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\n\n\n\n\\def\\begin{equation}{\\begin{equation}}\n\\def\\end{equation}{\\end{equation}}\n\n\n\\def\\begin{displaymath}{\\begin{displaymath}}\n\\def\\end{displaymath}{\\end{displaymath}}\n\n\n\\def\\begin{eqnarray}{\\begin{eqnarray}}\n\\def\\end{eqnarray}{\\end{eqnarray}}\n\\def\\nonumber{\\nonumber}\n\n\n\\begin{document}\n\\DeclareGraphicsExtensions{.jpg,.pdf,.mps,.eps,.png}\n\\begin{flushright}\n\\baselineskip=12pt\nEFI-09-30 \\\\\nANL-HEP-PR-09-102\n\\end{flushright}\n\n\\begin{center}\n\\vglue 1.5cm\n\n\n\n\n{\\Large\\bf Prospects for Higgs Searches at the Tevatron and LHC in the MSSM with Explicit $CP$-violation } \\vglue 2.0cm {\\Large Patrick Draper$^{a}$, Tao Liu$^{a}$, and Carlos E.M. Wagner$^{a,b,c}$}\n\\vglue 1cm {\n$^a$ Enrico Fermi Institute and\n$^b$ Kavli Institute for Cosmological Physics, \\\\\nUniversity of Chicago, 5640 S. Ellis Ave., Chicago, IL\n60637\\\\\\vglue 0.2cm\n$^c$ HEP Division, Argonne National Laboratory,\n9700 Cass Ave., Argonne, IL 60439\n}\n\\end{center}\n\n\n\n\n\\vglue 1.0cm\n\\begin{abstract}\nWe analyze the Tevatron and Large Hadron Collider (LHC) reach for the Higgs sector\nof the Minimal Supersymmetric Standard Model (MSSM) in the\npresence of explicit $CP$-violation. Using the most recent studies\nfrom the Tevatron and LHC collaborations, we examine the CPX\nbenchmark scenario for a range of $CP$-violating phases in the soft\ntrilinear and gluino mass terms and compute the exclusion\/discovery\npotentials for each collider on the $(M_{H^+}, \\tan\\beta)$ plane.\nProjected results from Standard Model (SM)-like, non-standard, and\ncharged Higgs searches are combined to maximize the statistical\nsignificance. We exhibit complementarity between the SM-like Higgs searches at the LHC with low luminosity and the Tevatron, and estimate the combined reach of the two colliders in the early phase of LHC running.\n\\end{abstract}\n\n\\newpage\n\n\\section{Introduction}\n\nThe origin of electroweak symmetry breaking remains a principal open question in high-energy physics. In the Standard Model (SM), the breakdown of the electroweak symmetry is induced by the vacuum\nexpectation value of a scalar field, which transforms non-trivially\nunder the $SU(2)_L\\times U(1)_Y$ symmetry. A consequence of this\nmechanism of spontaneous symmetry breaking is the presence of a physical\nscalar Higgs particle, with well-defined couplings to fermions and\ngauge bosons. The search for such a particle at lepton and hadron\ncolliders is therefore a paramount goal in particle\nphysics. The LEP experiments have already excluded at 95\\% C.L. the presence of a SM-like\nHiggs with mass below 114.4~GeV~\\cite{Barate:2003sz}.\nIn the coming years, Higgs searches will be performed at\nhadron colliders.\n\nThe Tevatron collider at Fermilab has an active Higgs search program and\nhas already excluded a SM-like Higgs at 95\\% C.L. in the mass range 160--170~GeV~\\cite{:2009pt}.\nThe Tevatron is expected to operate until the end of 2011. It is likely that by this time\nthe CDF and D0 experiments will collect 10--12~$\\mbox{ fb}^{-1}$\napiece and achieve\nsome improvements in the analysis, yielding a significant chance that they\nwill be able to probe the entire SM Higgs mass range\n$110-190\\mbox{ GeV}$. A similar conclusion is reached for the parameter\nspace of the Higgs sector in the $CP$-conserving MSSM, provided that\nthe limits derived from SM-like Higgs searches are statistically combined\nwith limits from direct searches for the non-standard Higgs\nbosons~\\cite{Draper:2009fh}.\n\nThe Large Hadron Collider (LHC) at CERN will begin collisions at\nthe end of 2009\nwith an expected center-of-mass energy of several TeV, and the ATLAS and\nCMS experