -s\\delta_M s^{-1}$. Then\n  $s: (M, \\delta_{M}) \\to (\\Pi M, \\delta_{\\Pi M})$ is a cycle in $(\\Hom {} {\\Pi M} M, \\delta_{\\operatorname{Hom}}).$\n\n\\item We use the sign conventions that when $x,y$ are permuted, a\n  factor of\n  $(-1)^{|x||y|}$ is introduced, and when applying a tensor product of\n  morphisms, we have\n  $(f \\otimes g)(x \\otimes y) = (-1)^{|g||x|}f(x) \\otimes g(y)$.\n \n \n \n \n\n\\item For complexes $M$ and $N$, the maps\n  $(\\Pi M) \\otimes N \\to \\Pi (M \\otimes N), [m] \\otimes n \\mapsto [m\n  \\otimes n]$ and\n  $M \\otimes (\\Pi N) \\to \\Pi (M \\otimes N),m \\otimes [n] \\mapsto\n  (-1)^{|m|}[m \\otimes n]$\n  are isomorphisms of complexes, as are the maps\n  $\\Pi \\Hom {} M N \\to \\Hom {} {\\Pi^{-1} M} N, [f] \\mapsto\n  (-1)^{|f|}fs$ and\n  $\\Pi \\Hom {} M N \\to \\Hom {} M {\\Pi N}, [f] \\mapsto sf$.\n\n\\item String diagrams are used to represent morphisms between\n  tensor products of graded modules. A line represents a graded\n  module, parallel lines represent a tensor product of graded modules, and a rectangle\n  represents a morphism. Lines may be decorated to\n  distinguish graded modules, for instance, let\n  $\\tikz[baseline={([yshift = -.1cm]current bounding\n    box.center)}]{\\draw[thick] (0,0) -- (0,.5)}$ represent a graded module $A$ and\n  $\\begin{tikzpicture}[baseline={([yshift = -.1cm]current bounding\n      box.center)}]\\draw[bmod, line width = .5mm] (0,0) --\n    (0,.5);\\end{tikzpicture}$ represent a graded module $B$. Then, e.g., $\n  \\begin{tikzpicture}\n    \\newBoxDiagram [arm height = .5cm, width = .25cm, decoration\n    yshift label = .05cm, box height = .5cm, label = {\\tiny{f}}]{f};\n    \\print[\/bmod]{f}; \\printArm{f}{0};\n  \\end{tikzpicture}$ represents a morphism $f: A^{\\otimes 3} \\to B.$\n  The utility of the diagrams becomes apparent when\n  composing such morphisms.\n\\item For unexplained conventions or definitions related to\n  differential graded Lie algebras, see \\cite{MR0195995}, and for\n  graded coalgebras, see Chapters 1 and 5 of \\cite{MR1243637}.\n\\end{enumerate}\n\n\\section{Nonunital $\\text{A}_{\\infty}$-algebras}\nIn this section we recall the definitions of nonunital $\\text{A}_{\\infty}$-algebra, $\\text{A}_{\\infty}$-morphism, bar construction, and dg-Lie algebra of\nHochschild cochains. We use the approach of Proute \\cite{MR2844537}\nand Getzler\n\\cite{MR1261901}, and the string diagram notation of Hinich\n\\cite{MR1978336}. Fix graded modules $A, B$ throughout the section. In string diagrams $\\tikz[baseline={([yshift = -.1cm]current bounding\n  box.center)}]{\\draw[thick] (0,0) -- (0,.5)}$ will denote $\\Pi A$ and $\\tikz[baseline={([yshift = -.1cm]current bounding\n  box.center)}]{\\draw[bmod, line width = .5mm] (0,0) -- (0,.5)}$ will denote $\\Pi B.$\n\n\\begin{defn}\\label{defn:bar_and_cc}\n  Set\n  $\\hhn n A B = \\Hom {} {(\\Pi A)^{\\otimes n}} {\\Pi B},$ and\n  \\begin{align*}\n    \\hhc A B & = \\prod_{n \\geq 1} \\Hom {} {(\\Pi A)^{\\otimes n}}\n               {\\Pi B}.\n  \\end{align*}\n  We write $f = (f^n) \\in \\hhc A B$ with $f^n \\in \\hhn n A B$ and call\n  $f^n$ the $n$th tensor homogeneous component.\n\nSince $A$ and $B$ are graded,\nso is $\\hhc A B$, using \\ref{sect:notation}.(1). The\n$i$th homogeneous component of this grading is denoted $\\hhc A B_{i}$.\n\\end{defn}\n\nThe module $\\hhc A A$ has a very intricate algebraic structure. In\nparticular it is a graded Lie algebra under the commutator of the\nfollowing.\n\\begin{defn}\n The \\emph{Gerstenhaber product} of $g = (g^n) \\in \\hhc A B_{i}$ and $f = (f^{n}) \\in \\hhc A\n  A_{j}$, denoted $g \\circ f \\in \\hhc A B_{i+j},$ has $n$th tensor homogeneous component given by the following:\n  \\begin{align*}\\label{def:gerst_prod}\n    (g \\circ f)^n\n   \n   \n                  &= \\sum_{i=1}^{n} \\sum_{j=0}^{i-1} \\hspace{.2cm}\n                    \\begin{tikzpicture}[baseline={(current bounding box.center)}]\n                     \n                      \\def1.2cm{1.5cm}\n                     \n                     \n                      \\def.52cm{.55cm}\n                     \n                     \n                     \n                      \\def0{-.1}\n                     \n                     \n                     \n                      \\def.6{.5}\n                     \n                     \n                      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n                     \n                     \n                      \\def.08}\\def.35{.35{.08}\n                     \n                     \n                     \n                      \\def.35{.35}\n                     \n                      \\pgfkeys{lowBoxStyle\/.style = {box height =\n                          .52cm, output height = .52cm, width = 1.2cm,\n                          coord =\n                          {($(0,0)$)}, decoration yshift label = .2cm,\n                          decoration yshift label = .05cm, arm height\n                          = 3*.52cm, label = g^{i}}}\n                     \n                      \\newBoxDiagram [\/lowBoxStyle]{lowBox};\n                      {\\print[\/bmod]{lowBox};}\n                     \n                      \\lowBox.coord at top(highcoor, 0);\n                     \n                      \\pgfkeys{highBoxStyle\/.style = {width =\n                          1.2cm*.6, box height = .52cm, arm height =\n                          .52cm, output height = .52cm, coord =\n                          {(highcoor)}, output height = .52cm, label =\n                          {f^{n-i+1}}}}\n                     \n                      \\newBoxDiagram[\/highBoxStyle]{highBox};\n                      \\print{highBox}; \\printArmSep{highBox};\n                     \n                     \n                     \n                      \\pgfmathsetmacro{\\var}{0 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n                      \\printArm{lowBox}{\\var}; \\printDec[\/decorate,\n                      decorate left = -1, decorate right =\n                      \\var]{lowBox}{j};\n                     \n                      \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n                      \\pgfmathsetmacro{\\vara}{0 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm\n                        - .08}\\def.35{.35} \\printArmSep[arm sep left = \\var,\n                      arm sep right = \\vara]{lowBox}\n                     \n                     \n                      \\pgfmathsetmacro{\\var}{0 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n                      \\printArm{lowBox}{\\var} \\printDec[\/decorate,\n                      decorate left = \\var, decorate right =\n                      1]{lowBox}{ i - j-1}\n                     \n                      \\pgfmathsetmacro{\\varb}{1-.08}\\def.35{.35}\n                      \\pgfmathsetmacro{\\vara}{\\var+.08}\\def.35{.35}\n                      \\printArmSep[arm sep left = \\vara, arm sep right\n                      = \\varb]{lowBox}\n                                          \\end{tikzpicture}\n                    = \\scalebox{.9}{$\\displaystyle \\sum_{i=1}^{n} \\sum_{j=0}^{i-1} g^{i}(1^{\\otimes j} \\otimes f^{n - i + 1} \\otimes 1^{\\otimes i - j - 1}).$}\n  \\end{align*}\n\\end{defn}\n\nThis was defined in \\cite{MR0161898} where it was shown to be a\npre-Lie algebra structure (Corollary to Theorem 2,\napplied to Example 5.5)\\footnote{A \\emph{pre-Lie algebra structure}\n  on a graded module $G$ is a degree zero morphism\n  $\\circ : G \\otimes G \\to G$ such that for all $f, g, h \\in G$, the\n  following equation holds,\n  \\[ (f \\circ g) \\circ h - f \\circ (g \\circ h) = (-1)^{|g||h|} \\left (\n      (f \\circ h) \\circ g - f \\circ (h \\circ g) \\right ),\\] i.e., the\n  associator is symmetric in the last two values. In particular, every\n  associative algebra is a pre-Lie algebra, since the\n  associator is zero. See e.g., \\cite[\\S 1.4]{MR2954392} for more\n  information.}, and by \\cite[Theorem 1]{MR0161898},\nthe commutator of any pre-Lie algebra, defined to be\n$[x,y] = x \\circ y - (-1)^{|x||y|} y \\circ x$, is a graded Lie algebra.\n\n Using string diagrams, it\n  is a relatively easy exercise to show $\\hhc A A$ is a pre-Lie algebra\n  (see \\cite[p. 20, Figure 1]{KellerDefTheory} for\n  details). Performing this exercise, one will see that\n  care must be taken with signs and string diagrams. The conventions\n  we use\n  for signs and string diagrams are formalized below\n  (we encourage the reader to skip ahead and return when a sign issue\n  occurs).\n\\begin{rem}\\label{rem:signs_for_diags} \n  \\mbox{}\\\\\n  \\begin{enumerate}\n  \\item Morphisms will always be grouped into horizontal lines,\n    i.e., the projections of any two boxes onto the left side of the page are either\n    disjoint or equal.\n    \n\\item If all morphisms are on the same line,\n    we visualize inputs feeding into the diagram from the\n    right, along the front, and use sign convention \\ref{sect:notation}.(3), e.g.,\n    \\begin{displaymath}\n\\begin{tikzpicture}[baseline={([yshift = -.3cm]current bounding\n          box.center)}]\n        \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm} \\def1.2cm{.5cm}\n        \\def.3cm} \\def\\hgt{.55cm{.4cm}  \\def.52cm{.5cm}\n        \\newBoxDiagram [arm height = 2*.52cm, box\n        height = .52cm, output height = .52cm, width = 1.2cm, coord =\n        {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + 1.2cm,\n          0)$)}, decoration yshift label = .2cm, label = f^{n}]{f};\n        \\newBoxDiagram\n        [box height = 0cm, output height = 0cm, arm height =4*.52cm,\n        width = .3cm} \\def\\hgt{.55cm, coord = {( 2*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*1.2cm + .3cm} \\def\\hgt{.55cm,\n          0)}, decoration yshift label = .2cm]{right};\n\\print{f}; \\printArmSep{f};\n        \\print{right}; \\printArmSep{right};\n        \\printDec[\/decorate]{right}{l}; \\newBoxDiagram [arm height =\n        2*.52cm, box height = .52cm, output height = .52cm, width = 1.3*1.2cm,\n        coord = {($(3*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + 3.3*1.2cm + 2*.3cm} \\def\\hgt{.55cm,\n          0)$)}, decoration yshift label = .2cm, label = g^{m}]{g};\n        \\print{g}; \\printArmSep{g};\n        \\g.coord at top(coor, 1);\n        \\begin{scope}[thick, color = black!40!white, decoration={\n    markings,\n    mark=at position 0.55 with {\\arrow{>}}}]\n  \\draw[postaction = {decorate}] ($(coor) + (.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm, 1.8*.52cm)$) arc (10:150:2.1 and -0.2);\n\\end{scope}\n\\end{tikzpicture} \\, (\\bs x \\otimes \\bs y \\otimes \\bs z) = (-1)^{|\\bs\n  x||g| + |\\bs y||g|} f^{n}(\\bs x) \\otimes \\bs y \\otimes g^{m}(\\bs z),\n\\end{displaymath}\nwhere $\\bs x = x_{1} \\otimes \\ldots \\otimes x_{n}, \\bs y = y_{1}\n\\otimes \\ldots \\otimes y_{l}, \\bs z = z_{1} \\otimes \\ldots \\otimes z_{m}.$\n    \n\\item If there are multiple lines of morphisms, we visualize the\n  output from each line\n  coming out behind the diagram,\n  needing to be twisted around to the front, where the next line of\n  morphisms is applied as in Step 2; e.g.,\n\\begin{displaymath}\n\\begin{tikzpicture}[baseline={([yshift = -.3cm]current bounding\n          box.center)}]\n        \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm} \\def1.2cm{.5cm}\n        \\def.3cm} \\def\\hgt{.55cm{.4cm}  \\def.52cm{.55cm}\n        \\newBoxDiagram [arm height = 2*.52cm, box\n        height = .52cm, output height = 4*.52cm, width = 1.2cm, coord =\n        {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + 1.2cm,\n          0)$)}, decoration yshift label = .2cm, label = f^{n}]{f};\n        \\newBoxDiagram\n        [box height = .52cm, output height = .52cm, arm height =5*.52cm,\n        width = .3cm} \\def\\hgt{.55cm, coord = {( 2*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*1.2cm + .3cm} \\def\\hgt{.55cm,\n          0)}, decoration yshift label = .2cm, label = h^{l}]{right};\n\\print{f}; \\printArmSep{f};\n        \\print{right}; \\printArmSep{right};\n        \\newBoxDiagram [arm height =\n        2*.52cm, box height = .52cm, output height = 4*.52cm, width = 1.3*1.2cm,\n        coord = {($(3*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + 3.3*1.2cm + 2*.3cm} \\def\\hgt{.55cm,\n          0)$)}, decoration yshift label = .2cm, label = g^{m}]{g};\n        \\print{g}; \\printArmSep{g};\n        \\g.coord at top(coor, 1);\n                \\begin{scope}[thick, color = black!40!white, decoration={\n    markings,\n    mark=at position 0.55 with {\\arrow{>}}}]\n  \\draw[postaction = {decorate}] ($(coor) + (.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm, 1.8*.52cm)$) arc (10:150:2.1 and -0.2);\n  \\end{scope}\n\\begin{scope}[thick, color = black!40!white, decoration={\n    markings,\n    mark=at position 0.65 with {\\arrow{<}}}]\n    \\draw [postaction = {decorate}, dashed] ($(coor) + (.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm, -2*.52cm)$) arc (0:140:2.1 and 0.25);\n  \\end{scope}\n        \\begin{scope}[thick, color = black!40!white, decoration={\n    markings,\n    mark=at position 0.565 with {\\arrow{>}}}]\n  \\draw[postaction = {decorate}] ($(coor) + (.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm, -2*.52cm)$) arc (0:140:2.1 and -0.25);\n  \\end{scope}\n      \\end{tikzpicture}  \\, (\\bs x \\otimes \\bs y \\otimes \\bs z) = (-1)^{|\\bs\n  x||g| + |\\bs y||g|+(|\\bs x| + |f|)|h|} f^{n}(\\bs x) \\otimes h^{l}(\\bs y) \\otimes g^{m}(\\bs z).\n\\end{displaymath}\nMoving the line a morphism is on only changes the diagram by a\nsign. Sign rules for vertical moves of a morphism are:\n  \\item up one line: multiply by $-1$ to the\n    degree of the morphism times the sum of the degrees of the morphims\n    \\emph{to the left on the new line.}\n  \\item down one line: multiply by $-1$ to\n    the degree of the morphism times the sum of the degrees of the morphisms\n    \\emph{to the right on the new line.}\n  \\end{enumerate}\n  In particular, we have\n  \\begin{displaymath}\n  \\begin{tikzpicture}\n      \\def.1{.2cm} \\newBoxDiagram [output height = 1.5cm] {f};\n      \\newBoxDiagram [coord = {(1cm + .1, 0)}, arm height = 1.75cm]\n      {g}; \\print{f}; \\printArmSep{f}; \\print{g};\n      \\printArmSep{g};\n    \\end{tikzpicture} = (-1)^{|f||g|} \\hspace{.1cm}\n    \\begin{tikzpicture}\n      \\def.1{.2cm} \\newBoxDiagram {f}; \\newBoxDiagram [coord = {(1cm\n        + .1, 0)}] {g}; \\print{f}; \\printArmSep{f}; \\print{g};\n      \\printArmSep{g};\n    \\end{tikzpicture}\n    = (-1)^{|f||g|} \\hspace{.1cm}\n    \\begin{tikzpicture}\n      \\def.1{.2cm} \\newBoxDiagram [arm height = 1.75cm] {f};\n      \\newBoxDiagram [coord = {(1cm + .1, 0)}, output height = 1.5cm]\n      {g}; \\print{f}; \\printArmSep{f}; \\print{g};\n      \\printArmSep{g};\n      \\printPeriod{g};\n    \\end{tikzpicture}\n  \\end{displaymath}\n\\end{rem}\n\n\\begin{defn}\n The \\emph{nonunital tensor coalgebra\n    on a graded module $V$} is $\\Tco V = \\bigoplus_{n \\geq 1} V^{\\otimes\n    n}$ with\n  comultiplication the linear extension of\n  \\[ \\Delta( v_1 \\otimes \\ldots \\otimes v_n ) = \\sum_{i = 1}^{n-1} (v_1\n    \\otimes \\ldots \\otimes v_i) \\otimes (v_{i+1} \\otimes \\ldots\n    \\otimes v_n).\\] Note that\n  $\\hhc A B = \\Hom {} {\\Tco {\\Pi A}} {\\Pi B}.$\n\nA \\emph{graded coderivation} of a graded coalgebra $C$ with comultiplication\n  $\\Delta$ is a homogeneous\n  endomorphism $d$ of $C$ such that\n  $(d \\otimes 1 + 1 \\otimes d) \\Delta = \\Delta d$. We write\n  $\\Coder(C,C)$ for the set of coderivations. This is a graded Lie\n  subalgebra of the commutator bracket on $\\Hom {} C C$.\n\\end{defn}\n\n\\begin{lem}\\label{lem:isom_hhc_coder}\nThe tensor coalgebra satisfies the following universal properties.\n  \\begin{enumerate}\n  \\item The canonical projection $\\pi_1: \\Tco {\\Pi A} \\to \\Pi A$\n    induces an isomorphism,\n    \\[ \\Phi = (\\pi_1)_*: {\\Coder}( \\Tco {\\Pi A},\n      \\Tco {\\Pi A}) \\xra{\\cong} \\Hom {} {\\Tco {\\Pi A}} {\\Pi A} = \\hhc\n      A A.\\] This is an isomorphism of graded Lie algebras, where the\n    bracket on the source is the commutator, and the bracket on the\n    target is the Gerstenhaber bracket. The inverse applied to\n    $f =(f^n) \\in \\hhc A A$ is given by\n    \\begin{align*}\n      \\pi_{n-i+1} \\Phi^{-1}(f)|_{(\\Pi A)^{\\otimes n}}\n     \n     \n                                           &\\begin{tikzpicture}[inner sep =0mm]\n                                                     \n                                                     \n                                                     \n                                                      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm}\n                                                      \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.4cm}\n                                                      \\def1.2cm{.5cm}\n                                                      \\def.3cm} \\def\\hgt{.55cm{.6cm}\n                                                      \\node at\n                                                      (-1.3,.8)\n                                                      {$=\n                                                       \\displaystyle\n                                                        \\sum_{j\n                                                          =0}^{n-i}$};\n                                                      \\newBoxDiagram\n                                                      [box height =\n                                                      0cm, output\n                                                      height = 0cm,\n                                                      arm height =\n                                                      1.5cm, width =\n                                                      .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm,\n                                                      decoration\n                                                      yshift label =\n                                                      .2cm]{left};\n                                                      \\newBoxDiagram\n                                                      [arm height =\n                                                      .5cm, width =\n                                                      1.2cm, coord =\n                                                      {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm +\n                                                        .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n                                                        0)$)},\n                                                      decoration\n                                                      yshift label =\n                                                      .2cm, label =\n                                                      f^i]{f};\n                                                      \\newBoxDiagram\n                                                      [box height =\n                                                      0cm, output\n                                                      height = 0cm,\n                                                      arm height =\n                                                      1.5cm, width =\n                                                      .3cm} \\def\\hgt{.55cm, coord =\n                                                      {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm +\n                                                        2*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*1.2cm +\n                                                        .3cm} \\def\\hgt{.55cm, 0)},\n                                                      decoration\n                                                      yshift label =\n                                                      .2cm]{right};\n                                                      \\print{left};\n                                                      \\printArmSep{left};\n                                                      \\printDec[\/decorate]{left}{j};\n                                                      \\print{f};\n                                                      \\printArmSep{f};\n                                                      \\print{right};\n                                                      \\printArmSep{right};\n                                                \\printDec[\/decorate]{right}{n-i-j};\n                                                      \\printPeriod{right};\n                                                    \\end{tikzpicture}\n    \\end{align*}\n\n    \n\n  \\item The canonical projection $\\pi_1: \\Tco {\\Pi B} \\to \\Pi B$\n    induces an isomorphism,\n    \\[ \\Psi = (\\pi_1)_*: \\Hom {\\Coalgk} {\\Tco {\\Pi A}} {\\Tco {\\Pi B}}\n      \\xra{\\cong}\\hhc A B_{0}.\\]\n    The inverse applied to $g =(g^n) \\in \\hhc A B_{0}$ is\n    given by:\n    \\[ \\pi_k \\Psi^{-1}(g)|_{(\\Pi A)^{\\otimes n}} = \\sum_{i_1 + \\ldots +\n        i_k = n} \\hspace{.1cm}\n      \\begin{tikzpicture}\n        \\def.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm{.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm}\n        \\def.32cm{.45cm} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1cm} \\def.52cm{.43cm}\n        \\def.55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm{.54cm} \\newBoxDiagram[width = .4cm} \\def.30cm{.35cm} \\def.33cm{.45cm, box height =\n        .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height = .52cm, label =\n        g^{i_1}]{g1} \\print[\/bmod]{g1} \\printArmSep{g1}\n        \\newBoxDiagram[width = .30cm, box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height\n        = .52cm, output height = .52cm, coord =\n        {($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+.30cm,\n          0)$)}, label = g^{i_2}]{g2} \\print[\/bmod]{g2} \\printArmSep{g2}\n        \\node at\n        ($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + 2*.30cm + .32cm, .52cm +\n        .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm\/2)$) {\\ldots}; \\newBoxDiagram[width = .33cm, box height\n        = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height = .52cm, coord =\n        {($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*.30cm+2*.32cm+.33cm,\n          0)$)}, label = g^{i_k}]{gk};\\print[\/bmod]{gk}; \\printArmSep{gk};\n        \\printPeriod{gk};\n      \\end{tikzpicture}\\]\n  \\end{enumerate}\n\\end{lem}\n\nFor  $g \\in \\hhc A B$ and $f \\in \\hhc A A$, it follows from \\ref{lem:isom_hhc_coder}.(1) that $g \\circ f = g \\Phi^{-1}(f).$ We define an analogous product using $\\Psi^{-1}.$\n\\begin{defn}\\label{rem:defining-start-and-gerst-produc}\n  For $g \\in \\hhc A B_{0}$ and $h\\in \\hhc B B_{i}$, set\n  \\[h * g = h \\Psi^{-1}(g) \\in \\hhc A B_{i}.\\]\n\\end{defn}\n\n\\begin{rem}\\label{rem:summation-convention}\n  If there are no superscripts on the morphisms of a string diagram,\n  the diagram represents an element $\\xi$ of $\\hhc A B$ with $\\xi^{n}$\n  given by the summing over all diagrams of the given shape that have $n$\n  inputs. For example, if $h \\in \\hhc B B$ and $g \\in \\hhc A B_{0}$,\nwe write\n{$h * g = \\begin{tikzpicture}[scale=0.75, every node\/.style={transform shape}] \\def.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm{.33cm} \n    \\def.30cm{.30cm}\n \\def.33cm{.33cm}\n \\def.32cm{.32cm}\n \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1cm}\n\\def.52cm{.5cm}\n \\def.75cm{.55cm}\n        \\newBoxDiagram[width = .4cm} \\def.30cm{.35cm} \\def.33cm{.45cm, box height = .52cm, arm height\n        = .52cm, coord = {(0,0)}, output height = .52cm, label =\n        {g}]{g1};\n        \\print[\/bmod]{g1};\n \\printArmSep{g1};\n\n        \\printDec[color = white]{g1}{\\phantom{test}}\n\n        \\node at\n        ($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .32cm, .52cm +\n        .52cm\/2)$) {\\ldots};\n \\newBoxDiagram[width = .33cm, box height\n        = .52cm, arm height = .52cm, output height = .52cm, coord =\n        {($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*.32cm+.33cm,\n          0)$)}, label = {g}]{gk};\n \\print[\/bmod]{gk}; \n\\printArmSep{gk};\n\\newlength{\\var}\n\\newlength{\\vara}\n        \\setlength{\\var}{.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm\/2+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm\/2+.32cm+.33cm\/2}\n        \\setlength{\\vara}{-.52cm - .52cm}\n \\newBoxDiagram[width = \\var,\n        box height = .52cm, arm height = .52cm, output height = .52cm,\n        coord = {(\\var, \\vara)}, label = {h}]{mut}\n        \\printNoArms[\/bmod]{mut}\n      \\end{tikzpicture}$}\nto mean\n\\begin{displaymath} \n\\def.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm{.4cm} \n\\def.30cm{.35cm}\n \\def.33cm{.45cm}\n \\def.32cm{.45cm}\n \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1cm}\n\\def.52cm{.4cm}\n\\def.55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm{.55cm}\n(h*g)^{n} = \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm}\n\\begin{tikzpicture}[baseline={(current bounding\n        box.center)}]\n        \\newBoxDiagram[width = .4cm} \\def.30cm{.35cm} \\def.33cm{.45cm, box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height\n        = .52cm, output height = .52cm, label = g^{i_1}]{g1};\n        \\print[\/bmod]{g1};\n \\printArmSep{g1};\n\n \\newBoxDiagram[width = .30cm,\n        box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height = .52cm,\n        coord = {($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+.30cm,\n          0)$)}, label = g^{i_2}]{g2} \\print[\/bmod]{g2} \\printArmSep{g2};\n\n        \\node at\n        ($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + 2*.30cm + .32cm, .52cm +\n        .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm\/2)$) {\\ldots};\n \\newBoxDiagram[width = .33cm, box height\n        = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height = .52cm, coord =\n        {($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*.30cm+2*.32cm+.33cm,\n          0)$)}, label = g^{i_k}]{gk};\n \\print[\/bmod]{gk}; \n\\printArmSep{gk};\n\n        \\setlength{\\var}{.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm\/2+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm\/2+.30cm+.32cm+.33cm\/2}\n        \\setlength{\\vara}{-.55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm - .52cm} \\newBoxDiagram[width = \\var,\n        box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height = .52cm,\n        coord = {(\\var, \\vara)}, label = {h^{k}}]{mut}\n        \\printNoArms[\/bmod]{mut}\n       \n      \\end{tikzpicture}\n    \\end{displaymath}\n    and analogously,\n    $\\displaystyle g \\circ f =\n    \\sum_{j}\\hspace{.2cm}\\begin{tikzpicture}[baseline={(current\n        bounding box.center)}, scale=0.75, every node\/.style={transform shape}]\n      \\def1.2cm{1cm} \\def.3}\\def\\relw{.6{-.1} \\def.6{.4} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n      \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 1.5cm, box\n      height = .5cm,\n      label = g]{f}; \\print[\/bmod]{f}; \\f.coord at top(gcoor, .3}\\def\\relw{.6);\n      \\newBoxDiagram[width = 1.2cm*.6, arm height = .5cm, coord =\n      {(gcoor)}, output height = .5cm, label = f]{g};\n      \\print{g}; \\printArmSep{g};\n      \\def.08}\\def.35{.35{.08}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n      decorate right = \\var]{f}{j};\n      \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var} \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n      \\printPeriod{f}.\n    \\end{tikzpicture}$ (We also extend this notation to tensor products of\n    elements of $\\hhc A A$ in the proof below.)\n\\end{rem}\n\n\n\\begin{proof}[Proof of Lemma \\ref{lem:isom_hhc_coder}]\nFor proofs that $\\Phi$ and $\\Psi$ are isomorphisms of modules see\ne.g., \\cite[2.16, 2.19]{MR2844537}. We\nwill show that $\\Phi^{-1}$ is a morphism of graded Lie algebras (this\nis also presumably well known, but string diagrams give an easy proof). Let $f, g \\in \\hhc A A$, and set\n  $d = \\Phi^{-1}(f), e = \\Phi^{-1}(g)$. We then have\n  \\[ d e = \\Phi^{-1}(f) \\Phi^{-1}(g) = \\hspace{.1cm} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm}\n    \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.2cm} \\def1.2cm{.25cm} \\def.3cm} \\def\\hgt{.55cm{.3cm} \\left\n      ( \\begin{tikzpicture}[baseline={([yshift = -.3cm]current\n          bounding box.center)}] \\node at (-.75,.75)\n        {$\\displaystyle\n          \\sum_{j}$}; \\newBoxDiagram [box height = 0cm, output height\n        = 0cm, arm height = 1.5cm, width = .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm, decoration yshift\n        label = .2cm]{left};\n        \\newBoxDiagram [arm height = .5cm, width = 1.2cm, coord =\n        {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n          0)$)}, decoration yshift label = .2cm]{f};\n        \\newBoxDiagram [box height = 0cm, output height = 0cm, arm\n        height = 1.5cm, width = .3cm} \\def\\hgt{.55cm, coord = {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm +\n          2*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*1.2cm + .3cm} \\def\\hgt{.55cm, 0)}, decoration yshift label =\n        .2cm]{right};\n        \\print{left}; \\printArmSep{left};\n        \\printDec[\/decorate]{left}{j}; \\print{f}; \\printArmSep{f};\n        \\print{right}; \\printArmSep{right};\n      \\end{tikzpicture} \\right ) \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n    \\def1.2cm{.25cm} \\def.3cm} \\def\\hgt{.55cm{.2cm} \\left\n      ( \\begin{tikzpicture}[baseline={([yshift = -.3cm]current\n          bounding box.center)}] \\node at (-.75,.75)\n        {$\\displaystyle\n          \\sum_{k}$}; \\newBoxDiagram [box height = 0cm, output height\n        = 0cm, arm height = 1.5cm, width = .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm, decoration yshift\n        label = .2cm]{left};\n        \\newBoxDiagram [arm height = .5cm, width = 1.2cm, coord =\n        {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n          0)$)}, decoration yshift label = .2cm, label = g]{f};\n        \\newBoxDiagram [box height = 0cm, output height = 0cm, arm\n        height = 1.5cm, width = .3cm} \\def\\hgt{.55cm, coord = {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm +\n          2*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*1.2cm + .3cm} \\def\\hgt{.55cm, 0)}, decoration yshift label =\n        .2cm]{right};\n        \\print{left}; \\printArmSep{left};\n        \\printDec[\/decorate]{left}{k}; \\print{f}; \\printArmSep{f};\n        \\print{right}; \\printArmSep{right};\n      \\end{tikzpicture} \\right ).\n  \\]\n  When composing terms, $g$ is inserted to the left of, into, or to\n  the right of, $f$, so\n  \\[ d e = \\sum_{j,k} \\left (\n      \\begin{tikzpicture}[baseline={([yshift = -.3cm]current bounding\n          box.center)}]\n        \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.25cm} \\def1.2cm{.25cm}\n        \\def.3cm} \\def\\hgt{.55cm{.2cm} \\def\\rightrightw{.25cm} \\def.52cm{.5cm}\n        \\hspace{.1cm} \\newBoxDiagram [box height = 0cm, output height\n        = 0cm, arm height = 5*.52cm, width = .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm, decoration yshift\n        label = .2cm]{left}; \\newBoxDiagram [arm height = .52cm, box\n        height = .52cm, output height = 3*.52cm, width = 1.2cm, coord =\n        {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n          0)$)}, decoration yshift label = .2cm]{g}; \\newBoxDiagram\n        [box height = 0cm, output height = 0cm, arm height =5*.52cm,\n        width = .3cm} \\def\\hgt{.55cm, coord = {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 2*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*1.2cm + .3cm} \\def\\hgt{.55cm,\n          0)}, decoration yshift label = .2cm]{right};\n        \\print{left}; \\printArmSep{left};\n        \\printDec[\/decorate]{left}{k}; \\print{g}; \\printArmSep{g};\n        \\print{right}; \\printArmSep{right};\n        \\printDec[\/decorate]{right}{j}; \\newBoxDiagram [arm height =\n        3*.52cm, box height = .52cm, output height = .52cm, width = 1.2cm,\n        coord = {($(3*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 3*1.2cm + 2*.3cm} \\def\\hgt{.55cm,\n          0)$)}, decoration yshift label = .2cm]{f}; \\newBoxDiagram\n        [box height = 0cm, output height = 0cm, arm height =5*.52cm,\n        width = \\rightrightw, coord = {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 4*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + 4*1.2cm +\n          2*.3cm} \\def\\hgt{.55cm+\\rightrightw, 0)}, decoration yshift label=\n        .2cm]{rightright}; \\print{f}; \\printArmSep{f};\n        \\print{rightright}; \\printArmSep{rightright};\n      \\end{tikzpicture}\n      \\quad + \\quad\n      \\begin{tikzpicture}[baseline={([yshift = -.3cm]current bounding\n          box.center)}]\n        \\def1.2cm{.75cm} \\def.3}\\def\\relw{.6{.1} \\def.6{.4} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n        \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm} \\def.3cm} \\def\\hgt{.55cm{.3cm}\n        \\def.52cm{.5cm} \\newBoxDiagram [box height = 0cm, output height\n        = 0cm, arm height = 2*.52cm, width = .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm, decoration yshift\n        label = .05cm, arm height = 5*.52cm]{left};\n        \\newBoxDiagram [box height = .52cm, output height = .52cm, width\n        = 1.2cm, coord = {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n          0)$)}, decoration yshift label = .2cm, decoration yshift\n        label = .05cm, arm height = 3*.52cm]{f};\n        \\newBoxDiagram [box height = 0cm, output height = 0cm, arm\n        height = 3*.52cm, width = .3cm} \\def\\hgt{.55cm, coord = {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm +\n          2*.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm+2*1.2cm + .3cm} \\def\\hgt{.55cm, 0)}, decoration yshift label =\n        .2cm, arm height = 5*.52cm]{right}; \\print{f}; \\f.coord at\n        top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, box\n        height = .52cm, arm height = .52cm, output height = .52cm, coord\n        = {(gcoor)}, output height = .52cm]{g}; \\print{g};\n        \\printArmSep{g};\n\n        \\def.08}\\def.35{.35{.08}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n        decorate right = \\var]{f}{j};\n        \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}\n       \n       \n        \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n        \\print{left}; \\printArmSep{left};\n        \\printDec[\/decorate]{left}{k}; \\print{right};\n        \\printArmSep{right};\n      \\end{tikzpicture}\n      \\quad+\\quad\n      \\begin{tikzpicture}[baseline={([yshift = -.3cm]current bounding\n          box.center)}]\n        \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.2cm} \\def1.2cm{.25cm}\n        \\def.3cm} \\def\\hgt{.55cm{.3cm} \\def\\rightrightw{.25cm} \\def.52cm{.5cm}\n        \\newBoxDiagram [box height = 0cm, output height = 0cm, arm\n        height = 5*.52cm, width = .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm, decoration yshift label =\n        .2cm]{left}; \\newBoxDiagram [arm height = 3*.52cm, box height =\n        .52cm, output height = .52cm, width = 1.2cm, coord =\n        {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n          0)$)}, decoration yshift label = .2cm, label = f]{g};\n        \\newBoxDiagram [box height = 0cm, output height = 0cm, arm\n        height =5*.52cm, width = .3cm} \\def\\hgt{.55cm, coord = {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm +\n          2*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*1.2cm + .3cm} \\def\\hgt{.55cm, 0)}, decoration yshift label =\n        .2cm]{right};\n        \\print{left}; \\printArmSep{left};\n        \\printDec[\/decorate]{left}{j}; \\print{g}; \\printArmSep{g};\n        \\print{right}; \\printArmSep{right};\n        \\printDec[\/decorate]{right}{k}; \\newBoxDiagram [arm height =\n        .52cm, box height = .52cm, output height = 3*.52cm, width = 1.2cm,\n        coord = {($(3*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 3*1.2cm + 2*.3cm} \\def\\hgt{.55cm,\n          0)$)}, decoration yshift label = .2cm, label = g]{f};\n        \\newBoxDiagram [box height = 0cm, output height = 0cm, arm\n        height =5*.52cm, width = \\rightrightw, coord = {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm +\n          4*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + 4*1.2cm + 2*.3cm} \\def\\hgt{.55cm+\\rightrightw, 0)}, decoration\n        yshift label = .2cm]{rightright};\n \n        \\print{f}; \\printArmSep{f}; \\print{rightright};\n        \\printArmSep{rightright}\n      \\end{tikzpicture}\n      \\hspace{.2cm}\\right ) \\] We then have, see\n  \\ref{rem:signs_for_diags} for signs,\n  \\[ [d,e] = de - (-1)^{|d||e|}ed = \\sum_{j,k} \\left (\n      \\begin{tikzpicture}[baseline={([yshift = -.3cm]current bounding\n          box.center)}]\n        \\def1.2cm{.75cm} \\def.3}\\def\\relw{.6{.1} \\def.6{.4} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n        \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm} \\def.3cm} \\def\\hgt{.55cm{.3cm}\n        \\def.52cm{.5cm} \\newBoxDiagram [box height = 0cm, output height\n        = 0cm, arm height = 2*.52cm, width = .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm, decoration yshift\n        label = .05cm, arm height = 5*.52cm]{left};\n        \\newBoxDiagram [box height = .52cm, output height = .52cm, width\n        = 1.2cm, coord = {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n          0)$)}, decoration yshift label = .2cm, decoration yshift\n        label = .05cm, arm height = 3*.52cm]{f};\n        \\newBoxDiagram [box height = 0cm, output height = 0cm, arm\n        height = 3*.52cm, width = .3cm} \\def\\hgt{.55cm, coord = {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm +\n          2*.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm+2*1.2cm + .3cm} \\def\\hgt{.55cm, 0)}, decoration yshift label =\n        .2cm, arm height = 5*.52cm]{right}; \\print{f}; \\f.coord at\n        top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, box\n        height = .52cm, arm height = .52cm, output height = .52cm, coord\n        = {(gcoor)}, output height = .52cm]{g}; \\print{g};\n        \\printArmSep{g};\n\n        \\def.08}\\def.35{.35{.08}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n        decorate right = \\var]{f}{j};\n        \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}\n       \n       \n        \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n        \\print{left}; \\printArmSep{left};\n        \\printDec[\/decorate]{left}{k}; \\print{right};\n        \\printArmSep{right};\n      \\end{tikzpicture}\n      -(-1)^{|d||e|} \\begin{tikzpicture}[baseline={([yshift =\n          -.3cm]current bounding box.center)}] \\def1.2cm{.75cm}\n        \\def.3}\\def\\relw{.6{-.1} \\def.6{.4} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n        \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.35cm} \\def.3cm} \\def\\hgt{.55cm{.25cm}\n        \\def.52cm{.5cm} \\newBoxDiagram [box height = 0cm, output height\n        = 0cm, arm height = 2*.52cm, width = .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm, decoration yshift\n        label = .05cm, arm height = 5*.52cm]{left};\n        \\newBoxDiagram [box height = .52cm, output height = .52cm, width\n        = 1.2cm, coord = {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n          0)$)}, decoration yshift label = .2cm, decoration yshift\n        label = .05cm, arm height = 3*.52cm, label = g]{f};\n        \\newBoxDiagram [box height = 0cm, output height = 0cm, arm\n        height = 3*.52cm, width = .3cm} \\def\\hgt{.55cm, coord = {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm +\n          2*.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm+2*1.2cm + .3cm} \\def\\hgt{.55cm, 0)}, decoration yshift label =\n        .2cm, arm height = 5*.52cm]{right}; \\print{f}; \\f.coord at\n        top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, box\n        height = .52cm, arm height = .52cm, output height = .52cm, coord\n        = {(gcoor)}, output height = .52cm, label = f]{g};\n        \\print{g}; \\printArmSep{g};\n\n        \\def.08}\\def.35{.35{.08}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n        decorate right = \\var]{f}{k};\n        \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}\n       \n       \n        \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n        \\print{left}; \\printArmSep{left};\n        \\printDec[\/decorate]{left}{j};\n\n        \\print{right}; \\printArmSep{right};\n      \\end{tikzpicture} \\right )\n  \\]\n  \\[ = \\Phi^{-1}(f \\circ g - (-1)^{|f||g|} f \\circ g) =\n    \\Phi^{-1}[f,g].\\qedhere\\]\n\\end{proof}\n\nWe will need the following in a later section. It follows from the\nexplicit formulas for $\\Phi^{-1}$ and $\\Psi^{-1}$ given in Lemma \\ref{lem:isom_hhc_coder}.\n\\begin{cor}\\label{cor:morphisms-commuting-with-coderivations}\n  A graded coalgebra morphism $\\gamma: \\Tco {\\Pi A} \\to \\Tco {\\Pi B}$\n  commutes with coderivations $d_{A}$ and $d_{B}$, of $\\Tco {\\Pi A}$ and\n  $\\Tco {\\Pi B}$, respectively, if and only if\n  $\\pi_{1} \\gamma d_{A} = \\pi_{1} d_B \\gamma.$\n\\end{cor}\n\n\\begin{defn}\\label{defn-A-infinity-algebra-morphism} Let $A, B$ be\n  graded modules.\n  \\begin{enumerate}\n  \\item A \\emph{nonunital $\\text{A}_{\\infty}$-algebra structure} on $A$ is an element\n    $\\nu \\in \\hhc A A_{-1}$ such that $\\nu \\circ \\nu = 0.$ For\n    $\\nu = (\\nu^n)$, this is equivalent to\n    \\[\n      \\sum_{\\substack{1 \\leq i \\leq n\\\\0 \\leq j \\leq i - 1}} \\hspace{.1cm} \\begin{tikzpicture}[baseline={(current\n          bounding box.center)}] \\def1.2cm{1.5cm} \\def.3}\\def\\relw{.6{-.1}\n        \\def.6{.5} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\newBoxDiagram[width\/.expand once\n        = 1.2cm, arm height = 1.5cm, box height = .55cm, label =\n        \\nu^i]{f}; \\print{f}; \\f.coord at top(gcoor, .3}\\def\\relw{.6);\n        \\newBoxDiagram[width = 1.2cm*.6, arm height = .5cm, coord =\n        {(gcoor)}, output height = .5cm, label = \\nu^{n-i+1}]{g};\n        \\print{g}; \\printArmSep{g};\n\n        \\def.08}\\def.35{.35{.08}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n        decorate right = \\var]{f}{j};\n        \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var} \\printDec[\/decorate, decorate left = \\var,\n        decorate right = 1]{f}{ i - j-1}\n        \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n      \\end{tikzpicture} = 0 \\text{ \\quad for all } n \\geq 1.\n    \\]\n  \\item An \\emph{$\\text{A}_{\\infty}$-morphism} $(A, \\nu_A) \\to (B, \\nu_B)$ between\n    nonunital $\\text{A}_{\\infty}$-algebras is an element $g \\in \\hhc A B_{0}$ such\n    that $ \\nu_B * g = g \\circ \\nu_A,$ where $*$ is defined in\n    \\ref{rem:defining-start-and-gerst-produc}. In diagrams this means,\n    \\[\n      \\def.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm{.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm}\n      \\def.32cm{.45cm} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1cm} \\def.52cm{.55cm}\n      \\def.55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm{.55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm}\n      \\begin{tikzpicture}\n        \\newBoxDiagram[width = .4cm} \\def.30cm{.35cm} \\def.33cm{.45cm, box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height\n        = .52cm, output height = .52cm, label = g^{i_1}]{g1}\n        \\print[\/bmod]{g1} \\printArmSep{g1} \n\n        \\printDec[color = white]{g1}{test}\n\n\\newBoxDiagram[width = .30cm,\n        box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height = .52cm,\n        coord = {($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+.30cm,\n          0)$)}, label = g^{i_2}]{g2} \\print[\/bmod]{g2} \\printArmSep{g2}\n        \\node at\n        ($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + 2*.30cm + .32cm, .52cm +\n        .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm\/2)$) {\\ldots}; \\newBoxDiagram[width = .33cm, box height\n        = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height = .52cm, coord =\n        {($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*.30cm+2*.32cm+.33cm,\n          0)$)}, label = g^{i_k}]{gk} \\print[\/bmod]{gk} \\printArmSep{gk}\n        \\setlength{\\var}{.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm\/2+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm\/2+.30cm+.32cm+.33cm\/2}\n        \\setlength{\\vara}{-.55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm - .52cm} \\newBoxDiagram[width = \\var,\n        box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height = .52cm,\n        coord = {(\\var, \\vara)}, label = {\\nu^{k}_{B}}]{mut}\n        \\printNoArms[\/bmod]{mut}\n       \n      \\end{tikzpicture}\n      =\\sum_{\\substack{1 \\leq i \\leq n\\\\0 \\leq j \\leq i - 1}} \\hspace{.1cm}\n      \\begin{tikzpicture}\n        \\def1.2cm{1.5cm} \\def.3}\\def\\relw{.6{-.1} \\def.6{.5} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n        \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 2*.52cm +\n        .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, output height = .52cm, box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, label =\n        g^i]{f}; \\print[\/bmod]{f}; \\f.coord at top(gcoor, .3}\\def\\relw{.6);\n        \\newBoxDiagram[box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, width = 1.2cm*.6, arm\n        height = .52cm, coord = {(gcoor)}, output height = .52cm, label\n        = \\nu^{n-i+1}_A]{g}; \\print{g}; \\printArmSep{g};\n        \\def.08}\\def.35{.35{.08}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n        decorate right = \\var]{f}{j};\n        \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}\n       \n       \n        \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n      \\end{tikzpicture} \\text{ for all } n \\geq 1.\n    \\]\n  \\item The \\emph{bar construction} of a nonunital $\\text{A}_{\\infty}$-algebra\n    $(A, \\nu)$ is the dg-coalgebra $\\Bar A = (\\Tco {\\Pi A}, \\Phi^{-1}(\\nu)).$ Since\n    $[\\nu, \\nu] = 0$, and $\\Phi^{-1}$ is a morphism of Lie algebras, it follows that $[\\Phi^{-1}(\\nu),\\Phi^{-1}(\\nu)] = 0$ and so\n    $\\Phi^{-1}(\\nu)^2 = 0$ (assuming $1\/2 \\in k$; or one can modify the proof of\n    \\ref{lem:isom_hhc_coder}). This is functorial with respect to\n    $\\text{A}_{\\infty}$-morphisms, using \\ref{lem:isom_hhc_coder}.(2).\n \n \n \n \n \n \n \n \n    \n  \\item The \\emph{Hochschild cochains} of a nonunital $\\text{A}_{\\infty}$-algebra\n    $(A, \\nu)$ is the dg-Lie algebra\n    $(\\hhc A A, [\\nu, -])$.\\footnote{This is a slightly non-standard\n      version of the Hochschild\n      cochains; the standard definition is $\\Sigma\n      \\hhc A A \\oplus A.$ To see the Lie algebra structure and\n      differential agree in the classical case when $A$ is a\n      $k$-algebra, see equation 23 on\n  page 280 of \\cite{MR0161898} and \\cite{MR1239562}.}\n  \\end{enumerate}\n\\end{defn}\n\n\\begin{rem}\\label{ex:dg-alg-is-ainf-alg}\nIt is often convenient to pass from a family of degree -1 maps\n$\\nu^{n}: (\\Pi A)^{\\otimes n} \\to \\Pi A$ to a family of degree $n -2$\nmaps $m^{n}: A^{\\otimes n} \\to A,$ and vice versa. We use the convention that\n\\begin{align*}\n  &m^{n} = s^{-1} \\nu^{n}s^{\\otimes n},\\\\\n  \\intertext{and since $(s^{\\otimes n})^{-1} = (-1)^{\\frac{n(n-1)}{2}}(s^{-1})^{\\otimes\n    n}$, it follows that}\n  &\\nu^{n} = (-1)^{\\frac{n(n-1)}{2}} s m^{n} (s^{-1})^{\\otimes n}.\n\\end{align*}\n(Proute \\cite{MR2844537} uses the convention that $\\nu^{n} =\n-sm^{n}(s^{-1})^{\\otimes n}.$) If $(A, (\\nu^{n}))$ is an $\\text{A}_{\\infty}$-algebra, then,\nin low tensor degrees, the corresponding maps $m^{n}$ satisfy:\n\\begin{equation*}\n  \\label{eq:1}\n  \\begin{array}{lccr}\n    n = 1 && m^{1} m^{1} = 0\\\\\n    n = 2 && m^{1} m^{2} = m^{2}(m^{1} \\otimes 1 + 1 \\otimes m^{1})\\\\\nn = 3 && m^{2}(1 \\otimes m^{2} - m^{2} \\otimes 1) = m^{1}m^{3} + m^{3}\n         \\circ m^{1}.\n  \\end{array}\n\\end{equation*}\nThus, $(A, m^{1})$ is a complex, $m^{2}$ satisfies the Leibniz rule\nwith respect to $m^{1},$ and the associator of $m^{2}$ is a boundary\nin the Hom-complex $(\\Hom {} {A^{\\otimes 3}} A, \\delta_{\\operatorname{Hom}})$ between the\ncomplexes $(A^{\\otimes 3}, \\delta_{\\otimes})$ and\n$(A, m^{1}).$\n\nIt follows easily from the above that a dg-algebra, i.e., a complex $(A, m^{1})$ with a\ncompatible associative multiplication $m^{2},$ uniquely determines an\n$\\text{A}_{\\infty}$-algebra $(A, (\\nu^{n}))$ with $\\nu^{n} = 0$ for all $n \\geq 3.$\nConversely, such an $\\text{A}_{\\infty}$-algebra uniquely determines a dg-algebra.\n\\end{rem}\n\n\\begin{rems}\nIn this section we could replace the category of graded $k$-modules\nwith the category of graded objects in an arbitrary symmetric monoidal\ncategory with coproducts, such that a finite coproduct is also a\nproduct, and such that the coproduct behaves as expected with respect\nto the tensor product. Indeed, given an object $V$ in such a category, set $\\Tco V =\n\\bigoplus_{n \\geq 1} V^{\\otimes n}$, and define a comultiplication $\\Tco\nV \\to \\Tco V \\otimes \\Tco V$ on the component $V^{\\otimes n}$ to be\nthe map $V^{\\otimes n} \\to \\bigoplus_{i = 1}^{n-1}(V^{\\otimes i} \\otimes\nV^{\\otimes n-i}) \\to \\Tco V \\otimes \\Tco V,$ which has components $V^{\\otimes n}\n\\xra{\\cong} V^{\\otimes i} \\otimes V^{\\otimes n - i}.$ Then $\\Tco V$ is\na coalgebra object in the category, and satisfies the formal\nproperties of \\ref{lem:isom_hhc_coder}, and so the definitions of\n\\ref{defn-A-infinity-algebra-morphism} make sense in this context.\n\\end{rems}\n\n\\section{Deformation theory of $\\text{A}_{\\infty}$-algebras}\nIn this section we recall how the Hochschild\ncochains control the infinitesimal deformation theory of an\n$\\text{A}_{\\infty}$-algebra. A goal is to give context and motivation for the\ndefinition and use of Maurer-Cartan elements of dg-Lie algebras. The\nreader uninterested in deformation theory only needs Definition\n\\ref{defn:MC-things}. There are no new results, and the approach follows \n\\cite{MR981619, KSDefTheory, KellerDefTheory}; see also\n\\cite{MR1364455,MR1935035, MR1239562}. We assume that $\\frac{1}{2} \\in\nk$.\\footnote{This assumption can be removed by treating $\\circ$ as a quadratic squaring map, as\n  in \\cite[\\S 2]{MR0195995}.}\n\n  \\begin{defn} Let $l$ be a commutative $k$-algebra.\n    \\begin{enumerate}\n    \\item \\emph{An $l$-family of\n        $\\text{A}_{\\infty}$-algebra structures on $A$} is an $l$-linear $\\text{A}_{\\infty}$-algebra structure on\n      $A \\otimes l$. We set\n\\[\\hhca l {A \\otimes l} {A \\otimes l} = \\prod_{n \\geq 1} \\Hom {l}\n  {(\\Pi (A \\otimes l))^{\\otimes n}} {\\Pi (A \\otimes l)}.\\] Using the isomorphism $\\hhca l {A \\otimes l} {A \\otimes l}\n  \\cong \\hhc A {A \\otimes l},$  and denoting $l$ in string\n  diagrams as $\\tikz[baseline={([yshift = -.1cm]current bounding\n    box.center)}]{\\draw[parameter] (0,0) -- (0,.5)}$, \nwe can write an $l$-family as\n$\\begin{tikzpicture}[inner sep =0mm]\n                                                     \n                                                     \n                                                     \n                                                      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm}\n                                                      \\def1.2cm{.4cm}\n                                                      \n                                                      \\newBoxDiagram\n                                                      [arm height =\n                                                      .5cm, width =\n                                                      1.2cm, coord =\n                                                      {($(1.2cm,\n                                                        0)$)},\n                                                      decoration\n                                                      yshift label =\n                                                      .2cm, label =\n                                                      {\\bs \\nu}]{f};\n                                                      \\printBox{f};\n                                                      \\printArm{f}{-1};\n                                                      \\printArm{f}{1};\n                                                      \\printArmSep{f};\n                                                      \\printOutput[\/parameter\n                                                      ]{f}{.4}\n                                                      \\printOutput{f}{-.4}\n                                                    \\end{tikzpicture}.$\n      \n\\item If $\\alpha: l \\to l'$ is a morphism of commutative\n      algebras, represented by a string diagram $\\begin{tikzpicture}\n    \\newBoxDiagram [arm height = .5cm, width = .25cm, decoration\n    yshift label = .05cm, box height = .5cm, label = {\\tiny{\\alpha}}]{f};\n    \\printBox{f}; \\printOutput[\/secparameter]{f}{0};\\printArm[\/parameter]{f}{0};\n  \\end{tikzpicture},$ then an $l$-family $\\bs \\nu$ gives rise to the\n  $l'$-family\n  $\\begin{tikzpicture}[inner sep =0mm]\n    \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm}\n    \\def1.2cm{.4cm}\n    \n    \\newBoxDiagram\n    [arm height =\n    .5cm, width =\n    1.2cm,\n    decoration\n    yshift label =\n    .2cm, output\n    height = 1.5cm, label =\n    {\\bs \\nu}]{f};\n   \n    \\printBox{f};\n    \\printArm{f}{-1};\n    \\printArm{f}{1};\n    \\printArmSep{f};\n   \n   \n    \\printOutput{f}{-.4}\n    \\newBoxDiagram [arm height = .5cm, width = .25cm,  coord =\n    {(.4cm,\n      0cm)},\n    decoration yshift label = .05cm, box height = .5cm, label = {\\tiny{\\alpha}}]{g};\n    \\printBox{g}; \\printOutput[\/secparameter]{g}{0};\\printArm[\/parameter]{g}{0};\n  \\end{tikzpicture}.\n  $\n Thus there is a functor,\n      \\begin{displaymath}\n        \\Aifam A: \\Algk \\to \\text{Set}\n      \\end{displaymath}\n      \\begin{displaymath}\n        l \\mapsto \\{ \\text{$l$-families of $\\text{A}_{\\infty}$-algebra structures on $A$} \\}.\n      \\end{displaymath}\n    \\end{enumerate}\n  \\end{defn}\n\n\n   \\begin{lem}[Yoneda]\\label{lem:Yoneda-for-deformations}\n    If the functor $\\Aifam A$ is representable by a $k$-algebra $l_{\\operatorname{u}}$ and an isomorphism of\n    functors\n  $\\zeta: \\Hom {\\Algk} {l_{\\operatorname{u}}} {-} \\xra{\\cong} \\Aifam A,$\n  then $\\operatorname{Spec} {l_{\\operatorname{u}}}$ is the moduli space of $\\text{A}_{\\infty}$-algebra structures on $A$\n  and $\\bs \\nu_{\\operatorname{univ}}  = \\zeta(1_{l_{\\operatorname{u}}})$ is the universal family of\n  $\\text{A}_{\\infty}$-algebra structures.\n\\end{lem}\nIndeed, for any commutative $k$-algebra $l$ and any $l$-family\n  $\\bs \\nu \\in \\hhca {l} {A \\otimes l} {A \\otimes l}$, there exists a unique morphism\n  $f: l_{\\operatorname{u}} \\to l$ such that $\\bs \\nu = \\Aifam A (f)(\\bs \\nu_{\\operatorname{univ}})$ (one\n  can take this as the definition of moduli space and universal family). In\n  particular, the set of $\\text{A}_{\\infty}$-algebra structures on $A$ corresponds to the\n  set of $k$-morphisms $l_{\\operatorname{u}} \\to k.$\n\n  If $A$ is a finitely generated\n  graded projective $k$-module, and is concentrated in non-negative\n  degrees, or in degrees at most $-2$, then\n  $\\Aifam A$ is representable. Indeed, set $L^{i} = \\hhc A\n  A_{-i}$ and $b = [-,-]: L^{1} \\otimes L^1 \\to L^2.$ By the\n  assumptions on $A$, $L^1$ is a finitely generated projective\n  $k$-module. Thus, writing $(-)^*$ for the $k$-dual, the natural map\n  $\\iota: L_1^* \\otimes L_1^* \\to (L_1 \\otimes L_1)^*$ is an\n  isomorphism. If we denote by $\\operatorname{sq}^*: L_2^* \\to \\operatorname{Sym}^2(L_1^*)$ the\n  map $L_2^* \\xra{b^*} (L_1 \\otimes L_1)^* \\xra{\\iota^{-1}} L_1^*\n  \\otimes L_1^* \\twoheadrightarrow \\operatorname{Sym}^2(L_1^*),$ then the algebra $l_{\\operatorname{u}} :=\n  \\operatorname{Sym}^{\\bullet}(L_1^*)\/(\\operatorname{sq}^*(L_2^*))$ represents $\\Aifam A$. If $k$\n  is an algebraically closed field, then the closed points of\n  $\\operatorname{Spec}{l_{\\operatorname{u}}}$ correspond to the Maurer-Cartan elements of $L^1 =\n  \\hhc A A_{-1}$, i.e., $\\text{A}_{\\infty}$-algebra structures on $A$.\n\nRegardless of whether the functor $\\Aifam A$ is representable, we can\nview the functor\nas a generalized scheme. By Yoneda's Lemma,\\footnote{A more general (and standard)\n  version than the one quoted in Lemma\n  \\ref{lem:Yoneda-for-deformations}.} an $l$-family $\\bs \\nu$\ncorresponds to\nthe natural transformation $\\bs \\nu_*: h^l \\to \\Aifam A$ that sends\n$\\beta \\in h^l(l') = \\Hom\n{\\Algk} l {l'}$ to $\\Aifam A(\\beta)(\\bs\n\\nu) \\in \\Aifam A(l')$. Given an $\\text{A}_{\\infty}$-structure $\\nu$ on $A$, we say\nthe $l$-family $\\bs \\nu$ contains $\\nu$ if there is a natural\ntransformation $\\epsilon^{*}: h^{k} \\to h^{l}$ such that the following diagram\nis commutative:\n\\begin{displaymath}\n  \\begin{tikzcd}\n    h^{k} \\ar[dr, \"\\nu_{*}\"] \\ar[dd, dashed, \"\\epsilon^{*}\"']&\\\\[-15pt]\n    &\\Aifam A.\\\\[-15pt]\n    h^{l} \\ar[ur,\"\\bs \\nu_{*}\"']&[-15pt]\n  \\end{tikzcd}\n\\end{displaymath}\nBy Yoneda again, the transformation $\\epsilon^{*}$ is determined by a\n$k$-algebra morphism $\\epsilon: l \\to k.$ Unraveling this gives an algebraic definition of $l$-family that contains a\nmarked $k$-point $\\nu.$\n\\begin{defn}\\label{defn:deformation}\n Let $(A, \\nu)$ be a nonunital $\\text{A}_{\\infty}$-algebra and $\\epsilon: l \\to k$ a\n morphism of commutative $k$-algebras. An \\emph{$(l, \\epsilon)$-deformation\n  of $(A, \\nu)$} is an $l$-family\n  $\\bs \\nu$ such that $\\Aifam A(\\epsilon)(\\bs \\nu) = \\nu.$ In\n  diagrams, this means,\n  \\begin{displaymath}\n\\begin{tikzpicture}[baseline={(current bounding box.center)}]\n                     \n                      \\def1.2cm{.75cm}\n                     \n                     \n                      \\def.52cm{.42cm}\n                     \n                     \n                     \n                      \\def0{0}\n                     \n                     \n                     \n                      \\def.6{1}\n                     \n                     \n                      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n                     \n                     \n                      \\def.08}\\def.35{.35{.08}\n                     \n                     \n                     \n                      \\def.35{.35}\n                     \n                      \\pgfkeys{lowBoxStyle\/.style = {box height =\n                          .52cm, output height = .52cm, width = 1.2cm,\n                          coord =\n                          {($(0,0)$)}, decoration yshift label = .2cm,\n                          decoration yshift label = .05cm, arm height\n                          = 3*.52cm, label = 1 \\otimes \\epsilon}}\n                     \n                      \\newBoxDiagram [\/lowBoxStyle]{lowBox};\n                      \\printBox{lowBox};\n                     \n                      \\lowBox.coord at top(highcoor, 0);\n                     \n                      \\pgfkeys{highBoxStyle\/.style = {width =\n                          1.2cm*.6, box height = .52cm, arm height =\n                          .52cm, output height = .52cm, coord =\n                          {(highcoor)}, output height = .52cm, label =\n                          {\\bs \\nu}}}\n                     \n                      \\newBoxDiagram[\/highBoxStyle]{highBox};\n                      \\printBox{highBox}; \\printArmSep{highBox};\n                        \\printArm{highBox}{-1};\n                      \\printArm{highBox}{1};\n                      \\printOutput{highBox}{-.4};\n                      \\printOutput[\/parameter]{highBox}{.4};\n                      \\printOutput{lowBox}{0};\n                    \\end{tikzpicture}\n                    \\hspace{.2cm} = \\hspace{.2cm}\n                    \\begin{tikzpicture}[baseline={(current bounding box.center)}]\n                     \n                      \\def1.2cm{.75cm}\n                     \n                     \n                      \\def.52cm{.42cm}\n                     \n                     \n                     \n                      \\def0{0}\n                     \n                     \n                     \n                      \\def.6{1}\n                     \n                     \n                      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n                     \n                     \n                      \\def.08}\\def.35{.35{.08}\n                     \n                     \n                     \n                      \\def.35{.35}\n                       \\pgfkeys{lowBoxStyle\/.style = {box height =\n                          .52cm, output height = .52cm, width = 1.2cm,\n                          coord =\n                          {($(0,0)$)}, decoration yshift label = .2cm,\n                          decoration yshift label = .05cm, arm height\n                          = 3*.52cm, label = \\nu}}\n                     \n                      \\newBoxDiagram [\/lowBoxStyle]{lowBox};\n                      \\print{lowBox};\n                      \\printArmSep{lowBox};\n                      \\printPeriod{lowBox};\n                      \\end{tikzpicture}\n\\end{displaymath}\nWe denote\nby $\\AugAlgk$ the category with objects pairs $(l, \\epsilon)$ as\nabove, and\nmorphisms the algebra morphisms commuting with the augmentations. Set $\\Aidef {(A, \\nu)}: \\AugAlgk\n\\to \\text{Set}$ to be the functor that sends $(l, \\epsilon)$ to the set of $(l, \\epsilon)$-deformations of $(A, \\nu)$.\n\\end{defn}\n\nIf $\\Aifam A$ is represented by $l_{\\operatorname{u}}$, and\n    $\\alpha: l_{\\operatorname{u}} \\to k$ is the morphism corresponding to $\\nu$,\n    then one checks\n    the augmented $k$-algebra $(l_{\\operatorname{u}}, \\alpha)$ represents\n    $\\Aidef {(A, \\nu)}$. Regardless of the representability of\n    $\\Aidef {(A, \\nu)},$ we can view the functor as describing the\n    generalized scheme $\\Aifam A$ near the $k$-point corresponding to $\\nu.$ We can focus\n  attention on the (generalized) infinitesimal neighborhoods of the $k$-point by restricting the\n  domain of $\\Aidef {(A,\\nu)}$ to $\\finAlgk$,  the full\n  subcategory of $\\AugAlgk$ with objects $(l, \\epsilon),$ such that $l$\n  is a finitely generated projective $k$-module and $(\\ker\n  \\epsilon)^{N} = 0$ for some $N \\geq 1$ (the last condition follows\n  from the first if\n  $k$ is a field,\n  and $l$ is local).\n\n\\begin{defn}\nAn \\emph{infinitesimal deformation} of a nonunital $\\text{A}_{\\infty}$-algebra $(A, \\nu)$ is\n  an $(l, \\epsilon)$-deformation with $(l, \\epsilon)$ an object of\n  $\\finAlgk$. The corresponding functor is denoted $\\infAidef {(A,\\nu)} = \\Aidef {(A,\\nu)}|_{\\finAlgk}: \\finAlgk \\to \\text{Set}.$\n\\end{defn}\n\nIf $\\Aidef {(A, \\nu)}$ is represented by an augmented $k$-algebra\n$(l_{\\operatorname{u}}, \\epsilon),$ such that $l_{\\operatorname{u}}\/(\\ker \\epsilon)^{n}$ is in\n$\\finAlgk$ for all $n$ (this holds when $k$ is a field and\n$l_{\\operatorname{u}}$ is noetherian), then\n$\\infAidef {(A,\\nu)}$ is pro-represented by the completion\n$\\displaystyle \\underleftarrow{\\lim}_{n \\geq 0} \\, l_{\\operatorname{u}}\/(\\ker\n\\epsilon)^{n}$. Indeed, the canonical morphism\nof functors,\n\\[\\displaystyle \\underrightarrow{\\operatornamewithlimits{colim}}_{n} \\Hom {\\Algk}\n{l_{\\operatorname{u}}\/(\\ker \\epsilon)^{n}} {-} \\to \\Hom {\\Algk} l_{\\operatorname{u}} -,\\] is\neasily checked to be an\nisomorphism on $\\finAlgk$, and this is the definition of\npro-representability (see e.g., \\cite[\\S 2]{MR1603480}).\nIf $l_{\\operatorname{u}}$ is Noetherian, then $\\operatorname{Spf}$ of the completion\nis the formal completion of $\\Aidef {(A, \\nu)}$ along the\n$k$-point of $\\nu$.\n\nSince $\\infAidef {(A, \\nu)}$ preserves limits, a result of Grothendieck, \\cite[Corollary to\n3.1]{MR1603480}, shows that it\nis pro-representable, but the result does not describe the\npro-representing object. Drinfeld showed \\cite{MR3285856}, in case $A$ is concentrated in non-negative\n  degrees, or in degrees at most $-2$, and degreewise\nfinitely generated, that the degree\nzero Lie algebra cohomology of the Hochschild cochains pro-represents\n$\\infAidef {(A, \\mu)}$. He put the answer in the following more general\ncontext, which shows where the finiteness assumptions on $A$\nenter. First note that $(\\finAlgk)^{\\text{op}} = \\finCoalgk,$ the category\nof cocomplete cocommutative coalgebras that are finitely generated projective\n$k$-modules. The ind-completion of this category is equivalent to $\\AugCoalgk,$ the category of\nall cocommutative cocomplete coalgebras that are projective\n$k$-modules\\footnote{This holds since the cocomplete coalgebras are\n  closed under colimits, and every element in a cocomplete coalgebra\n  is contained in a sub-coalgebra that is a finitely generated\n  projective $k$-module; see e.g., \\cite[Chapter 1, \\S 6]{MR0344261}, \\cite{MR0252485}}. It follows that\n$(\\AugCoalgk)^{\\text{op}}$ is equivalent to the pro-completion of $\\finAlgk.$\nAny functor $\\finAlgk \\to \\text{Set}$ that preserves limits extends uniquely\nto a limit preserving functor on the pro-completion, and thus such a functor is\npro-representable exactly when the corresponding functor on coalgebras\nis representable. The dual of the coalgebra is then a\npro-representing object. Kontsevich and Soibelman\ndevelop this point of view extensively in \\cite{KSDefTheory}.\n\nThe functor $\\infAidef {(A, \\mu)}$\nextends to a functor\non the category of dg-coalgebras (whose\nunderlying graded coalgebra is in $\\AugCoalgk$). We denote this\ncategory by $\\dgAugCoalgk$. When $k$ is a field\nof characteristic zero, Quillen \\cite{MR0258031}, assuming certain\nboundedness conditions later removed by\nHinich \\cite{HinichJPAA01}, defined a model category structure on\n$\\dgAugCoalgk,$ and showed there is an equivalence,\n\\begin{displaymath}\n\\xymatrix{ \\operatorname{Ho}(\\dgAugCoalgk) \\ar@<-1.2ex>[rr]_{\\cobar{}} &&\n\\operatorname{Ho}(\\dgLieAlgk), \\ar@<-1.2ex>[ll]_{\\Bar {}}^{\\cong}}\n\\end{displaymath}\nbetween the homotopy\ncategory of this model category and the homotopy category of dg-Lie\nalgebras. The equivalence is given by the commutative versions of the bar and\ncobar constructions. The functor $\\infAidef {(A, \\mu)}$ induces a functor $\\operatorname{Ho}(\\dgAugCoalgk)^{\\text{op}} \\to \\text{Set}.$ Such a\nfunctor is\nrepresentable exactly when its representable by $\\Bar L,$ for some\ndg-Lie algebra $L,$ by the above equivalence. Unwinding definitions, the functor represented by\n$\\Bar L,$ restricted to $\\finAlgk,$ is\nthe following (see\n\\cite[\\S 2.7]{KellerDefTheory} for details of the unwinding).\n\n\\begin{defn}\\label{defn:MC-things} Let $(L, \\delta)$ be a dg-Lie algebra.\n  \\begin{enumerate}\n  \\item The \\emph{Maurer-Cartan elements} are\n\\begin{displaymath}\n\\operatorname{MC} ( L, \\delta )\n        := \\{ v \\in L_{-1} \\, | \\, \\delta(v) + \\frac{1}{2}[ v, v] = 0 \\}.\n\\end{displaymath}\n  \\item The \\emph{Maurer-Cartan functor} is\n    \\begin{displaymath}\n      \\begin{gathered}\n        \\MCFunct {(L, \\delta)} : \\finAlgk \\to \\text{Set}\\\\\n        (l, \\epsilon) \\mapsto \\operatorname{MC} ( L \\otimes \\o l, \\delta \\otimes 1)\n        := \\{ v \\in (L \\otimes \\o l)_{-1} \\, | \\, (\\delta \\otimes\n        1)(v) + \\frac{1}{2}[ v, v] = 0 \\},\n      \\end{gathered}\n    \\end{displaymath}\n    where $\\o l = \\ker \\epsilon$ and $L \\otimes \\o l$ has the induced\n    bracket $[v \\otimes x, v' \\otimes x'] = [v, v'] \\otimes xx'$.\n    \n\\item A functor $F: \\finAlgk \\to \\text{Set}$ is \\emph{controlled by the dg-Lie\n      algebra $(L, \\delta)$} if there is an equivalence\n    $\\MCFunct {(L,\\delta)} \\xra{\\cong} F.$\n  \\end{enumerate}\n\\end{defn}\n\nWe assumed that $k$ was a characteristic zero field in the paragraph above, but Definition \\ref{defn:MC-things} makes sense over any\ncommutative ring (with $1\/2 \\in k$). The most natural context for this\nstory is derived algebraic\ngeometry, see \\cite{MR2827833, 1401.1044}. Staying at a more concrete level,\nSchectmann shows \\cite[Theorem 2.5]{math\/9802006} that if $L$ is concentrated in strictly\npositive cohomological degrees, and $L^{1}$ is finite dimensional, then the\nzeroth cohomology of the bar construction of $L$ pro-represents the Maurer-Cartan\nfunctor.\n\nOne motivation behind the Maurer-Cartan approach to deformation theory is that often there is an\napparent dg-Lie algebra controlling a given functor, for instance\n$\\infAidef {(A, \\nu)}$. We now show that the dg-Lie algebra of Hochschild\ncochains controls it (this is classical, but we give details for lack\nof a reference at this level of generality). Paired with  \\cite[Theorem\n2.5]{math\/9802006}, it recovers Drinfeld's description of the\npro-representing object of $\\infAidef {(A, \\mu)},$ assuming certain\nfiniteness conditions.\n\n\\begin{prop}\\label{prop:MC-equiv-to-defs}\n Let $(A, \\nu)$ be a nonunital $\\text{A}_{\\infty}$-algebra and\n  $(\\hhc A A, [\\nu,-])$ the Hochschld cochains. The following is an equivalence:\n  \\begin{align*}\n    \\MCFunct {(\\hhc A A, [\\nu, -])} \\to \\infAidef\n    {(A,\\nu)}\\\\\n    \\t \\nu \\mapsto \\theta_{l}(\\t \\nu + \\nu \\otimes 1) = \\bs \\nu,\n  \\end{align*}\n  where $\\theta_{l}$ is the canonical morphism of graded Lie algebras,\n  \\begin{displaymath}\n   \\begin{gathered}\n    \\theta_{l}: \\hhc A A \\otimes l \\to \\hhca l {A \\otimes l} {A\n      \\otimes l}\\\\\n    (\\nu^{n}) \\otimes y \\mapsto ([a_1 \\otimes x_1 | \\ldots | a_n\n    \\otimes x_n] \\mapsto \\nu^n[a_1|\\ldots|a_n] \\otimes yx_1\\ldots x_n\n    ),\n  \\end{gathered}\n\\end{displaymath}\nwith the induced bracket\non the source and the Gerstenhaber bracket on the target.\n\\end{prop}\n\n\n\\begin{proof}\n  Let $(l, \\epsilon)$ be an object of $\\finAlgk$ and set\n  $\\o l = \\ker \\epsilon.$ By definition,\n$$\\MCFunct {(\\hhc A A, [\\nu, -])}(l, \\epsilon)\n= \\operatorname{MC}( \\hhc A A \\otimes \\o l, [\\nu \\otimes 1, -]),$$ and one checks\nthe following is a bijection,\n\\begin{align*}\n  \\operatorname{MC}( \\hhc A A \\otimes \\o l, [\\nu \\otimes 1, -]) &\\xra{\\cong} \\operatorname{MC}(1\n                                                    \\otimes \\epsilon)^{-1}(\\nu) \\subseteq \\operatorname{MC}(\\hhc A\n                                                    A \\otimes l, 0)\\\\\n  \\t \\nu &\\mapsto \\t \\nu + \\nu \\otimes 1.\n\\end{align*}\nSince $l$ is a finite rank projective $k$-module, $\\theta_{l}$ is an\nisomorphism, and thus induces a bijection $ \\operatorname{MC}(\\hhc A A \\otimes l, 0)\n\\xra{\\cong} \\operatorname{MC}(\\hhca l {A \\otimes l} {A \\otimes l},0).$ This restricts to a bijection\n$\\operatorname{MC}(1 \\otimes \\epsilon)^{-1}(\\nu) \\xra{\\cong} \\Aifam\nA(\\epsilon)^{-1}(\\nu) = \\infAidef {(A,\\nu)}(l).$\n\\end{proof}\n\nOne is most often interested in families and deformations modulo the\nfollowing.\n\n\\begin{defn}\n  An \\emph{isomorphism between $l$-families} is an $l$-linear $\\text{A}_{\\infty}$-isomorphism.\n  An \\emph{equivalence of deformations} is an isomorphism of families\n  that reduces to the identity on $A$.\n\\end{defn}\n\n\nOne can consider isomorphism and equivalence classes using\nthe following group functors. For $l$ a commutative $k$-algebra, set $H_{A}(l)= \\{ \\t g = (\\t g^n) \\in \\hhca l {A \\otimes l} {A \\otimes\n      l}_{0} \\, | \\, \\t g^1 \\text{ is an isomorphism}\\}$. There is an equality $\\operatorname{Aut}_{\\Coalgk\n  [l]}(\\Tco{\\Pi A \\otimes l}) =  \\Psi^{-1}(H_{A}(l))$, where $\\Psi^{-1}$ is defined in \\ref{lem:isom_hhc_coder}.(2); see\n\\cite[Proposition 2.5]{MR1989615} for a proof. Thus $H_{A}$ is a group functor\n$ H_{A}: \\Algk \\to \\text{Group}$. Using this, set\n\\begin{displaymath}\n  \\begin{gathered}\n    G_A: \\AugAlgk \\to \\text{Group}\\\\\n    G_A(l,\\epsilon) = \\{ \\t g \\in H_A(l) \\, | \\,(1 \\otimes\n    \\epsilon)_{*}(\\t g) = 1 \\}.\n  \\end{gathered}\n\\end{displaymath}\nThere is an action $H_A \\times \\Aifam A \\to \\Aifam A$, defined using\nthe isomorphisms of \\ref{lem:isom_hhc_coder}, \nwhose quotient functor sends $l$ to the set of isomorphism classes\n  of $l$-families of $\\text{A}_{\\infty}$-structures on $A$. If $(A, \\nu)$ is an\n  $\\text{A}_{\\infty}$-structure, the action of $H_{A}$ restricts to an action $G_A \\times \\Aidef {(A,\\nu)} \\to \\Aidef {(A,\\nu)}$\n  whose quotient functor sends $(l, \\epsilon)$ to the set of equivalence classes of\n        $(l,\\epsilon)$-deformations of $(A,\\nu)$.\n\n\\begin{cor}\nLet $(A, \\nu)$ be a nonunital $\\text{A}_{\\infty}$-algebra. The following is a\nbijection,\n\\begin{displaymath}\n  \\begin{gathered}\nH^{1}(\\hhc A A,[\\nu, -]) \\to {\\infAidef {(A,\n  \\nu)}(k[t]\/(t^{2}))}\/{\\sim},\\\\\n\\nu \\mapsto \\theta(\\nu \\otimes t + \\nu \\otimes 1),\n\\end{gathered}\n\\end{displaymath}\nwhere the right side is the set of equivalence classes of $k[t]\/(t^{2})$-deformations.\n\\end{cor}\n\n\\begin{proof}\nLet $Z^{1}$ be the cohomological degree 1 cycles of the complex $(\\hhc\nA A, [\\nu, -]).$ The assignment $\\nu \\mapsto \\nu \\otimes t$\nis a bijection $Z^{1} \\to \\MCFunct {(A, \\nu)}(k[t]\/(t^{2})) = \\operatorname{MC}(\\hhc A A\n\\otimes kt, [\\nu \\otimes 1, -])$. Thus by \\ref{prop:MC-equiv-to-defs},\nthe assignment $\\nu \\mapsto \\theta(\\nu \\otimes t + \\nu \\otimes 1)$ is a\nbijection $Z^{1} \\xra{\\cong} \\infAidef {(A, \\nu)}(k[t]\/(t^{2})).$\n\nWe now claim that for $\\nu, \\nu' \\in Z^{1}$, the deformations\n$\\theta(\\nu \\otimes t + \\nu \\otimes 1)$ and $\\theta(\\nu' \\otimes t +\n\\nu \\otimes 1)$ are equivalent if and only if $\\theta((\\nu' - \\nu)\n\\otimes t) = \\theta([\\mu, \\alpha] \\otimes t),$ for some $\\alpha \\in\n\\hhc A A_{0}.$ The claim finishes the proof, since then $\\theta(\\nu \\otimes t + \\nu \\otimes 1)$ and $\\theta(\\nu' \\otimes t +\n\\nu \\otimes 1)$ are equivalent if and only if $\\nu' - \\nu = [\\mu,\n\\alpha]$ for some $\\alpha,$ using that\n$\\theta$ is a bijection. This last condition says exactly that $[\\nu]\n= [\\nu'] \\in H^{1}(\\hhc A A,[\\nu, -]).$ To see the claim, note there is a\nbijection $\\xi: \\hhc A A_{0} \\xra{\\cong} G_{A}(k[t]\/(t^{2})), \\alpha\n\\mapsto \\theta(1 \\otimes 1 + \\alpha \\otimes t),$ and the action of\n$\\xi(\\alpha)$ on $\\theta(\\nu \\otimes t + \\nu\n\\otimes 1)$ is $\\theta(\\nu \\otimes t + (\\nu\\circ \\alpha - \\alpha \\circ\n\\nu) \\otimes t + \\nu \\otimes 1).$\n\\end{proof}\n\n\\begin{rems}\nLet $(L, \\delta)$ be a dg-Lie algebra, and assume that $k$ contains\n$\\mathbb{Q}.$ For any $(l, \\epsilon) \\in \\finAlgk,$ the graded Lie algebra $L\n\\otimes \\o l$ is nilpotent, and thus we can define its exponential,\nwhich makes it a\ngroup. This gives a functor\n$\\finAlgk \\to \\text{Group}.$ This functor acts on the\nMaurer-Cartan functor of $(L, \\delta)$, and is usually the group\nfunctor one hopes to quotient by (in case the dg-Lie algebra is the Hochschild\ncochains, the group functor agrees with $G_{A}|_{\\finAlgk}$). This is another advantage of\nthe Maurer-Cartan formalism (in characteristic zero): the group we\nhope to quotient by is built into the Lie algebra.  This\npoint of view is due to Deligne, see \\cite{MR972343, KSDefTheory}.\n\\end{rems}\n\n\\section{Strictly unital $\\text{A}_{\\infty}$-algebras}\nIn this section, given a graded module $A$, we construct a dg-Lie\nalgebra whose Maurer-Cartan elements are the strictly unital\n$\\text{A}_{\\infty}$-structures on $A$. We first use this to recover Positselski's\nconstruction of a functorial curved bar construction from a strictly unital\n$\\text{A}_{\\infty}$-algebra, and then use it to show that the reduced Hochschild\ncochains control infinitesimal strictly unital deformations.\n\n\\subsection{Characterization of strictly unital structures}\n\\begin{defn}\n  Let $A, B$ be graded modules with fixed elements\n  $1 \\in A_0, 1 \\in B_{0}.$\n  \\begin{enumerate}\n  \\item An element $\\nu = (\\nu^n) \\in \\hhc A A_{-1}$ is \\emph{strictly\n      unital} (with respect to $1 \\in A_{0}$) if\n    \\[\\nu^2[1|a] = a = (-1)^{|a|} \\nu^2 [a|1]\\] and\n    $\\nu^n [a_1 | \\ldots |a_{i}|1|a_{i+1}|\\ldots|a_{n-1}] = 0$ for all\n    $a, a_1, \\ldots, a_{n-1} \\in A,$ where $n \\neq 2$ and\n    $0 \\leq i \\leq n$. If $\\nu$ is also an $\\text{A}_{\\infty}$-algebra structure, we say\n    $(A, \\nu)$ is a strictly unital $\\text{A}_{\\infty}$-algebra.\n  \n  \\item An element $f = (f^{n}) \\in \\hhc A B_{0}$ is \\emph{strictly\n      unital} if $f^{1}[1] = [1]$ and\n    $f^{n}[a_1|\\ldots |a_i|1|a_{i+1}|\\ldots|a_{n-1}] = 0$ for all\n    $a_1, \\ldots, a_{n-1} \\in A,$  $n \\geq 2$. If $f$ is also an $\\text{A}_{\\infty}$-morphism, we say it is a\n    strictly unital $\\text{A}_{\\infty}$-morphism.\n  \\end{enumerate}\n\\end{defn}\n\nFor our main results we need to place a further assumption on the pair $(A, 1)$ (that is automatically\nsatisfied when $k$ is a field).\n\n\\begin{defn}\n   A \\emph{split element} of a graded module $A$ is an element that\n  generates a rank one free module. A \\emph{graded module with split\n    element} is a pair $(A, 1)$ with $1$ a split element in $A$, and a fixed (unlabeled) splitting $A \\to k$ of the\n  inclusion $k \\to A, 1 \\mapsto 1.$ An \\emph{$\\text{A}_{\\infty}$-algebra with split\n    unit} is a triple $(A,1, \\nu)$, such that $(A, 1)$ is a\n  graded module with split element, and $(A, \\nu)$ is a strictly unital $\\text{A}_{\\infty}$-algebra  (with respect to\n  $1$). If $(A, 1)$ is a graded module with split element, we set $\\o A =\nA\/(k \\cdot 1).$ We  consider this as a submodule $\\o A\n\\subseteq A$ via the fixed splitting of $1.$\n\\end{defn}\n\nIf $(A,1)$ and $(B,1)$ are modules with\nsplit elements, then strictly unital elements $f \\in \\hhc {A} B_0$ are\nassumed to preserve the fixed splittings.\nIn string diagrams,\n$\\tikz[baseline={([yshift = -.1cm]current bounding\n  box.center)}]{\\draw[thick] (0,0) -- (0,.5)}$ represents $\\Pi \\o A$\n(previously it denoted $\\Pi A$), $\\tikz[baseline={([yshift = -.1cm]current bounding\n  box.center)}]{\\draw[bmod, line width = .5mm] (0,0) -- (0,.5)}$ represents $\\Pi \\o B,$ and\n$\\begin{tikzpicture}[baseline={([yshift = -.1cm]current bounding\n    box.center)}]\\draw[unit, line width = .35mm] (0,0) -- (0,.5);\\end{tikzpicture}$\nrepresents $\\Pi k$.\n  \n\\begin{defn}\\label{def-msu} An \\emph{$\\text{A}_{\\infty}$-algebra with split unit} is a\n  triple $(A,1,\\nu)$ with $(A,1)$ a module with split element and\n  $(A,\\nu)$ a strictly unital $\\text{A}_{\\infty}$-algebra with respect to\n  $1$. The \\emph{trivial $\\text{A}_{\\infty}$-algebra with split unit},\n  denoted $(A, 1, \\mu_{\\operatorname{su}})$, is defined by $\\mu_{\\operatorname{su}}^{n} = 0$ for $n\\neq 2$ and\n  \\begin{displaymath}\n    \\newcommand{.55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm}{.5cm}\n    \\newcommand{.52cm}{.6cm}\n    \\newcommand{.75cm}{.6cm}\n    \\mu_{\\operatorname{su}}^2 =\n    \\begin{tikzpicture}\n      \\newlength{\\lengg} \\setlength{\\lengg}{0pt}\\newBoxDiagram[width =\n      .75cm\/2, box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height =\n      .52cm, label = s^{-1}, coord = {($(0,\n        \\lengg)$)}]{s} \\printNoArms{s}\n      \\printLabelOnArm{s}{1}{\\Pi}\\setlength{\\lengg}{.55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm + .52cm}\n      \\newBoxDiagram[width = .75cm, box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height =\n      .52cm, output height = .52cm,output label = \\Pi, label =\n      \\small{(\\cong)}, coord = {($(0,\n        \\lengg)$)}]{isom} \\printNoArms{isom} \\printArm{isom}{1}\n      \\printArm[\/unit]{isom}{-1}\n    \\end{tikzpicture} \\hspace{.1cm} - \\hspace{.1cm}\n    \\begin{tikzpicture}\n      \\setlength{\\lengg}{0pt}\\newBoxDiagram[width = .75cm\/2, box\n      height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height = .52cm, label\n      = s^{-1}, coord = {($(0,\n        \\lengg)$)}]{s} \\printNoArms{s}\n      \\printLabelOnArm{s}{1}{\\Pi}\\setlength{\\lengg}{.55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm + .52cm}\n      \\newBoxDiagram[width = .75cm, box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height =\n      .52cm, output height = .52cm,output label = \\Pi, label =\n      \\small{(\\cong)}, coord = {($(0,\n        \\lengg)$)}]{isom} \\printNoArms{isom} \\printArm{isom}{-1}\n      \\printArm[\/unit]{isom}{1}\n    \\end{tikzpicture} \\hspace{.1cm} + \\hspace{.1cm}\n    \\begin{tikzpicture}\n      \\setlength{\\lengg}{0pt}\\newBoxDiagram[width = .75cm\/2, box\n      height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height = .52cm, output height = .52cm, label\n      = s^{-1}, coord = {($(0,\n        \\lengg)$)}]{s} \\printNoArms[\/unit output]{s}\n      \\printLabelOnArm{s}{1}{\\Pi}\\setlength{\\lengg}{.55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm + .52cm}\n      \\newBoxDiagram[width = .75cm, box height = .55cm} \\sum_{i_1 + \\ldots + i_k = n} \\hspace{.1cm, arm height =\n      .52cm, output height = .52cm,output label = \\Pi, label =\n      \\small{(\\cong)}, coord = {($(0,\n        \\lengg)$)}]{isom} \\printNoArms[\/unit output]{isom}\n      \\printArm[\/unit]{isom}{1} \\printArm[\/unit]{isom}{-1}\n    \\end{tikzpicture}\n    \\hspace{.1cm} = \\hspace{.1cm}\n    \\begin{tikzpicture}\n      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.4cm} \\def1.2cm{.5cm} \\def.3cm} \\def\\hgt{.55cm{.6cm}\n                                                  \n      \\newBoxDiagram [arm height = .5cm, width = 1.2cm, coord =\n      {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n        0)$)}, decoration yshift label = .2cm, label = \\mu_{\\operatorname{su}}^2]{f};\n       \\printNoArms{f};\n                                                      \\printArm[\/unit]{f}{-1};\n                                                      \\printArm{f}{1};\n                                                    \\end{tikzpicture}\n                                                    \\hspace{.1cm} + \\hspace{.1cm}\n    \\begin{tikzpicture}\n      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.4cm} \\def1.2cm{.5cm} \\def.3cm} \\def\\hgt{.55cm{.6cm}\n      \\newBoxDiagram [arm height = .5cm, width = 1.2cm, coord =\n      {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n        0)$)}, decoration yshift label = .2cm, label = \\mu_{\\operatorname{su}}^2]{f};\n       \\printNoArms{f};\n                                                      \\printArm{f}{-1};\n                                                      \\printArm[\/unit]{f}{1};\n                                                    \\end{tikzpicture}\n                                                                                                        \\hspace{.1cm} + \\hspace{.1cm}\n    \\begin{tikzpicture}\n      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.4cm} \\def1.2cm{.5cm} \\def.3cm} \\def\\hgt{.55cm{.6cm}\n      \\newBoxDiagram [arm height = .5cm, width = 1.2cm, coord =\n      {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n        0)$)}, decoration yshift label = .2cm, label = \\mu_{\\operatorname{su}}^2]{f};\n       \\printNoArms[\/unit output]{f};\n                                                      \\printArm[\/unit]{f}{-1};\n                                                  \\printArm[\/unit]{f}{1};\n    \\end{tikzpicture}\n    \\in \\hhn 2 A A_{-1},\n  \\end{displaymath}\n  where $(\\cong)$ denotes the following canonical isomorphisms, respectively: $\\Pi k\n  \\otimes \\Pi \\o A \\xra{\\cong} \\Pi(k \\otimes \\Pi \\o A) = \\Pi^{2} \\o\n  A$; $\\Pi \\o A \\otimes \\Pi k \\xra{\\cong} \\Pi( \\Pi \\o A \\otimes k) =\n  \\Pi^{2} \\o A;$ $\\Pi k \\otimes \\Pi k \\xra{\\cong} \\Pi(k \\otimes \\Pi k)\n  = \\Pi^{2} k$ (see\n  \\ref{sect:notation}.(4) for signs). One checks (carefully,\n  evaluating on elements) that\n  $\\mu_{\\operatorname{su}} \\circ \\mu_{\\operatorname{su}} = 0,$ and that $\\mu_{\\operatorname{su}}$ is strictly unital, thus $(A,1,\\mu_{\\operatorname{su}})$ is an $\\text{A}_{\\infty}$-algebra with split\n  unit.  If $B$ is a graded module with fixed element $1 \\in B_{0}$,\n  the \\emph{trivial strictly unital morphism $g_{\\operatorname{su}}: A \\to B$} is\n  $g_{\\operatorname{su}}^{1} = \\Pi A \\twoheadrightarrow \\Pi k \\to \\Pi B$ and $g_{\\operatorname{su}}^n = 0$\n  for $n \\geq 2.$\n\\end{defn}\n\n\\begin{lem}\n  \\label{lem:desc-of-strictly-unital}\n  Let $(A, 1)$ be a module with split element. Every strictly unital element in\n  $\\hhc A A_{-1}$ is of the form $\\mu + \\mu_{\\operatorname{su}}$ for a unique\n  $\\mu \\in \\hhc {\\o A} {A}_{-1}.$ If $B$ is another graded module with\n  a fixed element $1 \\in B_{0}$, every strictly unital element in\n  $\\hhc A B_{0}$ is of the form $g + g_{\\operatorname{su}}$ for a unique\n  $g \\in \\hhc {\\o A} B_{0}$.\n\\end{lem}\n\n\\begin{proof}\nFor a stricty unital element $\\nu \\in \\hhc A A_{-1},$ set $\\mu = \\nu -\n\\mu_{\\operatorname{su}} \\in \\hhc A A_{-1}.$ By definition, $\\mu$ is zero on any term\ncontaining a $1,$ and thus $\\mu \\in \\hhc {\\o A} A_{-1}.$ The proof for\nmorphisms is similar (and easier).\n\\end{proof}\n\n\\begin{rems}\nIf we replace the category of $k$-modules by a symmetric monoidal\ncategory, we can define $\\mu_{\\operatorname{su}}$ using the diagrams above (where $k$ is\nthe unit of the category), and\nuse the lemma to define strictly unital elements of $\\hhc A\nA_{-1}$ and $\\hhc A B_{0}$, when $A, B$ are objects in the category.\n\\end{rems}\n\nWe will use without remark that if $(A, 1)$ is a graded module\nwith split element, the splitting $A = \\o A \\oplus k$ induces a\nsplitting $\\hhc {\\o A} A = \\hhc {\\o A} {\\o A} \\oplus \\hhc {\\o A} k.$\n\n\\begin{defn}\n  A strictly unital element $\\mu + \\mu_{\\operatorname{su}} \\in \\hhc {A} A_{-1},$ with\n  $\\mu = \\o \\mu + h \\in \\hhc {\\o A}{\\o A}_{-1} \\oplus \\hhc {\\o A} k_{-1} = \\hhc\n  {\\o A} A_{-1},$ is \\emph{augmented} if $h = 0,$ i.e., if $\\mu$ is in $\\hhc\n  {\\o A} {\\o A}.$ (In this case, if $\\mu + \\mu_{\\operatorname{su}}$ is an $\\text{A}_{\\infty}$-algebra structure, the fixed\n  splitting $A \\to k$ is a strict $\\text{A}_{\\infty}$-morphism, called the\n  augmentation.)\n\\end{defn}\n\nWe note the term $h$ measuring the lack of augmentation is in\n\\[\\displaystyle\n\\hhc {\\o A} k_{-1} = \\prod_{n \\geq 1} \\Hom {} {(\\Pi \\o A)^{\\otimes n}}\n{\\Pi k}_{-1} = \\prod_{n \\geq 1}\n\\operatorname{Hom} \\left ( \\left ( \\left (\\Pi \\o\n  A\\right)^{\\otimes n}\\right )_{2},k \\right ).\\]\n\n\\begin{ex}\\label{ex:reslns-aug}\nLet $(A, 1)$ be a graded module with split element such that $A_{i}\n= 0$ for $i < 0,$ $A_{0} = k,$ and $1 \\in A_{0}$ is the unit in $k.$\nLet $\\mu = \\o \\mu + h \\in \\hhc {\\o A} A$ be an element such that $(A,\n1, \\nu = \\mu + \\mu_{\\operatorname{su}})$ is an $\\text{A}_{\\infty}$-algebra with split unit.\nSince $\\o A = A_{\\geq 1},$ it follows that $\\left ( \\left (\\Pi \\o\n  A\\right)^{\\otimes n}\\right )_{2} = 0$ for $n \\geq 2.$ Thus $h^{n} =\n0$ for all $n \\geq 2;$ the map $h^{1}$ makes the following diagram commutative:\n\\begin{displaymath}\n  \\begin{tikzpicture}\n      \\matrix (m) [matrix of math nodes,row sep=3em,column sep=4em,minimum width=2em]{\n     (\\Pi A)_{2} & (\\Pi k)_{1} \\\\\n    A_{1} & A_{0}. \\\\\n  };\n \\path[-stealth, auto] (m-1-1) edge node {$\\cong$}\n (m-2-1);\n  \\path[-stealth, auto]     (m-1-1)   edge node[swap] {$s^{-1}$} (m-2-1);\n \\path[-stealth, auto] (m-1-2) edge node[swap] {$\\cong$}\n (m-2-2);\n   \\path[-stealth, auto]     (m-1-2)   edge node {$s^{-1}$} (m-2-2);\n \\path[-stealth, auto]     (m-1-1)  edge node {$h^1$} (m-1-2);\n  \\path[-stealth, auto]     (m-2-1) edge node[swap] {$(m^1)_1$} (m-2-2);\n  \\end{tikzpicture}\n\\end{displaymath}\nHere $m^1$ is $s^{-1} \\nu^1 s$, see \\ref{ex:dg-alg-is-ainf-alg}. Note\nthat the image of $(m^1)_1$ is an ideal $I$ in $k = A_0$. The $\\text{A}_{\\infty}$-algebra $(A, 1, \\mu + \\mu_{\\operatorname{su}})$ is\naugmented exactly when $h^1 = 0,$ i.e., $I = 0.$\n\\end{ex}\n\nBy Lemma \\ref{lem:diagrams-for-sual} below, $[\\o \\mu, \\mu_{\\operatorname{su}}] = 0$ for all\n$\\o \\mu \\in \\hhc {\\o A} {\\o A}_{-1}$, and it follows that Maurer-Cartan elements of\n$\\hhc {\\o A} {\\o A}$ (i.e., nonunital $\\text{A}_{\\infty}$-algebra structures on\n$\\o A$) correspond  to\naugmented $\\text{A}_{\\infty}$-algebra structures on $A$, via the map $\\o \\mu \\mapsto\n\\o \\mu + \\mu_{\\operatorname{su}}$. The following generalizes this to all\nstrictly unital $\\text{A}_{\\infty}$-algebras.\n\n\\begin{thm}\\label{thm:main-thm-sunal}\n  Let $(A, 1)$ be a module with split element, and $\\mu_{\\operatorname{su}} \\in \\hhc A A_{-1}$ the trivial\n  strictly unital $\\text{A}_{\\infty}$-algebra structure defined in \\ref{def-msu}. The submodule $\\hhc {\\o A} A$\n  is a graded Lie subalgebra of $\\hhc A A$, and the derivation\n  $[\\mu_{\\operatorname{su}}, -]$ of $\\hhc A A$ preserves $\\hhc {\\o A} A$. The\n  Maurer-Cartan elements of the resulting dg-Lie algebra\n  $(\\hhc {\\o A} A, [\\mu_{\\operatorname{su}}, -])$ correspond to the strictly unital\n  $\\text{A}_{\\infty}$-structures on $(A,1)$ via\n  \\begin{displaymath}\n    \\o \\mu + h \\mapsto \\o \\mu + h + \\mu_{\\operatorname{su}}.\n  \\end{displaymath}\n  (See \\ref{defn:MC-things} for the definition of Maurer-Cartan\nelements.)\n\\end{thm}\n\n We need the following lemma for the proof of Theorem \\ref{thm:main-thm-sunal}.\n\n\\begin{lem}\n  \\label{lem:diagrams-for-sual} Let\n  $\\o \\mu, \\o \\mu' \\in \\hhc {\\o A} {\\o A}_{-1}$ and $h, h' \\in \\hhc {\\o A}\n  {k}_{-1}$ be arbitrary elements. The following hold:\n  \\begin{enumerate}\n  \\item $[\\o \\mu, \\mu_{\\operatorname{su}}] = 0 = [h, h'].$ \\vspace{.1in}\n  \\item\n    $[\\mu_{\\operatorname{su}}, h] = \\begin{tikzpicture}[baseline={(current bounding\n        box.center)}] \\def1.2cm{1cm} \\def.3}\\def\\relw{.6{-.35} \\def.6{.6}\n      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\newBoxDiagram[width\/.expand once = 1.2cm, arm height\n      = 1.5cm, label = \\mu_{\\operatorname{su}}^{2}]{f}; \\printNoArms{f}; \\f.coord at\n      top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, arm height\n      = .5cm, coord = {(gcoor)}, output height = .5cm, label = h]{g};\n      \\print[\/unit output]{g}; \\printArmSep{g}; \\printArm{f}{.8}\n      \\def.08}\\def.35{.35{.08}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n     \n     \n      \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n     \n\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      %\n     \n     \n      \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n     \n    \\end{tikzpicture} \\hspace{.1cm} + \\hspace{.1cm}\n    \\begin{tikzpicture}[baseline={(current bounding box.center)}]\n      \\def1.2cm{1cm} \\def.3}\\def\\relw{.6{.35} \\def.6{.6} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n      \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 1.5cm,\n      label = \\mu_{\\operatorname{su}}^{2}]{f}; \\printNoArms{f}; \\f.coord at top(gcoor,\n      .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, arm height = .5cm,\n      coord = {(gcoor)}, output height = .5cm, label = h]{g};\n      \\print[\/unit output]{g}; \\printArmSep{g}; \\printArm{f}{-.8}\n      \\def.08}\\def.35{.35{.08}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n     \n     \n      \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n     \n\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      %\n     \n     \n      \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n     \n    \\end{tikzpicture}\n    \\in \\hhc {\\o A} {\\o A}.$\n  \n\n  \\item $\\displaystyle [\\o \\mu, \\o \\mu'] = \\sum_j \\,\n    \\, \\begin{tikzpicture}[baseline={(current bounding box.center)}]\n      \\def1.2cm{1.3cm} \\def.3}\\def\\relw{.6{-.1} \\def.6{.5} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n      \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 1.5cm,\n      label = \\o \\mu]{f}; \\print{f}; \\f.coord at top(gcoor,\n      .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, arm height = .5cm,\n      coord = {(gcoor)}, output height = .5cm, label = \\o \\mu']{g};\n      \\print{g}; \\printArmSep{g};\n\n      \\def.08}\\def.35{.35{.08}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n      decorate right = \\var]{f}{j};\n      \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var}\n     \n     \n      \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n    \\end{tikzpicture} \\hspace{.1cm} + \\hspace{.1cm} \\sum_j \\,\n    \\, \\begin{tikzpicture}[baseline={(current bounding box.center)}]\n      \\def1.2cm{1.3cm} \\def.3}\\def\\relw{.6{-.1} \\def.6{.5} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n      \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 1.5cm,\n      label = \\o \\mu']{f}; \\print{f}; \\f.coord at top(gcoor,\n      .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, arm height = .5cm,\n      coord = {(gcoor)}, output height = .5cm, label = \\o \\mu]{g};\n      \\print{g}; \\printArmSep{g};\n\n      \\def.08}\\def.35{.35{.08}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n      decorate right = \\var]{f}{j};\n      \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var}\n     \n     \n      \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n    \\end{tikzpicture} \\in \\hhc {\\o A} {\\o A}.$\n\n  \\item\n   \n   \n    $\\displaystyle [\\o \\mu, h] = \\sum_j \\,\n    \\, \\begin{tikzpicture}[baseline={(current bounding box.center)}]\n      \\def1.2cm{1.3cm} \\def.3}\\def\\relw{.6{-.1} \\def.6{.5} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n      \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 1.5cm,\n      label = h]{f}; \\print[\/unit output]{f}; \\f.coord at\n      top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, arm height\n      = .5cm, coord = {(gcoor)}, output height = .5cm, label = \\o\n      \\mu]{g}; \\print{g}; \\printArmSep{g};\n\n      \\def.08}\\def.35{.35{.08}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n      decorate right = \\var]{f}{j};\n      \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var}\n     \n     \n      \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n    \\end{tikzpicture} \\in \\hhc {\\o A} k.$\n  \\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nAll of the equalities are automatic except for the first half of (1),\n$[\\o \\mu, \\mu_{\\operatorname{su}} ] = 0,$ and (2). To show (1), one can first check that for\nall $j \\geq 1,$ the following holds:\n\\begin{displaymath}\n\\begin{tikzpicture}[baseline={([yshift = -.35cm]current bounding\n  box.center)}]\n                                                     \n                                                     \n                                                     \n                                                      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm}\n                                                      \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.4cm}\n                                                      \\def1.2cm{.5cm}\n                                                      \\def.3cm} \\def\\hgt{.55cm{.6cm}\n                                                   \n                                                      \\newBoxDiagram\n                                                      [box height =\n                                                      0cm, output\n                                                      height = 0cm,\n                                                      arm height =\n                                                      1.5cm, width =\n                                                      .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm,\n                                                      decoration\n                                                      yshift label =\n                                                      .2cm]{left};\n                                                      \\newBoxDiagram\n                                                      [arm height =\n                                                      .5cm, width =\n                                                      1.2cm, coord =\n                                                      {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm +\n                                                        .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n                                                        0)$)},\n                                                      decoration\n                                                      yshift label =\n                                                      .2cm, label =\n                                                      \\mu_{\\operatorname{su}}^2]{f};\n                                                      \\newBoxDiagram\n                                                      [box height =\n                                                      0cm, output\n                                                      height = 0cm,\n                                                      arm height =\n                                                      1.5cm, width =\n                                                      .3cm} \\def\\hgt{.55cm, coord =\n                                                      {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm +\n                                                        2*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*1.2cm +\n                                                        .3cm} \\def\\hgt{.55cm, 0)},\n                                                      decoration\n                                                      yshift label =\n                                                      .2cm]{right};\n                                                      \\print{left};\n                                                      \\printArmSep{left};\n                                                      \\printDec[\/decorate]{left}{j};\n                                                      \\printNoArms{f};\n                                                      \\printArm[\/unit]{f}{-1};\n                                                      \\printArm{f}{1};\n                                                      \\print{right};\n                                                      \\printArmSep{right};\n                                                             \\end{tikzpicture}\n  \\hspace{.2cm} + \\hspace{.2cm} \n    \\begin{tikzpicture}[baseline={([yshift = -.35cm]current bounding\n  box.center)}]\n                                                     \n                                                     \n                                                     \n                                                      \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.2cm}\n                                                      \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.4cm}\n                                                      \\def1.2cm{.5cm}\n                                                      \\def.3cm} \\def\\hgt{.55cm{.6cm}\n                                                   \n                                                      \\newBoxDiagram\n                                                      [box height =\n                                                      0cm, output\n                                                      height = 0cm,\n                                                      arm height =\n                                                      1.5cm, width =\n                                                      .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm,\n                                                      decoration\n                                                      yshift label =\n                                                      .2cm]{left};\n                                                      \\newBoxDiagram\n                                                      [arm height =\n                                                      .5cm, width =\n                                                      1.2cm, coord =\n                                                      {($(.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm +\n                                                        .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n                                                        0)$)},\n                                                      decoration\n                                                      yshift label =\n                                                      .2cm, label =\n                                                      \\mu_{\\operatorname{su}}^2]{f};\n                                                      \\newBoxDiagram\n                                                      [box height =\n                                                      0cm, output\n                                                      height = 0cm,\n                                                      arm height =\n                                                      1.5cm, width =\n                                                      .3cm} \\def\\hgt{.55cm, coord =\n                                                      {(.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm +\n                                                        2*.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*1.2cm +\n                                                        .3cm} \\def\\hgt{.55cm, 0)},\n                                                      decoration\n                                                      yshift label =\n                                                      .2cm]{right};\n                                                      \\print{left};\n                                                      \\printArmSep{left};\n                                                      \\printDec[\/decorate]{left}{j-1};\n                                                      \\printNoArms{f};\n                                                      \\printArm{f}{-1};\n                                                      \\printArm[\/unit]{f}{1};\n                                                      \\print{right};\n                                                      \\printArmSep{right};\n                                                    \n                                                             \\end{tikzpicture}\n                                                             \\hspace{.1cm}\n                                                             = 0.\n                                                           \\end{displaymath}\n(To check this one can evaluate both diagrams on the element\n$[a_{1}|\\ldots|a_{j}|1|a_{j+1}|\\ldots|a_{n}]$, using the sign\nconventions of \\ref{rem:signs_for_diags}.) The above implies that\n\\begin{displaymath}\n\\o \\mu \\circ \\mu_{\\operatorname{su}} = \\begin{tikzpicture}[baseline={(current bounding box.center)}]\n      \\def1.2cm{1.3cm} \\def.3}\\def\\relw{.6{-.5} \\def.6{.5} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n      \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 1.5cm,\n      label = \\o \\mu]{f}; \\printNoArms{f}; \\printArm{f}{1}; \\f.coord at top(gcoor,\n      .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, arm height = .5cm,\n      coord = {(gcoor)}, output height = .5cm, label = \\mu_{\\operatorname{su}}^{2}]{g};\n      \\printNoArms{g}; \\printArm[\/unit]{g}{-1}; \\printArm{g}{1};\n\n      \\def.08}\\def.35{.35{.08}\n     \n     \n     \n     \n     \n     \n     \n\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var}\n     \n     \n      \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n    \\end{tikzpicture} +\n    \\begin{tikzpicture}[baseline={(current bounding box.center)}]\n      \\def1.2cm{1.3cm} \\def.3}\\def\\relw{.6{.5} \\def.6{.5} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n      \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 1.5cm,\n      label = \\o \\mu]{f}; \\printNoArms{f}; \\printArm{f}{-1}; \\f.coord at top(gcoor,\n      .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, arm height = .5cm,\n      coord = {(gcoor)}, output height = .5cm, label = \\mu_{\\operatorname{su}}^{2}]{g};\n      \\printNoArms{g}; \\printArm{g}{-1}; \\printArm[\/unit]{g}{1}; \\printArmSep{g};\n\n      \\def.08}\\def.35{.35{.08}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var};\n     \n     \n      \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n\n     \n     \n     \n     \n     \n     \n     \n     \n    \\end{tikzpicture}\n  \\end{displaymath}\n  and one checks this is $-\\mu_{\\operatorname{su}} \\circ \\o \\mu$ (by evaluating on\n  elements as above). The proof of $(2)$ is similar.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:main-thm-sunal}.] \n  For $\\o \\mu + h, \\o \\mu' + h' \\in \\hhc {\\o A} A$, we have $[\\o\n  \\mu + h, \\o \\mu' + h'] = [\\o \\mu, \\o \\mu'] + [\\o \\mu', h] + [\\o \\mu,\n  h'] \\in \\hhc {\\o A} {A}$, using the previous lemma. Thus $\\hhc {\\o\n    A} A$ is a graded subalgebra of $\\hhc A A$. Again using the lemma,\n  we have $[\\mu_{\\operatorname{su}},\n  \\o \\mu + h] = [\\mu_{\\operatorname{su}}, h] \\in \\hhc {\\o A} {\\o A}$, and thus $\\hhc {\\o\n    A} A$ is preserved by $[\\mu_{\\operatorname{su}}, -]$.\n\n  A strictly unital element $\\o \\mu + h + \\mu_{\\operatorname{su}}$ in\n  $\\hhc A A_{-1}$ is an $\\text{A}_{\\infty}$-algebra structure exactly when $[\\o \\mu + h + \\mu_{\\operatorname{su}}, \\o \\mu + h + \\mu_{\\operatorname{su}}] = [\\o \\mu + h, \\o \\mu + h]\n    + 2[\\mu_{\\operatorname{su}}, \\o \\mu + h]$ is zero, i.e., $\\frac{1}{2}[\\o \\mu + h, \\o\n    \\mu + h] + [\\mu_{\\operatorname{su}}, \\o \\mu + h] = 0.$ And this is the definition of\n    $\\o \\mu + h$ being a Maurer-Cartan element of $(\\hhc {\\o A} A,\n    [\\mu_{\\operatorname{su}}, -]).$\n\\end{proof}\n\n\\begin{rems}\n  The dg-Lie algebra $(\\hhc {\\o A} A, [\\mu_{\\operatorname{su}}, -])$ is an adaptation \n of a construction of Schlessinger and Stasheff \\cite[\\S\n  2]{SchlStash}, who use the cofree Lie coalgebra where we use the\n  cofree coassociative \n  coalgebra $\\Tco {\\Pi \\o A}$.  To match the definitions, one can\n  check that the graded subalgebra\n  $\\hhc {\\o A} {\\o A}$ of $\\hhc A A$ acts on the $k$-module\n  $\\hhc {\\o A} k,$ via Lemma \\ref{lem:diagrams-for-sual}.(4), and the\n  resulting semi-direct product $\\hhc {\\o A} {\\o A} \\oplus \\hhc {\\o A} k$ is\n  isomorphic as a graded Lie algebra to $\\hhc {\\o A} A$; one then checks the derivations agree.\n\\end{rems}\n\n\\subsection{Curved bar construction}\n\n\\begin{defn}\\label{defn:ad-and-curved-coalg}\n  Let $C$ be a graded coalgebra and $\\xi \\in \\Hom {} C k$ a\n  homogeneous linear map. Define $\\operatorname{ad}{\\xi} \\in \\Hom {} C C_{|\\xi|}$ to\n  be\n  \\begin{displaymath}\n    \\operatorname{ad}{\\xi} := \\begin{aligned}\n      \\left( C \\xra{\\Delta} C \\otimes C \\xra{\\xi \\otimes 1} k \\otimes C\n        \\cong C \\right)\n      -\\left( C \\xra{\\Delta} C \\otimes C \\xra{1 \\otimes \\xi} C \\otimes k\n        \\cong C \\right).\n    \\end{aligned}\n  \\end{displaymath}\n  (One checks this is a coderivation of $C$.) A \\emph{curved\n    dg-coalgebra} is a triple $(C,d, \\xi)$, with $C$ a graded\n  coalgebra, $d: C \\to C$ a coderivation of degree $-1$, and\n  $\\xi: C \\to k$ a degree $-2$ linear map, such that\n  $d^{2} = \\operatorname{ad}{\\xi}$ and $\\xi d = 0.$ A dg-coalgebra is a curved\n  dg-coalgebra with $h = 0$ (so $d^{2} = 0$).\n\\end{defn}\n\nWhen $C = \\Tco {\\Pi A},$ we can calculate $\\operatorname{ad}{\\xi}$ using the\nGerstenhaber bracket and the trivial strictly unital $\\text{A}_{\\infty}$-structure.\n\n\\begin{lem}\\label{lem:ad-for-tensor-coalgebra}\n  If $\\xi \\in \\Hom {} {\\Tco {\\Pi A}} k$, then\n  $\\operatorname{ad}{\\xi} = \\Phi^{-1}([\\mu_{\\operatorname{su}}, s \\xi])$.\n\\end{lem}\n\n\\begin{proof}\n  Since $\\operatorname{ad}{\\xi}$ is a coderivation, it is equal to\n  $\\Phi^{-1} (\\pi_{1} \\operatorname{ad}{\\xi}),$ using Lemma\n  \\ref{lem:isom_hhc_coder}.(1) (where $\\Phi^{-1}$ is defined,\n  also). Thus it is enough to show that\n  $\\pi_{1} \\operatorname{ad}{\\xi}|_{\\Pi \\overbar A^{\\otimes n}} = [\\mu_{\\operatorname{su}}, s\n  \\xi]|_{\\Pi \\overbar A^{\\otimes n}}$ for all $n \\geq 1.$ If\n  $\\xi = (\\xi^{n}),$ then\n  \\begin{displaymath}\n    \\pi_{1} \\operatorname{ad}{\\xi}|_{\\Pi \\overbar A^{\\otimes n}} = (\\Pi \\o A^{\\otimes n}\n    \\xra{\\xi^{n-1} \\otimes 1} k \\otimes \\Pi \\o A \\cong \\Pi \\o A) - (\\Pi \\o A^{\\otimes n}\n    \\xra{1 \\otimes \\xi^{n-1}} \\Pi \\o A \\otimes k \\cong \\Pi \\o A).\n  \\end{displaymath}\n  By Lemma \\ref{lem:diagrams-for-sual}.(2) we have\n  $[\\mu_{\\operatorname{su}}, s \\xi]|_{\\Pi \\overbar A^{\\otimes n}} = \\mu_{\\operatorname{su}}^{2}(s \\xi^{n-1}\n  \\otimes 1 + 1 \\otimes s \\xi^{n-1})$. Using the definition of $\\mu_{\\operatorname{su}}$\n  in \\ref{def-msu} and the isomorphisms \\ref{sect:notation}.(4), one\n  checks these agree.\n\\end{proof}\n\n\n\\begin{cor}\\label{cor:sual-ai-alg-equiv-to-curved-bar}\n  Let $(A, 1)$ be a graded module with split element. A strictly unital element\n  $\\o \\mu + h + \\mu_{\\operatorname{su}}$ in $\\hhc A A_{-1}$ is an $\\text{A}_{\\infty}$-algebra structure\n  if and only if the triple\n  $(\\Tco {\\Pi \\o A}, \\Phi^{-1}(\\o \\mu), -s^{-1} h)$ is a curved\n  dg-coalgebra ($\\Phi^{-1}$ is defined in\n  \\ref{lem:isom_hhc_coder}). In diagrams, this is equivalent\n  to:\n  \\begin{equation}\n    \\begin{gathered}\n        \\sum_{j} \\begin{tikzpicture}[baseline={(current bounding\n          box.center)}] \\def1.2cm{1cm} \\def.3}\\def\\relw{.6{.1} \\def.6{.4}\n        \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\def.3cm} \\def\\hgt{.55cm{.3cm} \\def.52cm{.5cm} \\newBoxDiagram [box height =\n        .52cm, output height = .52cm, width = 1.2cm, coord =\n        {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm, 0)$)}, decoration yshift label =\n        .2cm, decoration yshift label = .05cm, arm height = 3*.52cm,\n        label = \\o \\mu]{f}; \\print{f}; \\f.coord at top(gcoor,\n        .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, box height = .52cm,\n        arm height = .52cm, output height = .52cm, coord = {(gcoor)},\n        output height = .52cm, label = \\o \\mu]{g}; \\print{g};\n        \\printArmSep{g}; \\def.08}\\def.35{.35{.08}\n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n        decorate right = \\var]{f}{j};\n        \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var} \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n      \\end{tikzpicture}       \\hspace{.1cm} + \\hspace{.1cm}\n      \\begin{tikzpicture}[baseline={(current bounding box.center)}]\n        \\def1.2cm{1cm} \\def.3}\\def\\relw{.6{-.35} \\def.6{.6} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n        \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 1.5cm,\n        label = \\mu_{\\operatorname{su}}^{2}]{f}; \\printNoArms{f}; \n\n        \\printDec[color = white]{f}{test}\n\n\\f.coord at\n        top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, arm\n        height = .5cm, coord = {(gcoor)}, output height = .5cm, label\n        = h]{g}; \\print[\/unit output]{g}; \\printArmSep{g};\n        \\printArm{f}{.8} \\def.08}\\def.35{.35{.08}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n       \n       \n        \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n       \n\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      %\n       \n       \n        \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n       \n      \\end{tikzpicture}\n \\hspace{.1cm} + \\hspace{.1cm}\n      \\begin{tikzpicture}[baseline={(current bounding box.center)}]\n        \\def1.2cm{1cm} \\def.3}\\def\\relw{.6{.35} \\def.6{.6} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n        \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 1.5cm,\n        label = \\mu_{\\operatorname{su}}^{2}]{f};\n\n        \\printDec[color = white]{f}{test}\n \\printNoArms{f}; \\f.coord at\n        top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width = 1.2cm*.6, arm\n        height = .5cm, coord = {(gcoor)}, output height = .5cm, label\n        = h]{g}; \\print[\/unit output]{g}; \\printArmSep{g};\n        \\printArm{f}{-.8} \\def.08}\\def.35{.35{.08}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n       \n       \n        \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n       \n\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      %\n       \n       \n        \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n       \n      \\end{tikzpicture}\n\\quad = \\quad 0\\\\\n      \\sum_{j} \\begin{tikzpicture}[baseline={(current bounding\n          box.center)}] \\def1.2cm{1.3cm} \\def.3}\\def\\relw{.6{-.1} \\def.6{.5}\n        \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\newBoxDiagram[width\/.expand once = 1.2cm, arm\n        height = 1.5cm, label = h]{f}; \\print[\/unit output]{f};\n        \\f.coord at top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width =\n        1.2cm*.6, arm height = .5cm, coord = {(gcoor)}, output height\n        = .5cm, label = \\o \\mu]{g}; \\print{g}; \\printArmSep{g};\n\n        \\def.08}\\def.35{.35{.08}\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n        decorate right = \\var]{f}{j};\n        \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n\n       \n        \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n        \\printArm{f}{\\var}\n       \n       \n        \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n        \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n        \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n      \\end{tikzpicture} \\quad = \\quad 0.\n    \\end{gathered}\\label{eqn:diagrams-for-sual-ainf-alg}\n  \\end{equation}\n\\end{cor}\n\n\\begin{proof}\n  Let $\\o \\mu + h + \\mu_{\\operatorname{su}}$ be a strictly unital element with\n  $\\o \\mu + h \\in \\hhc {\\o A} {\\o A} \\oplus \\hhc {\\o A} k = \\hhc {\\o\n    A} A$. By Theorem \\ref{thm:main-thm-sunal}, this is an $\\text{A}_{\\infty}$-algebra\n  structure if and only if it is a Maurer-Cartan element of\n  $(\\hhc {\\o A} {A}, [\\mu_{\\operatorname{su}}, -]).$ By Lemma\n  \\ref{lem:diagrams-for-sual}, this is equivalent to\n$$[\\mu_{\\operatorname{su}}, h] + \\frac{1}{2}[\\o \\mu, \\o\n\\mu] = 0 \\quad \\text{ and } \\quad [\\o \\mu, h] = 0,$$ and these are\nequivalent to the first and second equations of\n\\eqref{eqn:diagrams-for-sual-ainf-alg}, respectively.\n\nSet $\\o d = \\Phi^{-1}(\\o \\mu)$. The triple\n$(\\Tco {\\Pi \\o A}, \\o d, -s^{-1} h)$ is a curved dg-coalgebra if and\nonly if $\\o d^2 + \\operatorname{ad}{s^{-1}h} = 0$ and $s^{-1} h \\o d = 0.$ We have\n$h \\o d = h \\Phi^{-1}(\\o \\mu) = h \\circ \\o \\mu,$ so $s^{-1} h \\o d$ is zero\nexactly when the second equation of\n\\eqref{eqn:diagrams-for-sual-ainf-alg} holds. Since $\\o d^{2}$\nand $\\operatorname{ad}{s^{-1} h}$ are both coderivations of $\\Tco {\\Pi \\o A}$, $\\o\nd^2 = -\\operatorname{ad}{s^{-1}h}$ holds if and only if\n$\\pi_{1}\\o d^{2} = -\\pi_{1} \\operatorname{ad}{s^{-1} h}$ holds, by\n\\ref{lem:isom_hhc_coder}.(1). We have\n$\\pi_{1}\\o d^{2} = \\o \\mu \\circ \\o \\mu$, and\n$\\pi_{1} \\operatorname{ad}{s^{-1} h} = [\\mu_{\\operatorname{su}}, h]$ by Lemma\n\\ref{lem:ad-for-tensor-coalgebra}. Thus $\\o d^{2} + \\operatorname{ad}{s^{-1}h} = 0$ holds if\nand only if $\\o \\mu \\circ \\o \\mu + [\\mu_{\\operatorname{su}}, h] = 0$ holds, and this is exactly\nthe first equation of\n\\eqref{eqn:diagrams-for-sual-ainf-alg}.\n\\end{proof}\n\n\\begin{defn}\n  If $(A, 1, \\o \\mu + h + \\mu_{\\operatorname{su}})$ is an $\\text{A}_{\\infty}$-algebra with split unit,\n  the \\emph{curved bar construction},\n  denoted $\\Bar {\\o A}$, is the curved dg-coalgebra\n  $(\\Tco {\\Pi \\o A}, \\Phi^{-1}(\\o \\mu), -s^{-1} h)$.\n\\end{defn}\n\n\\begin{rems}\n  Note that $\\Bar {\\o A}$ is a dg-coalgebra if and only if\n  $h = 0$ if and only if $(A, 1, \\mu)$ is augmented.\n\\end{rems}\n\n\\begin{ex}\n Let $(A,1)$ be a graded module with split element as in Example\n \\ref{ex:reslns-aug}, and let $\\nu = \\o \\mu + h + \\mu_{\\operatorname{su}}$ be a strictly\n unital $\\text{A}_{\\infty}$-algebra structure on $(A,1)$. Set $\\o d = \\Phi^{-1}(\\o\n \\mu) \\in \\Coder(\\Tco{\\Pi \\o A}, \\Tco{\\Pi \\o A}).$ By Example\n \\ref{ex:reslns-aug}, $h^n = 0$ for $n \\geq 2$, thus $h \\circ \\o \\mu =\n h^1 \\o \\mu^1 = 0$ and $\\mu_{\\operatorname{su}}^2 \\circ h$ is concentrated in tensor\n degree two. It follows from Corollary\n \\ref{cor:sual-ai-alg-equiv-to-curved-bar} that $\\o\n d^2[a_1|\\ldots|a_n] = 0$ for $n \\neq 2$ and\n\\begin{displaymath}\n  \\o d^{2}[a_{1}|a_{2}] =\n    \\begin{cases}\n      \\phantom{-}0 & |a_{1}| \\neq 1 \\text{ and } |a_{2}| \\neq 1\\\\\n      \\phantom{-}m^{1}(a_{1})a_{2} & |a_{1}| = 1 \\text{ and } |a_{2}| \\neq 1\\\\\n -m^{1}(a_{2})a_{1} & |a_{1}| \\neq 1 \\text{ and } |a_{2}| = 1\\\\\n \\phantom{-}m^{1}(a_{1})a_{2} - m^{1}(a_{2})a_{1} & |a_{1}| = 1 \\text{ and } |a_{2}| = 1.\n\\end{cases}\n\\end{displaymath}\nThe $\\text{A}_{\\infty}$-algebra $(A, 1, \\nu)$ is augmented exactly when $h^{1} = 0,$\nwhich is equivalent to $(m^{1})_{1} = 0.$ Thus we see directly in this\ncase that $\\o d^{2} = 0$ if and only if $(A, 1, \\nu)$ is augmented.\n \\end{ex}\n\nThe smallest nontrivial case of the above is the following.\n\n\\begin{ex}\\label{ex:curved-bar-constr-of-koszul-cx}\n  Let $(A, 1, \\mu)$ be the Koszul complex on $f \\in k,$  so $\\mu^{n} = 0$ for $n \\geq 3$, $\\mu^{2} = \\mu_{\\operatorname{su}}$ and\n  $\\mu^{1} = (k \\cdot [e] \\xra{f} k \\cdot [1]) \\in \\hhc {\\o A} k_{-1}$.\n  Thus $\\o \\mu = 0$ and $h = \\mu^{1}$, so $A$ is augmented if\n  and only if $f = 0.$ We also have:\n\\begin{displaymath} \\Tco {\\Pi \\o A}\\hspace{.2cm} = \\hspace{.2cm}\n  \\begin{tikzpicture}[font=\\fontsize{7}{22.4}\\selectfont]\n    \\node[matrix] (my matrix) at (0,0) {\n      \\node {\\ldots}; & \\node{0};  & \\node{$k[e]^{\\otimes n}$};  &\n      \\node{0};  & \\node{\\ldots};  & \\node{0};  & \\node{$k[e|e]$};  &\n      \\node{0};  & \\node{$k[e]$};  & \\node{0}; 0  & \\node{0};\\\\\n      \\node {}; & \\node{$2n+1$};  & \\node{$2n$};  & \\node{$2n-1$};  &\n      \\node{};  & \\node{5};  & \\node{4};  & \\node{3};  & \\node{2};  &\n      \\node{1};  & \\node{0};  & \\node{};\\\\};\n  \\end{tikzpicture}\n\\end{displaymath}\nand $h_{1}([e]) = -f$ and $h_{n} = 0$ for $n \\geq 2$. If we set $T =\n[e] \\in \\Tco {\\Pi \\o A}_{2},$ then $\\Tco {\\o A} = k[T],$ the divided powers coalgebra on the\n1-dimensional free module generated by $T$. The $k$-dual is the symmetric\nalgebra $k[T^{*}]$, with curvature $-f T^{*}\n\\in k[T^{*}]_{-2}$.\n\\end{ex}\n\nWe now show the curved bar construction is functorial.\n\n\\begin{defn}\n  A \\emph{morphism of curved dg-coalgebras, $(C, d_{C}, h_{C}) \\to\n  (D, d_{D}, h_{D}),$} is a pair $(\\gamma, \\alpha),$ with\n  $\\gamma: C \\to D$ a graded coalgebra morphism and\n  $\\alpha : {C} \\to k$ a degree $-1$ linear map, such that the\n  following equations hold,\n  \\begin{align*}\n    d_D \\gamma = \\gamma d_{C} + \\gamma \\operatorname{ad}{\\alpha} &\\in \\Hom {} C D\\\\\n    h_D \\gamma - \\alpha^2 = \\alpha d_{C} + h_{C} &\\in \\Hom {} C k,\n  \\end{align*}\n  where $\\operatorname{ad} \\alpha$ is defined in \\ref{defn:ad-and-curved-coalg}, and\n  $\\alpha^2 = (C \\xra{\\Delta} C \\otimes C \\xra{\\alpha \\otimes \\alpha}\n  k \\otimes k \\cong k).$\n\\end{defn}\n\n\n\\begin{cor}\\label{cor-strictly-unital-morphisms-curved-morphisms}\n  Let $(A, 1, \\o \\mu_{A}+ h_{A} + \\mu_{\\operatorname{su}})$ and $(B, 1, \\o \\mu_B + h_{B}\n  + \\mu_{\\operatorname{su}})$ be\n  $\\text{A}_{\\infty}$-algebras with split units. A strictly unital element\n  $g + g_{\\operatorname{su}} \\in \\hhc A B_0$, with\n$$g = \\o g + a \\in \\hhc {\\o A} {\\o B} \\oplus \\hhc {\\o A} k = \\hhc {\\o\n  A} B,$$ is a morphism of $\\text{A}_{\\infty}$-algebras if and only if\n$$(\\Psi^{-1}(\\o g), -s^{-1}a): \\Bar {\\o A} \\to \\Bar {\\o B}$$ is a\nmorphism of the corresponding curved dg-coalgebras, where\n$\\Psi^{-1}$ is defined in \\ref{lem:isom_hhc_coder}.(2). In diagrams this\nis equivalent to:\n\\begin{equation}\n  \\label{eq:cor-for-strictly-unital-morphism}\n\\def.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm{.33cm} \n    \\def.30cm{.30cm}\n \\def.33cm{.33cm}\n \\def.32cm{.32cm}\n \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1cm}\n\\def.52cm{.52cm}\n\\def.75cm{.55cm}\n\\begin{gathered}\n\\begin{tikzpicture}\n        \\newBoxDiagram[width = .4cm} \\def.30cm{.35cm} \\def.33cm{.45cm, box height = .52cm, arm height\n        = .52cm, coord = {(0,0)}, output height = .52cm, label =\n        {\\o g}]{g1};\n        \\print[\/bmod]{g1};\n        \\printArmSep{g1};\n        \\printDec[color = white]{g1}{test}\n        \\node at\n        ($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .32cm, .52cm +\n        .52cm\/2)$) {\\ldots};\n \\newBoxDiagram[width = .33cm, box height\n        = .52cm, arm height = .52cm, output height = .52cm, coord =\n        {($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*.32cm+.33cm,\n          0)$)}, label = {\\o g}]{gk};\n \\print[\/bmod]{gk}; \n\\printArmSep{gk};\n        \\setlength{\\var}{.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm\/2+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm\/2+.32cm+.33cm\/2}\n        \\setlength{\\vara}{-.52cm - .52cm}\n \\newBoxDiagram[width = \\var,\n        box height = .52cm, arm height = .52cm, output height = .52cm,\n        coord = {(\\var, \\vara)}, label = {\\o \\mu_{B}}]{mut}\n        \\printNoArms[\/bmod]{mut}\n       \n      \\end{tikzpicture}\n\\hspace{.2cm} - \\hspace{.2cm}\n \\sum_{j}\\hspace{.2cm}\n\\begin{tikzpicture}\n      \\def1.2cm{1cm} \\def.3}\\def\\relw{.6{-.1} \\def.6{.4} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n      \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 3*.52cm, box\n      height = .52cm, coord = {(0,0)},\n      label = {\\o g}]{f}; \\print[\/bmod]{f};\n\n \\f.coord at top(gcoor, .3}\\def\\relw{.6);\n      \\newBoxDiagram[width = 1.2cm*.6, arm height = .52cm, coord =\n      {(gcoor)}, output height = .52cm, label = {\\o \\mu_{A}}]{g};\n      \\print{g}; \\printArmSep{g};\n      \\def.08}\\def.35{.35{.08}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n      decorate right = \\var]{f}{j};\n      \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var} \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n    \\end{tikzpicture}\n\\hspace{.2cm} +  \\hspace{.2cm}\n\\begin{tikzpicture}[baseline={([yshift = .2cm]current bounding\n  box.center)}]\n\n\\setlength{\\lengg}{0pt}\n\n      \\newBoxDiagram[width = .75cm, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label =\n      \\mu_{\\operatorname{su}}^2, coord = {(0,\n       0)}]{isom} \\printNoArms[\/bmod]{isom}\n      \\printArm[\/unit]{isom}{1};\n\n\\newlength{\\whighbox}\n\\setlength{\\whighbox}{.75cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n\\isom.coord at top(hghcoor, -1)\n\\newBoxDiagram[width\/.expand once = \\whighbox, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label = {\\o g}, coord = {(hghcoor)}]{gmap}\n       \\printNoArms[\/bmod]{gmap}\n\\printArmSep{gmap};\n\\printArm{gmap}{-1};\n\\printArm{gmap}{1};\n\n\\setlength{\\whighbox}{.75cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n\\isom.coord at top(hghcoor, 1)\n\\newBoxDiagram[width\/.expand once = \\whighbox, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label = {a}, coord = {(hghcoor)}]{amap}\n       \\printBox{amap}\n\\printArmSep{amap};\n\\printArm{amap}{-1};\n\\printArm{amap}{1};\n    \\end{tikzpicture} \n    %\n\\hspace{.2cm} + \\hspace{.2cm}\n %\n\\begin{tikzpicture}[baseline={([yshift = .2cm]current bounding\n  box.center)}]\n\\setlength{\\lengg}{.52cm + .52cm}\n      \\newBoxDiagram[width = .75cm, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label =\n      \\mu_{\\operatorname{su}}^2, coord = {(0,\n        0)}]{isom} \\printNoArms[\/bmod]{isom}\n      \\printArm[\/unit]{isom}{-1};\n\n\\setlength{\\whighbox}{.75cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n\\isom.coord at top(hghcoor, -1)\n\\newBoxDiagram[width\/.expand once = \\whighbox, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label = a, coord = {(hghcoor)}]{amap}\n       \\printBox{amap}\n\\printArmSep{amap};\n\\printArm{amap}{-1};\n\\printArm{amap}{1};\n\n\\setlength{\\whighbox}{.75cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n\\isom.coord at top(hghcoor, 1)\n\\newBoxDiagram[width\/.expand once = \\whighbox, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label = {\\o g}, coord = {(hghcoor)}]{gmap}\n       \\printNoArms[\/bmod]{gmap};\n\\printArmSep{gmap};\n\\printArm{gmap}{-1};\n\\printArm{gmap}{1};\n    \\end{tikzpicture}  \\hspace{.2cm} = \\hspace{.2cm} 0\\\\\n    %\n    %\n   \n    %\n    %\n   \\begin{tikzpicture}\n        \\newBoxDiagram[width = .4cm} \\def.30cm{.35cm} \\def.33cm{.45cm, box height = .52cm, arm height\n        = .52cm, coord = {(0,0)}, output height = .52cm, label =\n        {\\o g}]{g1};\n        \\print[\/bmod]{g1};\n        \\printArmSep{g1};\n        \\printDec[color = white]{g1}{test}\n\n        \\node at\n        ($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .32cm, .52cm +\n        .52cm\/2)$) {\\ldots};\n \\newBoxDiagram[width = .33cm, box height\n        = .52cm, arm height = .52cm, output height = .52cm, coord =\n        {($(.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm+2*.32cm+.33cm,\n          0)$)}, label = {\\o g}]{gk};\n \\print[\/bmod]{gk}; \n\\printArmSep{gk};\n        \\setlength{\\var}{.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm\/2+.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm\/2+.32cm+.33cm\/2}\n        \\setlength{\\vara}{-.52cm - .52cm}\n \\newBoxDiagram[width = \\var,\n        box height = .52cm, arm height = .52cm, output height = .52cm,\n        coord = {(\\var, \\vara)}, label = {h_{B}}]{mut}\n        \\printNoArms[\/unit output]{mut}\n       \n      \\end{tikzpicture}\n\\hspace{.2cm} - \\hspace{.2cm}\n \\sum_{j}\\hspace{.2cm}\n\\begin{tikzpicture}\n      \\def1.2cm{1cm} \\def.3}\\def\\relw{.6{-.1} \\def.6{.4} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1}\n      \\newBoxDiagram[width\/.expand once = 1.2cm, arm height = 3*.52cm, box\n      height = .52cm, coord = {(0,0)},\n      label = {a}]{f}; \\print[\/unit output]{f};\n\n \\f.coord at top(gcoor, .3}\\def\\relw{.6);\n      \\newBoxDiagram[width = 1.2cm*.6, arm height = .52cm, coord =\n      {(gcoor)}, output height = .52cm, label = {\\o \\mu_{A}}]{g};\n      \\print{g}; \\printArmSep{g};\n      \\def.08}\\def.35{.35{.08}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n      decorate right = \\var]{f}{j};\n      \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n     \n      \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n      \\printArm{f}{\\var} \\pgfmathsetmacro{\\var}{1-.08}\\def.35{.35}\n      \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n      \\printArmSep[arm sep left = \\vara, arm sep right = \\var]{f}\n    \\end{tikzpicture}\n\\hspace{.2cm} -  \\hspace{.2cm}\n\\begin{tikzpicture}[baseline={([yshift = .2cm]current bounding\n  box.center)}]\n\\setlength{\\lengg}{.52cm + .52cm}\n      \\newBoxDiagram[width = .75cm, box height = .52cm, arm height =\n      2*.52cm, output height = 2*.52cm, label =\n      h_{A}, coord = {($(0,\n        \\lengg)$)}]{hmap} \n      \\print[\/unit output]{hmap}\n      \\printArmSep{hmap}\n\\end{tikzpicture}\n\\hspace{.2cm} + \\hspace{.2cm}\n %\n\\begin{tikzpicture}[baseline={([yshift = .2cm]current bounding\n  box.center)}]\n\n\\setlength{\\lengg}{.52cm + .52cm}\n      \\newBoxDiagram[width = .75cm, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label =\n      \\mu_{\\operatorname{su}}^2, coord = {($(0,\n        \\lengg)$)}]{isom} \\printNoArms[\/unit output]{isom} \\printArm[\/unit]{isom}{1}\n      \\printArm[\/unit]{isom}{-1};\n\n\\setlength{\\whighbox}{.75cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n\\isom.coord at top(hghcoor, -1)\n\\newBoxDiagram[width\/.expand once = \\whighbox, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label = a, coord = {(hghcoor)}]{amap}\n       \\printBox{amap}\n\\printArmSep{amap};\n\\printArm{amap}{-1};\n\\printArm{amap}{1};\n\n\\setlength{\\whighbox}{.75cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n\\isom.coord at top(hghcoor, 1)\n\\newBoxDiagram[width\/.expand once = \\whighbox, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label = a, coord = {(hghcoor)}]{gmap}\n       \\printBox{gmap}\n\\printArmSep{gmap};\n\\printArm{gmap}{-1};\n\\printArm{gmap}{1};\n    \\end{tikzpicture} \\hspace{.2cm} = \\hspace{.2cm} 0.\n    \\end{gathered}\n \n\\end{equation}\n\\end{cor}\n\n\nWe need the following lemma for the proof. For later use, we assume that $B$\nhas a strict, but not necessarily split, unit; e.g., $B = k\/I$ for\nsome ideal $I$.\n\\begin{lem}\\label{lem:strictly-unital-morphisms}\n  Let $(A, 1, \\o \\mu_{A} + h_{A} + \\mu_{\\operatorname{su}})$ be an $\\text{A}_{\\infty}$-algebra with split unit\n  and $(B, \\nu_B)$ an $\\text{A}_{\\infty}$-algebra with strict unit $1 \\in B_{0}$. A\n  strictly unital element $g + g_{\\operatorname{su}}$, with $g \\in \\hhc {\\o A} B_{0}$\n  is an $\\text{A}_{\\infty}$-morphism if and only if $\\nu_B * g$ and $g \\circ \\o \\mu_A\n  + g_{\\operatorname{su}} \\circ h_{A}$ are equal,\nwhere $*$ is defined in \\ref{rem:defining-start-and-gerst-produc} and\n$\\circ$ is the Gerstenhaber product.\n\\end{lem}\n\n\\begin{proof}\n  By definition,\n  $g + g_{\\operatorname{su}}$ is an $\\text{A}_{\\infty}$-morphism exactly when\n  \\begin{equation}\\label{eqn-ainf-morphism}\n    \\nu_B * (g + g_{\\operatorname{su}}) = (g + g_{\\operatorname{su}}) \\circ (\\o \\mu_{A} + h_{A} + \\mu_{\\operatorname{su}}).\n  \\end{equation}\n  We claim this equation always holds for elements of $\\Tco{\\Pi A}\n  \\setminus \\Tco{\\Pi \\o A}.$ Assuming the claim, we now note that the\n  above equation holds on elements of $\\Tco {\\Pi \\o A}$ if and only if\n  $\\nu_{B}* g = (g + g_{\\operatorname{su}}) \\circ (\\o \\mu_{A} + h_{A}),$ since $g_{\\operatorname{su}}$ and $\\mu_{\\operatorname{su}}$ are\n  zero on $\\Tco {\\Pi \\o A}.$ Also, clearly $g \\circ h_{A} = 0$ and\n  $g_{\\operatorname{su}} \\circ \\o \\mu_{A} = 0.$ Thus \\eqref{eqn-ainf-morphism} holds on\n  $\\Tco {\\Pi \\o A}$ if and only if $\\nu_B * g = g \\circ \\o \\mu_A\n  + g_{\\operatorname{su}} \\circ h_{A}.$\n\nWe are left to prove the claim, i.e., that \\eqref{eqn-ainf-morphism}\nholds on any element of the form\n$\\bs a = [a_1|\\ldots|a_{l-1}|1|a_{l+1}|\\ldots|a_{n}].$  We first compute the\nleft side. Since $g + g_{\\operatorname{su}}$\nis strictly unital, we have\n\\begin{displaymath}\n\\nu_B * (g + g_{\\operatorname{su}})(\\bs a) =\n      \\nu_{B}\\left (\\Psi^{-1}(g+ g_{\\operatorname{su}})[a_1|\\ldots|a_{l-1}] \\otimes [1]\n      \\otimes \\Psi^{-1}(g+g_{\\operatorname{su}})[a_{l+1}|\\ldots|a_{n}]\\right ).\n  \\end{displaymath}\nUsing that $\\nu_{B}$ is strictly unital, we have: if $l = 1 = n$, the result is $0$; if $l = 1 < 2 = n$ and\n $a_{2} = 1,$ or if $l = 2 = n$ and $a_{1} = 1,$ the\n result is $[1];$ if $l = 1 < n,$ the result is\n $g^{n-1}[a_2|\\ldots|a_{n}];$ if $1 < l = n,$ the result is\n  $(-1)^{|a_{1}| + \\ldots + |a_{n-1}| + n -\n    1}g^{n-1}[a_1|\\ldots|a_{n-1}];$ all other cases are zero. One now\n  checks the same six cases on the right side.\n\\end{proof}\n\n\\begin{proof}[Proof of \\ref{cor-strictly-unital-morphisms-curved-morphisms}]\n \n \n \n \n \n \n \n \n \n  By the previous lemma, $g + g_{\\operatorname{su}}$ is an $\\text{A}_{\\infty}$-morphism if and only if\n  $(\\o \\mu_B +  h_{B} + \\mu_{\\operatorname{su}}) * g = g \\circ \\o \\mu_A + g_{\\operatorname{su}} \\circ h_{A}.$\n  Substituting $g = \\o g + a$, and using the equalities $\\o \\mu_{B}*g =\n  \\o \\mu_{B} * \\o g$ and $h_{B}*g = h_{B} * \\o g,$ this is\n  equivalent to:\n  \\begin{equation}\\label{eq:in-proof-for-ainf-morphism}\n    \\o \\mu_B * \\o g + h_B * \\o g + \\mu_{\\operatorname{su}}*(\\o g + a) - \\o g \\circ \\o \\mu_{A} - a\n    \\circ \\o \\mu_{A} - g_{\\operatorname{su}} \\circ h_{A} = 0.\n  \\end{equation}\n  We can match each term of \\eqref{eq:in-proof-for-ainf-morphism}\n  with a diagram in \\eqref{eq:cor-for-strictly-unital-morphism}: $\\o\n  \\mu_{B} * \\o g$ is the first diagram and $- \\o g\n  \\circ \\o \\mu_{A}$ is the second diagram, both in the first line; $h_B * \\o g, -a \\circ \\o \\mu_{A}$ and $-g_{\\operatorname{su}} \\circ h_{A}$\n  are the first, second, and third diagrams of the\n  second line; $\\mu_{\\operatorname{su}}*(\\o g +\n  a)$ is the sum of the third and fourth diagrams of the first line,\n  and the fourth diagram on the second line. It follows that $g +\n  g_{\\operatorname{su}}$ is an $\\text{A}_{\\infty}$-morphism if and only if the equations\n  \\eqref{eq:cor-for-strictly-unital-morphism} hold.\n\n  We now claim the equations\n  \\eqref{eq:cor-for-strictly-unital-morphism} hold if and only if\n  $(\\Psi^{-1}(\\o g), -s^{-1}a)$ is a morphism of curved dg-coalgebras,\n  i.e.,\n  \\begin{align*}\n    \\Phi^{-1}(\\o \\mu_{B}) \\Psi^{-1}(\\o g) = \\Psi^{-1}(\\o g) \\Phi^{-1}(\\o\n    \\mu_{A}) - \\psi^{-1}(\\o g) \\operatorname{ad} {s^{-1}a}\\\\\n    -s^{-1} h_B \\Psi^{-1}(\\o g) - (-s^{-1}a)^2 = -s^{-1} a \\Phi^{-1}(\\o\n    \\mu_{B}) - s^{-1}h_{A}.\n  \\end{align*}\n  Using \\ref{cor:morphisms-commuting-with-coderivations} to reduce the\n  first equation, and applying $-s$ to the second, we have\n  \\begin{align*}\n    \\o \\mu_{B} \\Psi^{-1}(\\o g) = \\o g \\Phi^{-1}(\\o\n    \\mu_{A}) - \\o g \\operatorname{ad} {s^{-1}a}\\\\\n    h_B \\Psi^{-1}(\\o g) + s(s^{-1}a)^2 = a \\Phi^{-1}(\\o\n    \\mu_{B}) + h_{A}.\n  \\end{align*}\n  Using \\ref{lem:ad-for-tensor-coalgebra}, one calculates that\n   $\\o g \\operatorname{ad} {s^{-1}a}$ is the third and fourth terms of the first equation,\n   and one checks $s(s^{-1}a)^2$ is the last diagram in the second equation.\n The other terms are easily matched to their counterparts in the\n  equations \\eqref{eq:cor-for-strictly-unital-morphism}, \n  which completes the proof.\n\\end{proof}\n\n\\subsection{Strictly unital deformation theory}\nWe will use without comment that if $(A,1)$ is a graded $k$-module\nwith split element and $l$ is a $k$-algebra, then\n$(A \\otimes l, 1 \\otimes 1)$ is a graded $l$-module with split\nelement, and $\\o{A \\otimes l} = \\o A \\otimes l$.\n\n\\begin{defn}\n  A \\emph{strictly unital\n    $(l, \\epsilon)$-deformation of an $\\text{A}_{\\infty}$-algebra with split unit $(A,\n    1, \\mu)$}, where $(l, \\epsilon)$ is an augmented algebra, is an\n  $l$-linear $\\text{A}_{\\infty}$-algebra with split unit of the form\n  $(A\\otimes l, 1 \\otimes 1, \\bs \\mu)$, such that\n  $(A \\otimes l, \\bs \\mu)$ is a nonunital $(l, \\epsilon)$-deformation\n  of $(A, \\mu)$. We denote the resulting functor\n  \\begin{displaymath}\n    \\begin{gathered}\n      \\suinfAidef {(A,1,\\mu)}: \\finAlgk \\to \\text{Set}.\n    \\end{gathered}\n  \\end{displaymath}\n  If $(A, 1, \\mu)$ is an augmented $\\text{A}_{\\infty}$-algebra, an\n  \\emph{augmented $(l, \\epsilon)$ deformation} is a strictly unital\n  deformation $(A \\otimes l, 1 \\otimes 1, \\bs \\mu)$ that is augmented;\n  the corresponding functor is denoted $\\auginfAidef {(A,1, \\mu)}$.\n \n \n\\end{defn}\n\nWe denote by $\\msul l \\in \\hhca l {A \\otimes l} {A \\otimes l}$ the\n$l$-linear trivial strictly unital algebra structure (see\n\\ref{def-msu} for the definition). It follows from\n\\ref{lem:desc-of-strictly-unital} that strictly unital elements of\n$\\hhca l {A \\otimes l} {A \\otimes l}$ are of the form\n$\\bs {\\mu} + \\msul l$, where\n$$ \\bs {\\mu} \\in \\hhca l {\\o{A \\otimes l}} {A \\otimes l} =\n\\hhca l {\\o A \\otimes l} {A \\otimes l} \\cong \\hhc {\\o A} {A \\otimes\n  l}.$$ Using the decomposition\n$\\hhc {\\o A} {A \\otimes l} = \\hhc {\\o A} {A \\otimes \\o l} \\oplus \\hhc\n{\\o A} A,$ the strictly unital element $\\bs \\mu$ is a deformation of\nan $\\text{A}_{\\infty}$-algebra structure with split unit $(A, 1, \\mu + \\mu_{\\operatorname{su}})$ if and\nonly if $\\bs \\mu = \\t{\\bs \\mu} + \\theta_{l}(\\mu \\otimes 1)$ for some\n$\\t{\\bs \\mu} \\in \\hhc {\\o A} {A \\otimes \\o l}.$\n\nUsing a different decomposition, we can write\n$$\\bs \\mu =\n\\o{\\bs {\\mu}} + \\bs h \\in \\hhc {\\o A} {\\o A \\otimes l} \\oplus \\hhc {\\o\n  A} l = \\hhc {\\o A} {A \\otimes l}.$$ The element $\\bs \\mu + \\msul l$\nis augmented if and only if $\\bs h = 0.$ If $(A, 1, \\o \\mu + \\mu_{\\operatorname{su}})$ is\nan augmented $\\text{A}_{\\infty}$-algebra (so $(\\o A, \\o \\mu)$ is a nonunital\n$\\text{A}_{\\infty}$-algebra) it now follows easily that there is a natural equivalence\nof functors,\n\\begin{align*}\n  \\infAidef {(\\overbar A, \\overbar \\mu)} &\\to \\auginfAidef {(A, 1, \\overbar \\mu + \\mu_{\\operatorname{su}})}\\\\\n  \\o{\\bs \\mu} &\\mapsto \\o{\\bs \\mu} + \\msul l.\n\\end{align*}\nThus by \\ref{prop:MC-equiv-to-defs}, the dg-Lie algebra\n$(\\hhc {\\o A} {\\o A}, [\\o \\mu, -])$ controls the infinitesimal\naugmented deformations of the augmented $\\text{A}_{\\infty}$-algebra\n$(A, 1, \\overbar \\mu + \\mu_{\\operatorname{su}})$.\n\n\\begin{defn}\n  The \\emph{reduced Hochschild cochains} of an $\\text{A}_{\\infty}$-algebra\n  with split unit $(A, 1, \\mu + \\mu_{\\operatorname{su}})$ is the dg-Lie algebra\n  $(\\hhc {\\o A} A, [\\mu + \\mu_{\\operatorname{su}}, -])$ (it follows from\n  \\ref{lem:diagrams-for-sual} that $[\\mu + \\mu_{\\operatorname{su}}, -]$ preserves the\n  subalgebra $\\hhc {\\o A} A$ of $\\hhc A A$).\n\\end{defn}\n\n\\begin{cor}\\label{cor:mc-repn-of-sual-and-aug-deforms}\n  Let $(A,1, \\mu + \\mu_{\\operatorname{su}})$ be $\\text{A}_{\\infty}$-algebra with split unit. The reduced Hochschild cochains control the infinitesimal strictly\n  unital deformation functor via the natural transformation\n  \\begin{displaymath}\n    \\begin{array}{ccc}\n      \\MCFunct {(\\hhc {\\overbar A} {A}, [\\mu + \\mu_{\\operatorname{su}}, -])} &\\to& \\suinfAidef{(A,1,\\mu+\\mu_{\\operatorname{su}})}\\\\\n      \\t \\mu & \\mapsto & \\theta_{l}(\\t \\mu +\\mu \\otimes 1 + \\mu_{\\operatorname{su}} \\otimes 1)\n                         = \\bs \\mu + \\msul l.\n    \\end{array}\n  \\end{displaymath}\n\\end{cor}\n\n  \\begin{proof}\n    Let $(l, \\epsilon)$ be an object of $\\finAlgk$ and set\n    $\\o l = \\ker \\epsilon.$ By definition,\n$$\\MCFunct {(\\hhc {\\overbar A} A, [\\mu + \\mu_{\\operatorname{su}}, -])}(l, \\epsilon)\n= \\operatorname{MC}( \\hhc {\\overbar A} A \\otimes \\o l, [\\mu \\otimes 1 + \\mu_{\\operatorname{su}} \\otimes\n1, -]),$$ and the following is seen to be a bijection,\\\\\n\\scalebox{.85}{\\parbox{.5\\linewidth}{\n  \\begin{align*}\n  \\operatorname{MC}( \\hhc {\\overbar A} A \\otimes \\o l, [\\mu \\otimes 1 + \\mu_{\\operatorname{su}} \\otimes\n  1, -]) &\\xra{\\cong} \\operatorname{MC}(1\n           \\otimes \\epsilon)^{-1}(\\mu) \\subseteq \\operatorname{MC}(\\hhc {\\overbar A}\n           A \\otimes l, [\\mu_{\\operatorname{su}} \\otimes 1, -])\\\\\n  \\t \\mu &\\mapsto \\t \\mu + \\mu \\otimes 1.\n\\end{align*}}}\\\\\nOne checks that $\\theta_{l}(\\mu_{\\operatorname{su}} \\otimes 1) = \\msul l,$ and thus\n$\\theta_{l}$ is a morphism of dg-Lie algebras $(\\hhc {\\overbar A}\n  A \\otimes l, [\\mu_{\\operatorname{su}} \\otimes 1, -]) \\to (\\hhca l {\\overbar{A \\otimes\n      l}} {A \\otimes l},[\\msul l, -])$. Since $l$ is a finitely\ngenerated projective $k$-module, $\\theta_{l}$ is an isomorphism,\nand thus induces a bijection between MC elements.\nThe target is the set of $\\text{A}_{\\infty}$-algebra structures on\n$A \\otimes l$ such that $1 \\otimes 1$ is a split unit by Theorem\n\\ref{thm:main-thm-sunal}. Finally, one checks the bijection restricts\nto a bijection\n$\\operatorname{MC}(1 \\otimes \\epsilon)^{-1}(\\mu) \\xra{\\cong} \\Aifam\nA(\\epsilon)^{-1}(\\mu) = \\infAidef {(A,\\mu)}(l).$\n\\end{proof}\n\n\\begin{rems}\nThe reduced Hochschild cochains are quasi-isomorphic to the standard\nHochschild complex, see \\cite[Theorem 4.4]{MR1989615}, but not as\ndg-Lie algebras. Indeed, the functors they control, infinitesimal\nstrictly unital deformations and infinitesimal nonunital deformations,\nare different.\n\\end{rems}\n\n\n\\section{Representations of $\\text{A}_{\\infty}$-algebras}\nIn this section we treat strictly unital\n$\\text{A}_{\\infty}$-modules. In particular, we give a proof of\nPositselski's result that strictly unital modules over a strictly\nunital $\\text{A}_{\\infty}$-algebra correspond to cofree curved dg-comodules over the\ncurved bar construction.\n\n\\subsection{Representations of nonunital $\\text{A}_{\\infty}$-algebras}\nIf $(M, \\delta_{M})$ is a complex of modules, $\\Hom {} M M$ is\n  a dg algebra with multiplication equal to composition and\n  differential $\\delta_{\\operatorname{Hom}} = [\\delta_{M}, -].$ We denote by $(\\operatorname{End} M,\n  \\mu_{\\operatorname{End}})$ the corresponding\n  $\\text{A}_{\\infty}$-algebra, see \\ref{ex:dg-alg-is-ainf-alg}. \n\\begin{defn}\n  A \\emph{representation of a nonunital $\\text{A}_{\\infty}$-algebra $(A, \\mu)$ on\n a complex $(M, \\delta_{M})$} is an $\\text{A}_{\\infty}$-morphism\n  $p = (p^{n}) \\in \\hhc {A} {\\operatorname{End} M}_{0}$ from $(A, \\mu)$ to $(\\operatorname{End} M,\n  \\mu_{\\operatorname{End}}).$\n\\end{defn}\n\n\\begin{defn}\n  Let $M, N$ be graded modules. The \\emph{adjoint} of an element\n  $p^{n}$ in $\\hhcn {A} {\\Hom {} M N}_{k}$ is $\\lambda^{n+1} =\n  \\operatorname{ev}(s^{-1}p^{n} \\otimes 1): (\\Pi A)^{\\otimes n} \\otimes M \\to N$, where $\\operatorname{ev}(f \\otimes m ) = f(m)$. Thus\n  $\\lambda^{n+1}$ is the image of $p^{n}$ under the following isomorphisms:\n\\begin{align*}\n\\hhcn A {\\Hom {} M N}_{k} & \\cong \\Pi \\Hom {} {(\\Pi A)^{\\otimes n}} {\\Hom\n                         {} M N}_{k}\\\\\n  &\\cong \\Pi \\Hom {} {(\\Pi A)^{\\otimes n} \\otimes M} N_{k} = \\Hom {} {(\\Pi A)^{\\otimes n} \\otimes M} N_{k-1}, \n\\end{align*}\nwhere the first isomorphism is from \\ref{sect:notation}.(4).\nIn string diagrams, $\\tikz[baseline={([yshift = -.1cm]current bounding\n  box.center)}]{\\draw[thick] (0,0) -- (0,.5)}$ denotes $\\Pi A$, $\\tikz[baseline={([yshift = -.1cm]current bounding\n    box.center)}]{\\draw[module] (0,0) -- (0,.5)}$ denotes $M$, $\\tikz[baseline={([yshift = -.1cm]current bounding\n  box.center)}]{\\draw[secmodule] (0,0) -- (0,.5)}$ denotes $N$, and $\\tikz[baseline={([yshift = -.1cm]current bounding\n  box.center)}]{\\draw[hom output] (0,0) -- (0,.5)}$ represents\n$\\Pi \\Hom {} M N.$ We then have:\n\\begin{displaymath}\n\\def.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm{.33cm} \n    \\def.30cm{.30cm}\n \\def.33cm{.33cm}\n \\def.32cm{.32cm}\n \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1cm}\n\\def.1{.1}\n\\def.52cm{.52cm}\n \\def.75cm{1cm}\n     \n     \n      \\def.35{.35}\n\\begin{tikzpicture}\n\\setlength{\\lengg}{.52cm + .52cm}\n      \\newBoxDiagram[width = .75cm, box height = .52cm, arm height =\n      3*.52cm, output height = 3*.52cm, label =\n      \\lambda^{n+1}, coord = {($(0,\n        \\lengg)$)}]{g} \n      \\print[\/modulemap]{g}\n\n      \\pgfmathsetmacro{\\rightsolidarm}{1 - .35}\n      \\printArm{hmap}{\\rightsolidarm} \n\\pgfmathsetmacro{\\vara}{-1 +\n        2*.1} \n\\pgfmathsetmacro{\\varb}{\\rightsolidarm - 2*.1}\n      \\printArmSep[arm sep left = \\vara, arm sep right =\n      \\varb]{g}\n      \\printArm{g}{\\rightsolidarm}\n      \\printDec[\/decorate, decorate right = \\rightsolidarm]{g}{n}\n\\end{tikzpicture}\n\\hspace{.3cm} = \\hspace{.3cm}\n\\begin{tikzpicture}\n\n\\setlength{\\lengg}{0pt}\n\\newBoxDiagram[width =\n      .75cm, box height = .52cm, arm height = 5*.52cm, output height =\n      .52cm, label = \\operatorname{ev}, coord = {($(0,\n        \\lengg)$)}]{eval}\n \\printNoArms[\/secmodule]{eval}\n\\printRightArm[\/module]{eval}{1}\n\n\\eval.coord at top(coor, -.6)\n      \\newBoxDiagram[width = .4cm} \\def.30cm{.35cm} \\def.33cm{.45cm, box height = .52cm, arm height =\n      .52cm, output height = .52cm,output label = \\Pi^{-1}, label =\n      {s^{-1}}, coord = {(coor)}]{s}\n \\printNoArms[\/hom output]{s} \n\n\n\\setlength{\\whighbox}{.75cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n\\s.coord at top(hghcoor, 0)\n\\newBoxDiagram[width\/.expand once = .82*.75cm, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label = {p^{n}}, coord = {(hghcoor)}]{p}\n       \\print[\/hom output]{p};\n       \\printArmSep{p}\n       \\printPeriod{eval}\n       \\printDec[\/decorate]{p}{n}\n    \\end{tikzpicture}\n\\end{displaymath}\n\\end{defn}\n\n\n\\begin{lemma}\\label{lem:represen-iff-diagrams}\nAn element $p = (p^{n}) \\in \\hhc {A} {\\operatorname{End} M}_{0}$ is a representation\nof $(A, \\mu = (\\mu^{n}))$ on $(M,\n\\delta_{M})$ if and\nonly if the adjoint family $(\\lambda^{n+1}),$ \nwith $\\lambda^{1} = \\delta_{M}$, satisfies:\n\\begin{displaymath} \\sum_{i = 2}^{n+1} \\sum_{j=0}^{i-2} \\, \\,\n\\begin{tikzpicture}[baseline={(current bounding\n            box.center)}]\n          \\def1.2cm{1.4cm};\n          \\def.3}\\def\\relw{.6{-.2}; \\def.6{.45}\n          \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\def.3cm} \\def\\hgt{.55cm{.3cm} \\def.52cm{.5cm}\n          \\newBoxDiagram [box height =\n          .52cm, output height = .52cm, width = 1.2cm, coord =\n          {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift label\n          = .05cm, arm height = 3*.52cm, label = \\lambda^{i}]{f};\n          \\print[\/module]{f};\n      \n          \\f.coord at top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width =\n          1.2cm*.6, box height = .52cm, arm height = .52cm, output\n          height = .52cm, coord = {(gcoor)}, output height = .52cm,\n          label = \\mu^{n-i+2}]{g}; \\print{g}; \\printArmSep{g};\n          \\def.08}\\def.35{.35{.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n          decorate right = \\var]{f}{j};\n          \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n          \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var}\n          \\pgfmathsetmacro{\\vara}{1-.35+.08}\\def.35{.35}\n          \\printDec[\/decorate, decorate left = \\var, decorate right = \\vara]{f}{i\n          - j - 2}\n\\pgfmathsetmacro{\\var}{1-.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n          \\pgfmathsetmacro{\\varb}{\\var+.08}\\def.35{.35} \\printArmSep[arm sep\n          left = \\vara, arm sep right = \\var]{f} \\printArm{f}{\\varb}\n        \\end{tikzpicture}\n\\hspace{.2cm} + \\hspace{.2cm}\n\\sum_{i = 1}^{n+1}\\, \\, \\begin{tikzpicture}[baseline={(current bounding\n            box.center)}] \\def1.2cm{1.3cm}; \\def.3}\\def\\relw{.6{.5};\n          \\def.6{.52} \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm}\n          \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm} \\def.3cm} \\def\\hgt{.55cm{.3cm} \\def.52cm{.5cm}\n          \\newBoxDiagram [box height = .52cm, output height = .52cm,\n          width = 1.2cm, coord = {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift\n          label = .05cm, arm height = 3*.52cm, label = {\\lambda^{i}}]{f};\n          \\printNoArms[\/module]{f}; \\printArm{f}{-1};\n      \n          \\def.08}\\def.35{.35{.08}\\def.35{.35} \\f.coord at top(gcoor,\n          .3}\\def\\relw{.6); \n\\newBoxDiagram[width = 1.2cm*.6, box height =\n          .52cm, arm height = .52cm, output height = .52cm, coord =\n          {(gcoor)}, output height = .52cm, decoration yshift\n          label = .05cm, label = {\\lambda^{n-i+2}}]{g};\n          \\print[\/module]{g}; \\pgfmathsetmacro{\\var}{1-.35}\n          \\printArm{g}{\\var} \n\\printDec[\/decorate, decorate right = \\var]{g}{n-i+1}\n\\printArmSep[arm sep right = .2]{g};\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var};\n          \\printDec[\/decorate, decorate right = \\var]{f}{i-1}\n\n\\pgfmathsetmacro{\\var}{-1+2*.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm -\n            2*.08}\\def.35{.35} \\printArmSep[arm sep left = \\var, arm sep\n          right = \\vara]{f}\n        \\end{tikzpicture} = 0.\n\\end{displaymath}\n\\end{lemma}\n\n\\begin{proof}\nBy the definition of $\\text{A}_{\\infty}$-morphism, $p$ is a\nrepresentation if and only if $p \\circ \\mu_{A} - \\mu_{\\operatorname{End}} * p = 0.$ This\nequation holds if and only if it holds in every tensor degree. Applying the isomorphism\n$\\hhcn A {\\operatorname{End} M}_{0} \\cong \\Hom {} {(\\Pi A)^{\\otimes n} \\otimes M} M_{-1}$,\none checks the equation in tensor degree $n$ is equivalent to the diagrams above, with $p \\circ\n\\mu_{A}$ corresponding to the left diagram and $-\\mu_{\\operatorname{End}} * p$ to the\nright diagram.\n\\end{proof}\n\nTo define a morphism of representations, we need to add a counit\nto the tensor coalgebra (else we would have to fix a morphism of\ncomplexes, and talk about morphisms of\nrepresentations over that fixed morphism of complexes). Set\n\\begin{align*}\n\\Tcou {\\Pi A} &= k \\times \\Tco {\\Pi A} = \\bigoplus_{n \\geq 0} (\\Pi\n  A)^{\\otimes n}\\\\\n  \\hhcu A B &= \\Hom {} {\\Tcou {\\Pi A}} {\\Pi B} \\cong \\hhc A B \\oplus\\Pi\nB.\n\\end{align*}\nUsing this isomorphism, given a representation $p$ on a complex $(M, \\delta_{M})$, we set\n$p_{M} = p + \\delta_{M} \\in \\hhcu A {\\operatorname{End} M}_{0}$. Conversely, we can view\n representations as elements $p_{M} \\in \\hhcu A {\\operatorname{End} M}_{0}$\nsuch that $p^{0}_{M} \\in \\operatorname{End} M$ is a differential and $p^{\\geq 1}_{M}$ is an\n$\\text{A}_{\\infty}$-morphism from $A$ to the endomorphism $\\text{A}_{\\infty}$-algebra of the complex\n$(M, p^{0}_{M}).$ \n\n\\begin{defn}\nLet $M, N, P$ be graded modules. We consider the action\n\\begin{gather*}\n  \\star:  \\hhcu A {\\Hom {} N P}_{k} \\otimes \\hhcu A {\\Hom {} M N}_{l} \\to \\hhcu A {\\Hom\n  {} M P}_{k + l -1 }\\\\\n  \\alpha \\otimes \\beta = (\\alpha^n) \\otimes (\\beta^n) \\mapsto \\left (\\gamma  \\sum_{j =\n  0}^{n} \\alpha^j\\otimes \\beta^{n-j}\\right) = \\alpha \\star \\beta,\n\\end{gather*}\nwhere $\\gamma = s c (s^{-1} \\otimes s^{-1}),$ with $c$ the\ncomposition map. If $a^{n+1}$ and $b^{n+1}$ are the adjoints of\n$\\alpha^{n}$ and $\\beta^{n}$, and $\\tikz[baseline={([yshift = -.1cm]current bounding\n  box.center)}]{\\draw[thirdmodule] (0,0) -- (0,.5)}$ represents\n$P$, then the adjoint of $(\\alpha \\star \\beta)^n$ is\n\\begin{displaymath}\n          \\def1.2cm{1.4cm}\n          \\def.3}\\def\\relw{.6{.4} \\def.6{.5}\n          \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\def.3cm} \\def\\hgt{.55cm{.3cm} \\def.52cm{.55cm}\n\\def.08}\\def.35{.35{.08} \\def.35{.35}\n (-1)^{|a|-1}\\sum_{i = 1}^{n+1} \\, \\,\n\\begin{tikzpicture}[baseline={(current bounding\n            box.center)}]\n          \\newBoxDiagram [box height = .52cm, output height = .52cm,\n          width = 1.2cm, coord = {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift\n          label = .05cm, arm height = 3*.52cm, label = {a^{i}}]{g};\n          \\printNoArms[\/secmodulemap]{g}; \\printArm{g}{-1};\n      \n         \\g.coord at top(gcoor,\n          .3}\\def\\relw{.6); \n\\newBoxDiagram[width = 1.2cm*.6, box height =\n          .52cm, arm height = .52cm, output height = .52cm, coord =\n          {(gcoor)}, output height = .52cm, decoration yshift\n          label = .05cm, label = {b^{n-i+2}}]{m};\n          \\print[\/modulemap]{m}; \\pgfmathsetmacro{\\var}{1-.35}\n          \\printArm{m}{\\var} \n\\printDec[\/decorate, decorate right = \\var]{m}{n-i+1}\n\\printArmSep[arm sep right = .2]{m};\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{g}{\\var};\n          \\printDec[\/decorate, decorate right = \\var]{g}{i-1}\n\n\\pgfmathsetmacro{\\var}{-1+2*.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm -\n            2*.08}\\def.35{.35} \\printArmSep[arm sep left = \\var, arm sep\n          right = \\vara]{g}\n          \\printPeriod{g}\n        \\end{tikzpicture}\n\\end{displaymath}\n\\end{defn}\n\n\\begin{defn}\n\\emph{A morphism of representations $(M, p_{M}) \\to (N, p_{N})$ of a nonunital\n$\\text{A}_{\\infty}$-algebra $(A, \\mu)$} is\n  an element $f \\in \\hhcu {A} {\\Hom {} M N}_{1}$ such that\n  $p_{N} \\star f + (f^{\\geq 1})\\circ \\mu +  f \\star p_{M} = 0.$ The\n  composition with a second morphism, $\\t f \\in \\hhcu {A} {\\Hom {} N\n    P}_{1}$ is $\\t f \\star f \\in \\hhcu A {\\Hom {} M N}_{1}.$\n\\end{defn}\n\n\\begin{lemma}\\label{lem:morphism-represen-iff-diagrams}\nLet $(M, p_{M})$ and $(N, p_{N})$ be representations of a nonunital\n$\\text{A}_{\\infty}$-algebra $(A, \\mu)$, with adjoints $\\lambda_{M}$ and $\\lambda_{N},$ respectively. An element $f = (f^{n}) \\in \\hhcu {A} {\\Hom {}\n  M N}_{1}$ is a morphism of representations\n if and\nonly if the adjoint family $(g^{n+1})$ satisfies the equations\n\\begin{displaymath}\n          \\def1.2cm{1.2cm}\n          \\def.3}\\def\\relw{.6{.5} \\def.6{.5}\n          \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\def.3cm} \\def\\hgt{.55cm{.3cm} \\def.52cm{.55cm}\n\\def.08}\\def.35{.35{.08} \\def.35{.35}\n \\sum_{i = 1}^{n+1} \\, \\,\n\\begin{tikzpicture}[baseline={(current bounding\n            box.center)}]\n\n          \\newBoxDiagram [box height =\n          .52cm, output height = .52cm, width = 1.2cm, coord =\n          {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift label\n          = .05cm, arm height = 3*.52cm, label = {\\lambda_{N}^{i}}]{f};\n          \\printNoArms[\/secmodule]{f};\n          \\printArm{f}{-1};\n     \n           \\f.coord at top(gcoor,\n          .3}\\def\\relw{.6);\n \\newBoxDiagram[width = 1.2cm*.6, box height =\n          .52cm, arm height = .52cm, output height = .52cm, coord =\n          {(gcoor)}, output height = .52cm, decoration yshift\n          label = .05cm, label = {g^{n-i+2}}]{g};\n\n          \\print[\/modulemap]{g}; \n\\pgfmathsetmacro{\\var}{1-.35}\n          \\printArm{g}{\\var} \n\\printDec[\/decorate, decorate right = \\var]{g}{n-i+1}\n\\printArmSep[arm sep right = .2]{g};\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var}; \n\\printDec[\/decorate, decorate right = \\var]{f}{i-1}\n\\pgfmathsetmacro{\\var}{-1+2*.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm -\n            2*.08}\\def.35{.35} \n\\printArmSep[arm sep left = \\var, arm sep\n          right = \\vara]{f}\n        \\end{tikzpicture} \n\\hspace{.2cm} - \\hspace{.2cm}\n\\sum_{i = 2}^{n+1} \\sum_{j=0}^{i-2} \\, \\,\n\\begin{tikzpicture}[baseline={(current bounding\n            box.center)}]\n          \\def1.2cm{1.45cm};\n          \\def.3}\\def\\relw{.6{-.2}; \\def.6{.45}\n          \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\def.3cm} \\def\\hgt{.55cm{.3cm} \\def.52cm{.55cm}\n          \\newBoxDiagram [box height =\n          .52cm, output height = .52cm, width = 1.2cm, coord =\n          {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift label\n          = .05cm, arm height = 3*.52cm, label = g^{i}]{f};\n          \\print[\/modulemap]{f};\n      \n          \\f.coord at top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width =\n          1.2cm*.6, box height = .52cm, arm height = .52cm, output\n          height = .52cm, coord = {(gcoor)}, output height = .52cm,\n          label = \\mu^{n-i+2}]{g}; \\print{g}; \\printArmSep{g};\n          \\def.08}\\def.35{.35{.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n          decorate right = \\var]{f}{j};\n          \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n          \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var}\n          \\pgfmathsetmacro{\\vara}{1-.35+.08}\\def.35{.35}\n          \\printDec[\/decorate, decorate left = \\var, decorate right = \\vara]{f}{i\n          - j - 2}\n\\pgfmathsetmacro{\\var}{1-.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n          \\pgfmathsetmacro{\\varb}{\\var+.08}\\def.35{.35} \\printArmSep[arm sep\n          left = \\vara, arm sep right = \\var]{f} \\printArm{f}{\\varb}\n        \\end{tikzpicture}\n\\hspace{.2cm} - \\hspace{.2cm}\n\\sum_{i = 1}^{n+1}\\, \\, \\begin{tikzpicture}[baseline={(current bounding\n            box.center)}]\\def.3}\\def\\relw{.6{.3}\\def.6{.6}\n          \\newBoxDiagram [box height = .52cm, output height = .52cm,\n          width = 1.2cm, coord = {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift\n          label = .05cm, arm height = 3*.52cm, label = {g^{i}}]{g};\n          \\printNoArms[\/modulemap]{g}; \\printArm{g}{-1};\n      \n         \\g.coord at top(gcoor,\n          .3}\\def\\relw{.6); \n\\newBoxDiagram[width = 1.2cm*.6, box height =\n          .52cm, arm height = .52cm, output height = .52cm, coord =\n          {(gcoor)}, output height = .52cm, decoration yshift\n          label = .05cm, label = {\\lambda^{n-i+2}_{M}}]{m};\n          \\print[\/module]{m}; \\pgfmathsetmacro{\\var}{1-.35}\n          \\printArm{m}{\\var} \n\\printDec[\/decorate, decorate right = \\var]{m}{n-i+1}\n\\printArmSep[arm sep right = .2]{m};\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{g}{\\var};\n          \\printDec[\/decorate, decorate right = \\var]{g}{i-1}\n\n\\pgfmathsetmacro{\\var}{-1+2*.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm -\n            2*.08}\\def.35{.35} \\printArmSep[arm sep left = \\var, arm sep\n          right = \\vara]{g}\n        \\end{tikzpicture} = 0.\n\\end{displaymath}\n\\end{lemma}\n\nIndeed, each of the three terms above is the adjoint of (-1) times the corresponding\nterm in the definition of morphism.\n\n\\subsection{Representations of strictly unital $\\text{A}_{\\infty}$-algebras}\n\n\n\\begin{defn}\nLet $(A,1, \\mu)$ be a strictly unital $\\text{A}_{\\infty}$-algebra. A \\emph{strictly\n  unital representation} on a complex $(M, \\delta_{M})$ is a strictly unital\n$\\text{A}_{\\infty}$-morphism $p \\in \\hhc A {\\operatorname{End} M}_{0}$. A \\emph{morphism of strictly unital\nrepresentations $(M, p_{M})\\to (N, p_{N})$} is a morphism of\nrepresentations $f$ such that $f \\in \\hhcu {\\o A}\n{\\Hom {}M N}_{1},$ i.e.,\n$$f^{n}([a_{1}|\\ldots|a_{i-1}|1|a_{i+1}|\\ldots|a_{n}]) = 0 \\quad \\text{for all $n\n\\geq 1$}.$$\n\\end{defn}\n\nIn string diagrams, $\\tikz[baseline={([yshift = -.1cm]current bounding\n  box.center)}]{\\draw[thick] (0,0) -- (0,.5)}$ will now denote $\\Pi \\o A$\n(previously it denoted $\\Pi A$), while  $\\tikz[baseline={([yshift = -.1cm]current bounding\n    box.center)}]{\\draw[module] (0,0) -- (0,.5)}$ continues to represent $M$, $\\tikz[baseline={([yshift = -.1cm]current bounding\n  box.center)}]{\\draw[secmodule] (0,0) -- (0,.5)}$ represents $N$, and $\\tikz[baseline={([yshift = -.1cm]current bounding\n    box.center)}]{\\draw[unit] (0,0) -- (0,.5)}$ represents $\\Pi k.$\n\n\\begin{lem}\\label{lem:diagrams-for-s-ual-reps}\nLet $(A, 1, \\o \\mu + h + \\mu_{\\operatorname{su}})$ be an $\\text{A}_{\\infty}$-algebra with split unit.\n\\begin{enumerate}\n\\item A strictly unital element $p = \\o p +\ng_{\\operatorname{su}} \\in \\hhc A {\\operatorname{End} M}_{0}$, with $\\o p \\in \\hhcu {\\o A} {\\operatorname{End} M}_{0}$, is a representation if and only if the\nadjoint family $\\o \\lambda = (\\o \\lambda^{n+1})$ of $\\o p$, where $\\o \\lambda^{1} = \\delta_{M}$, satisfies\n\\begin{displaymath}\n            \\def1.2cm{1.4cm}\n          \\def.3}\\def\\relw{.6{-.2} \\def.6{.45}\n          \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\def.3cm} \\def\\hgt{.55cm{.3cm} \\def.52cm{.5cm}\n \\sum_{i = 2}^{n+1} \\sum_{j=0}^{i-2} \\, \\,\n\\begin{tikzpicture}[baseline={(current bounding\n            box.center)}]\n          \\newBoxDiagram [box height =\n          .52cm, output height = .52cm, width = 1.2cm, coord =\n          {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift label\n          = .05cm, arm height = 3*.52cm, label = {\\o \\lambda}^{i}]{f};\n          \\print[\/module]{f};\n      \n          \\f.coord at top(gcoor, .3}\\def\\relw{.6); \n\\newBoxDiagram[width = \n          1.2cm*.6, box height = .52cm, arm height = .52cm, output\n          height = .52cm, coord = {(gcoor)}, output height = .52cm,\n          label = {\\o \\mu}^{n-i+2}]{g}; \\print{g}; \\printArmSep{g};\n          \\def.08}\\def.35{.35{.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n          decorate right = \\var]{f}{j};\n          \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n          \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var}\n          \\pgfmathsetmacro{\\vara}{1-.35+.08}\\def.35{.35}\n          \\printDec[\/decorate, decorate left = \\var, decorate right = \\vara]{f}{i\n          - j - 2}\n\\pgfmathsetmacro{\\var}{1-.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n          \\pgfmathsetmacro{\\varb}{\\var+.08}\\def.35{.35} \\printArmSep[arm sep\n          left = \\vara, arm sep right = \\var]{f} \\printArm{f}{\\varb}\n        \\end{tikzpicture}\n\\hspace{.2cm} + \\hspace{.2cm}\n\\sum_{i = 1}^{n+1}\\, \\, \\begin{tikzpicture}[baseline={(current bounding\n            box.center)}]\n  \\def1.2cm{1.1cm}; \\def.3}\\def\\relw{.6{.35};\n           \\def.6{.6}\n\n          \\newBoxDiagram [box height = .52cm, output height = .52cm,\n          width = 1.2cm, coord = {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift\n          label = .05cm, arm height = 3*.52cm, label = {\\o \\lambda^{i}}]{f};\n          \\printNoArms[\/module]{f}; \\printArm{f}{-1};\n      \n          \\def.08}\\def.35{.35{.08}\\def.35{.35} \\f.coord at top(gcoor,\n          .3}\\def\\relw{.6); \n\\newBoxDiagram[width = 1.2cm*.6, box height =\n          .52cm, arm height = .52cm, output height = .52cm, coord =\n          {(gcoor)}, output height = .52cm,decoration yshift\n          label = .05cm, label = {\\o \\lambda^{n-i+2}}]{g};\n          \\print[\/module]{g}; \\pgfmathsetmacro{\\var}{1-.35}\n          \\printArm{g}{\\var} \n\\printDec[\/decorate, decorate right = \\var]{g}{n-i+1}\n\\printArmSep[arm sep right = .2]{g};\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var};\n          \\printDec[\/decorate, decorate right = \\var]{f}{i-1}\n\n\\pgfmathsetmacro{\\var}{-1+2*.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm -\n            2*.08}\\def.35{.35} \\printArmSep[arm sep left = \\var, arm sep\n          right = \\vara]{f}\n        \\end{tikzpicture}\n\\hspace{.1cm} + \\hspace{.1cm}\n\\begin{tikzpicture}\n\n  \\def.3}\\def\\relw{.6{-.3}\n\\def.4cm} \\def.30cm{.35cm} \\def.33cm{.45cm{.33cm} \n    \\def.30cm{.30cm}\n \\def.33cm{.33cm}\n \\def.32cm{.32cm}\n \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1cm}\n \\def.75cm{.75cm}\n\\setlength{\\lengg}{0pt}\n\n      \\newBoxDiagram[width = .75cm, box height = .52cm, arm height =\n      3*.52cm, output height = .52cm, label =\n      \\small{(\\cong)}, coord = {($(0,\n        0)$)}]{isom}\n      \\printNoArms[\/module]{isom}\n      \\printRightArm[\/module]{isom}{1}\n\n\\setlength{\\whighbox}{.75cm - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n\\isom.coord at top(hghcoor,.3}\\def\\relw{.6)\n\\newBoxDiagram[width\/.expand once = \\whighbox, box height = .52cm, arm height =\n      .52cm, output height = .52cm, label = {s^{-1}h^{n}}, output label\n      = \\Pi^{-1}, coord = {(hghcoor)}]{gmap}\n       \\print[\/unit output]{gmap}\n\\printArmSep{gmap};\n\\printArm{gmap}{-1};\n\\printArm{gmap}{1};\n\\printDec[color = white]{gmap}{test}\n    \\end{tikzpicture} \n\\hspace{.1cm} = \\hspace{.1cm} 0.\n\\end{displaymath}\n\\item An\n  element $f \\in \\hhcu {\\o A} {\\Hom {} M N}_{1}$ is a morphism of\n  strictly unital\n  representations $(M,\\o p_{M}) \\to (N, \\o p_{N})$ if and only if the\n  following holds, where $g, \\o \\lambda_{M}, \\o \\lambda_{N}$ are the adjoint\n  families of $f, \\o p_{M}, \\o p_{N}:$\n\\begin{displaymath}\n          \\def1.2cm{1.2cm}\n          \\def.3}\\def\\relw{.6{.5} \\def.6{.5}\n          \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\def.3cm} \\def\\hgt{.55cm{.3cm} \\def.52cm{.55cm}\n\\def.08}\\def.35{.35{.08} \\def.35{.35}\n \\sum_{i = 1}^{n+1} \\, \\,\n\\begin{tikzpicture}[baseline={(current bounding\n            box.center)}]\n\n          \\newBoxDiagram [box height =\n          .52cm, output height = .52cm, width = 1.2cm, coord =\n          {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift label\n          = .05cm, arm height = 3*.52cm, label = {\\o \\lambda_{N}^{i}}]{f};\n          \\printNoArms[\/secmodule]{f};\n          \\printArm{f}{-1};\n     \n           \\f.coord at top(gcoor,\n          .3}\\def\\relw{.6);\n \\newBoxDiagram[width = 1.2cm*.6, box height =\n          .52cm, arm height = .52cm, output height = .52cm, coord =\n          {(gcoor)}, output height = .52cm, decoration yshift\n          label = .05cm, label = {g^{n-i+2}}]{g};\n\n          \\print[\/modulemap]{g}; \n\\pgfmathsetmacro{\\var}{1-.35}\n          \\printArm{g}{\\var} \n\\printDec[\/decorate, decorate right = \\var]{g}{n-i+1}\n\\printArmSep[arm sep right = .2]{g};\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var}; \n\\printDec[\/decorate, decorate right = \\var]{f}{i-1}\n\\pgfmathsetmacro{\\var}{-1+2*.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm -\n            2*.08}\\def.35{.35} \n\\printArmSep[arm sep left = \\var, arm sep\n          right = \\vara]{f}\n        \\end{tikzpicture} \n\\hspace{.2cm} - \\hspace{.2cm}\n\\sum_{i = 2}^{n+1} \\sum_{j=0}^{i-2} \\, \\,\n\\begin{tikzpicture}[baseline={(current bounding\n            box.center)}]\n          \\def1.2cm{1.45cm};\n          \\def.3}\\def\\relw{.6{-.2}; \\def.6{.45}\n          \\def.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm{.1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\def.3cm} \\def\\hgt{.55cm{.3cm} \\def.52cm{.55cm}\n          \\newBoxDiagram [box height =\n          .52cm, output height = .52cm, width = 1.2cm, coord =\n          {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift label\n          = .05cm, arm height = 3*.52cm, label = g^{i}]{f};\n          \\print[\/modulemap]{f};\n      \n          \\f.coord at top(gcoor, .3}\\def\\relw{.6); \\newBoxDiagram[width =\n          1.2cm*.6, box height = .52cm, arm height = .52cm, output\n          height = .52cm, coord = {(gcoor)}, output height = .52cm,\n          label = \\o \\mu^{n-i+2}]{g}; \\print{g}; \\printArmSep{g};\n          \\def.08}\\def.35{.35{.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var}; \\printDec[\/decorate, decorate left = -1,\n          decorate right = \\var]{f}{j};\n          \\pgfmathsetmacro{\\var}{-1+.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm - .08}\\def.35{.35}\n          \\printArmSep[arm sep left = \\var, arm sep right = \\vara]{f}\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{f}{\\var}\n          \\pgfmathsetmacro{\\vara}{1-.35+.08}\\def.35{.35}\n          \\printDec[\/decorate, decorate left = \\var, decorate right = \\vara]{f}{i\n          - j - 2}\n\\pgfmathsetmacro{\\var}{1-.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 + .6 + .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm + .08}\\def.35{.35}\n          \\pgfmathsetmacro{\\varb}{\\var+.08}\\def.35{.35} \\printArmSep[arm sep\n          left = \\vara, arm sep right = \\var]{f} \\printArm{f}{\\varb}\n        \\end{tikzpicture}\n\\hspace{.2cm} - \\hspace{.2cm}\n\\sum_{i = 1}^{n+1}\\, \\, \\begin{tikzpicture}[baseline={(current bounding\n            box.center)}]\\def.3}\\def\\relw{.6{.3}\\def.6{.6}\n          \\newBoxDiagram [box height = .52cm, output height = .52cm,\n          width = 1.2cm, coord = {($(.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm + .3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm + 1.2cm,\n            0)$)}, decoration yshift label = .2cm, decoration yshift\n          label = .05cm, arm height = 3*.52cm, label = {g^{i}}]{g};\n          \\printNoArms[\/modulemap]{g}; \\printArm{g}{-1};\n      \n         \\g.coord at top(gcoor,\n          .3}\\def\\relw{.6); \n\\newBoxDiagram[width = 1.2cm*.6, box height =\n          .52cm, arm height = .52cm, output height = .52cm, coord =\n          {(gcoor)}, output height = .52cm, decoration yshift\n          label = .05cm, label = {\\o \\lambda^{n-i+2}_{M}}]{m};\n          \\print[\/module]{m}; \\pgfmathsetmacro{\\var}{1-.35}\n          \\printArm{m}{\\var} \n\\printDec[\/decorate, decorate right = \\var]{m}{n-i+1}\n\\printArmSep[arm sep right = .2]{m};\n          \\pgfmathsetmacro{\\var}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm}\n          \\printArm{g}{\\var};\n          \\printDec[\/decorate, decorate right = \\var]{g}{i-1}\n\n\\pgfmathsetmacro{\\var}{-1+2*.08}\\def.35{.35}\n          \\pgfmathsetmacro{\\vara}{.3}\\def\\relw{.6 - .6 - .1} \\def.1cm} \\def\\leftw{.35cm} \\def\\rightw{.25cm{.1cm} \\def.3cm} \\def\\rightw{.3cm} \\def\\hgt{.5cm{.3cm -\n            2*.08}\\def.35{.35} \\printArmSep[arm sep left = \\var, arm sep\n          right = \\vara]{g}\n        \\end{tikzpicture} = 0.\n\\end{displaymath}\n\\end{enumerate}\n\\end{lem}\n\n\\begin{proof}\nBy Lemma\n\\ref{lem:strictly-unital-morphisms}, $\\o p + g_{\\operatorname{su}}$ is an $\\text{A}_{\\infty}$-morphism if and\nonly if $\\o p \\circ \\o \\mu - \\mu_{\\operatorname{End} {}} * \\o p  + g_{\\operatorname{su}} \\circ\nh_{A}= 0$. One checks that the adjoints of the three terms of this\nequation agree with the three families of displayed diagrams, and this\nproves part 1.\n\nFor part 2, the element $f$ is a morphism of representation if and only if\n$p_{N} \\star f + f^{\\geq 1} \\circ \\mu + f \\star p_{M} = 0.$ Using the\ndecompositions $\\mu = \\o \\mu + h + \\mu_{\\operatorname{su}}$, $p_{M} = \\o p_{M} + g_{\\operatorname{su}},$ and $p_{N} = \\o p_{N} +\ng_{\\operatorname{su}},$ together with the equality $g_{\\operatorname{su}} \\star f - f^{\\geq 1}\n\\circ \\mu_{\\operatorname{su}} + f \\star g_{\\operatorname{su}} = 0$, we see $f$ is a morphism if and\nonly if $\\o p_{N} \\star f + f^{\\geq 1}\n\\circ \\o \\mu + f \\star \\o p_{M} = 0.$\nEach of the three terms of this equation is the adjoint of $(-1)$ times the\ncorresponding term in the displayed equation.\n\\end{proof}\n\n\\begin{ex}\\label{ex:strictly-unital-modules-over-Koszul-complex}\n Let $(A, 1, \\mu)$ be the Koszul complex on $f \\in k,$ see\n \\ref{ex:curved-bar-constr-of-koszul-cx}, and $M$ a graded\n module. Let $\\o p_{M} \\in \\hhcu {\\o A} {\\operatorname{End} M}_{0}$ be an arbitrary element\n with adjoint family $(\\o \\lambda^{n})$ and set $\\sigma^{n}: M \\xra{\\cong}\n k[e]^{\\otimes n} \\otimes M \\xra{\\overbar \\lambda^{n}} M,$ a degree $2n -1$\n endomorphism of $M$. Since $\\o \\mu = 0,$ we see that $\\o p_{M}$ is a\n representation if and only if $\\sigma^{1} \\sigma^0 + \\sigma^0\n \\sigma^{1} = -f \\cdot 1_{M}$ and $\\sum_{i = 0}^n \\sigma^{n-i} \\sigma^i\n = 0$ for $n \\geq 2.$ Such a system of maps was first considered by\n Shamash \\cite{MR0241411}, who assumed that $M$ was the $k$-free\n resolution of a $k\/(f)$-module, and has since been important in\n the construction of free resolutions in commutative algebra; see e.g., \\cite[\\S 3.1]{MR1648664}\n and the references contained there.\n\\end{ex}\n\n\\subsection{Comodules}\nIf $(A, \\mu)$ is a nonunital $\\text{A}_{\\infty}$-algebra, set $\\Baru A$ to be the\ncoaugmented bar construction $(\\Tcou\n{\\Pi A}, d)$ (here $d|_{\\Tco {\\Pi A}} = \\Phi^{-1}(\\mu)$ and $d(1) = 0$).\n\n\\begin{defn}\n  Let $C$ be a graded coalgebra. The \\emph{cofree $C$-comodule on a\n    graded module $M$} has underlying graded module $C \\otimes M$ and\n  comultiplication $\\Delta_C \\otimes 1$. If $d$ is a graded\n  coderivation of $C$ and $P$ is a graded\n  $C$-comodule, a \\emph{coderivation} of $P$ (with respect to $d$) is\n  a homogeneous map $d_P: P \\to P,$ with $|d_P| = |d|,$ that satisfies\n  $(d \\otimes 1 + 1 \\otimes d_{P})\\Delta_{P} = \\Delta_P d_{P}.$ We\n  denote by $\\Coder^{d}(P, P)$ the set of coderivations of $P$. If\n  $(C, d)$ is a dg-coalgebra, a dg-comodule is a pair $(P, d_{P})$ with\n  $P$ a graded comodule and $d_{P}$ an element of $\\Coder^{d}(P, P)$ such that\n  $d_{P}^{2} = 0.$ A morphism of dg-comodules is a morphism of\n  comodules that commutes with the given coderivations. A\n  dg-comodule is \n  \\emph{cofree} if the underlying comodule is cofree.\n\\end{defn}\n\nCofree comodules satisfy the linear analogue of\n  \\ref{lem:isom_hhc_coder}.\n\\begin{lemma}\n  \\label{rem:univ-props-cofree-comods}\n Let $(C, \\epsilon)$ be a graded coalgebra with counit\n  $\\epsilon: C \\to k$, and $M$ a graded\n  module. The following hold.\n  \\begin{enumerate}\n  \\item  For any degree $n$ coderivation $d$ of $C$, the following is an isomorphism,\n    \\[ \\phi: \\operatorname{Coder}^{d}(C \\otimes M, C \\otimes M)\n      \\xra{\\cong} \\Homd {} {n} {C \\otimes M} {M}\\]\n    \\begin{displaymath}\n      d_{C \\otimes M} \\mapsto (\\epsilon \\otimes 1) d_M,\n    \\end{displaymath}\n    with\n    $\\phi^{-1}(m) = d \\otimes 1 + (1 \\otimes m)(\\Delta_C \\otimes 1).$ A\n    coderivation $\\phi^{-1}(m)$ is a differential, i.e., squares to\n    zero, if and only if $m \\phi^{-1}(m) = 0.$\n  \\item For any graded $C$-comodule $P$, the following is an isomorphism,\n    \\[ \\psi:\\Hom C P {C \\otimes M} \\xra{\\cong} \\Hom {} P M,\\]\n    \\begin{displaymath}\n      \\beta \\mapsto (\\epsilon \\otimes 1) \\beta,\n    \\end{displaymath}\n where\n    $\\Hom C {-} {-}$ denotes morphisms of graded $C$-comodules. The\n    inverse is given by $\\psi^{-1}(\\alpha) =  (1 \\otimes \\alpha) \\Delta_P$. A morphism\n $\\psi^{-1}(g)$ commutes with coderivations\n    $d_{P}$ of $P$ and $\\phi^{-1}(m)$ of $C \\otimes\n    M$ if and only if $g d_{P} = m_{N} \\psi^{-1}(g).$\n  \\end{enumerate}\n\\end{lemma}\n\nNote the above properties emphasize the need to adjoin a counit to $\\Bar\nA$.\n\n\\begin{prop}\n  Let $(A, \\mu)$ be a nonunital $\\text{A}_{\\infty}$-algebra with counital bar construction\n  $\\Baru A$, and $M, N$ graded modules.\n\n  \\begin{enumerate}\n  \\item An element $p_{M} \\in \\hhcu A {\\operatorname{End} M}_{0}$ is a\n    representation of $(A, \\mu)$ if and only if, for $\\lambda_{M}$ the\n    adjoint family, the pair $(\\Baru A \\otimes M,\n    \\phi^{-1}(\\lambda_{M}))$ is a dg-$\\Baru A$ comodule.\n\n    \\item An element $f \\in \\hhcu A {\\Hom {} M N}_{1}$ is a\n      morphism of representations $(M, p_{M}) \\to\n      (N,p_{N})$ if and only if $\\psi^{-1}(g): (\\Baru\n      A \\otimes M,\\phi^{-1}(\\lambda_{M})) \\to (\\Baru\n      A \\otimes N,\\phi^{-1}(\\lambda_{N}))$ is a morphism of dg-$\\Baru\n      A$ comodules, where $g, \\lambda_{M}, \\lambda_{N}$ are the adjoint families\n      of $f, p_{M}, p_{N}.$\n  \\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nThe pair $(\\Baru A \\otimes M,\n    \\phi^{-1}(\\lambda_{M}))$ is a dg $\\Baru A$-comodule if and only\n    if $(\\phi^{-1}(\\lambda_{M}))^{2} = 0.$ By\n    \\ref{rem:univ-props-cofree-comods}.(1), this is equivalent to the equation\n    $\\lambda_{M} \\phi^{-1}(\\lambda_{M}) = 0,$ and by the definition of\n    $\\phi^{-1},$ we see this is equivalent\n    to the equations of \\ref{lem:represen-iff-diagrams}.\n\nAnalogously, $\\psi^{-1}(g)$ is a morphism of dg-comodules if and only\nif it commutes with the coderivations $\\phi^{-1}(\\lambda_{M})$ and\n$\\phi^{-1}(\\lambda_{N})$. By \\ref{rem:univ-props-cofree-comods}.(2) this\nis equivalent to $g \\phi^{-1}(\\lambda_{M}) = \\lambda_{N} \\psi^{-1}(g)$, and from\nthe definitions of $\\phi^{-1}$ and $\\psi^{-1},$ this in\n equivalent to the equations of \\ref{lem:morphism-represen-iff-diagrams}.\n\\end{proof}\n\n\\begin{cor}\nLet $(A, \\mu)$ be a nonunital $\\text{A}_{\\infty}$-algebra. There is a functor from the\ncategory of\nrepresentations of $A$ to the category of dg $\\Baru A$ comodules, that sends $(M,\np_{M})$ to $(\\Baru A \\otimes M, \\phi^{-1}(\\lambda_{M}))$. This is fully faithful with\nimage the full subcategory of cofree dg comodules.\n\\end{cor}\n\nWe now assume that $A$ has a split unit, and construct the analogue of\nthe above for strictly unital representations of $A$. \n\n\\begin{defn}\nA \\emph{curved dg-comodule} over a curved dg-coalgebra $(C, d, \\xi)$\nis a pair $(P, d_{P}),$ with $P$ a graded $C$ comodule and $d_{P} \\in \\Coder^{d}(P, P)_{-1},$ that satisfies\n\\[d_P^{2} = \\left ( P \\xra{\\Delta} C \\otimes P \\xra{\\xi \\otimes 1} k\n    \\otimes P \\cong P \\right ) =: L_{\\xi}.\\]\nA \\emph{morphism of curved dg-comodules $(P, d_{P}) \\to (N, d_{N})$}\nis a degree zero morphism of graded $C$-comodules $f: P \\to N$ that satisfies $f d_{P} = d_{N} f.$\n\\end{defn}\n\nIf $(A, 1, \\o \\mu + h + \\mu_{\\operatorname{su}})$ is an $\\text{A}_{\\infty}$-algebra with split unit, we\ndenote by $\\Baru {\\o A}$ the counital curved bar construction $(\\Tcou {\\Pi \\o A}, \\Phi^{-1}(\\o \\mu),\n-s^{-1}h),$ where $\\Phi^{-1}(\\o \\mu)$ and $-s^{-1}h$ are extended by zero from $\\Tco {\\Pi \\o A}$\nto $\\Tcou {\\Pi  \\o A}.$\n\n\\begin{thm}\nLet $(A, 1, \\o \\mu + h + \\mu_{\\operatorname{su}})$ be an $\\text{A}_{\\infty}$-algebra with split unit and\ncounital curved bar construction $\\Baru {\\o A}.$\nLet $M, N$ be graded modules.\n\\begin{enumerate}\n\\item A strictly unital element $p = \\o p + g_{\\operatorname{su}}$, with $\\o p \\in\n  \\hhcu {\\o A} {\\operatorname{End} M}_{0}$, is a representation if and only if, for\n  $\\o \\lambda$ the adjoint family of $\\o p$, the pair\n  $(\\Baru {\\o A} \\otimes M, \\phi^{-1}(\\o \\lambda))$ is a curved dg-$\\Baru\n  {\\o A}$ comodule.\n \n    \\item  An element $f \\in \\hhcu {\\o A} {\\Hom {} M N}_{1}$ is a\n      morphism of strictly unital representations $(M, \\o p_{M}) \\to\n      (N,\\o p_{N})$ if and only if\n      \\begin{displaymath}\n\\psi^{-1}(g): (\\Baru\n      {\\o A} \\otimes M,\\phi^{-1}(\\o \\lambda_{M})) \\to (\\Baru\n      {\\o A} \\otimes N,\\phi^{-1}(\\o \\lambda_{N}))\n    \\end{displaymath}\n    is a morphism of curved dg-$\\Baru\n      {\\o A}$ comodules, where $g, \\o \\lambda_{M},\\o \\lambda_{N}$ are the adjoint families of\n      $f, \\o p_{M}, \\o p_{N}.$\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nThe pair $(\\Baru {\\o A} \\otimes M, \\phi^{-1}(\\o \\lambda))$ is a curved dg $\\Baru\n  {\\o A}$-comodule if and only if $\\phi^{-1}(\\o \\lambda)^{2} = L_{-s^{-1}\n    h}.$ Since $\\phi^{-1}(\\o \\lambda)$ is a coderivation with respect to $\\o\n  d = \\Phi^{-1}(\\o \\mu)$, $\\phi^{-1}(\\o \\lambda)^{2}$ is a coderivation with\n  respect to $\\o d^{2} = \\operatorname{ad} {-s^{-1}h},$ and one checks $L_{-s^{-1}h}$\n  is also a coderivation with respect to $\\operatorname{ad} {-s^{-1}h}$. Thus by\n  \\ref{rem:univ-props-cofree-comods}.(1), $\\phi^{-1}(\\o \\lambda)^{2} + L_{s^{-1}\n    h} = 0$ exactly when $\\o \\lambda\\phi^{-1}(\\o \\lambda) + (\\epsilon \\otimes\n  1)L_{s^{-1}h} = 0.$ The adjoint of $\\o \\lambda\\phi^{-1}(\\o \\lambda) = \\o \\lambda(\\o d\n  \\otimes 1 + (1 \\otimes \\o \\lambda)(\\Delta \\otimes 1))$ is equal to the\n  first two terms of the equation of\n  \\ref{lem:diagrams-for-s-ual-reps}.(1), while the adjoint of $(\\epsilon \\otimes\n  1)L_{s^{-1}h} = (\\Baru {\\o A} \\otimes M \\xra{s^{-1}h \\otimes 1} k\n  \\otimes M \\cong M)$ is the third term  of\n  \\ref{lem:diagrams-for-s-ual-reps}.(1). Now by  \\ref{rem:univ-props-cofree-comods}.(2), $\\psi^{-1}(g)$ is a\n  morphism of curved dg comodules if and only if $\\o \\lambda_{N}\n  \\psi^{-1}(g) - g\\phi^{-1}(\\o \\lambda_{M}) = 0$. The first term is\n  the adjoint of the first term of\n  \\ref{lem:diagrams-for-s-ual-reps}.(2), and the second term is the\n  adjoint of the second and third terms of \\ref{lem:diagrams-for-s-ual-reps}.(2).\n\\end{proof}\n\n\\begin{ex}\n  Let $(A, 1, \\mu)$ be the Koszul complex on $f \\in k$ with curved bar\n  construction $\\Bar \\o A = (k[T], f T^{*})$, see\n  \\ref{ex:curved-bar-constr-of-koszul-cx}, and let $(M, \\o \\lambda)$ be a\n  strictly unital representation, described in\n  \\ref{ex:strictly-unital-modules-over-Koszul-complex}. Set $d_{M} =\n  \\phi^{-1}(\\o \\lambda): k[T] \\otimes M \\to k[T] \\otimes M.$ For $x \\in M,$\n  $d_{M}(T^j \\otimes x) = \\sum_{k = 0}^{j} T^k \\otimes\n  \\sigma^{j-k}(x),$ where $\\sigma^{j-k}$ is the composition $M \\xra{\\cong}\n  k[e]^{\\otimes j - k} \\otimes M \\xra{\\overbar \\lambda^{j-k}} M.$ It follows\n  from \\ref{ex:strictly-unital-modules-over-Koszul-complex} that\n  $d_M^{2}(T^j\\otimes x) = -fT^{j-1} \\otimes x$.\n  \nDualizing gives a graded module over the polynomial ring $k[T^{*}]$\nand a degree -1 map on $k[T^{*}] \\otimes M^{*}$ whose square is\nmultiplication by $-fT^{*}.$ Sheafifying this, we get two $k$-modules\nand maps,\n$M^{\\operatorname{ev}} \\to M^{\\operatorname{odd}} \\to \\Pi^{-1} M^{\\operatorname{ev}},$ whose composition is\nmultiplication by $f \\in k.$ This is exactly a matrix factorization in\nthe sense of Eisenbud \\cite{Ei80}.\n\\end{ex}\n\n\\begin{cor}\nLet $(A, 1, \\o \\mu + h + \\mu_{\\operatorname{su}})$ be an $\\text{A}_{\\infty}$-algebra with split unit. There is a functor from the\ncategory of strictly unital\nrepresentations of $A$ to the category of curved dg $\\Baru {\\o A}$ comodules, that sends $(M,\n\\o p_{M})$ to $(\\Baru {\\o A} \\otimes M, \\phi^{-1}(\\o \\lambda_{M}))$. This is fully faithful with\nimage the full subcategory of cofree curved dg comodules.\n\\end{cor}\n\n\n\n\n\n\n\\def$'${$'$}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}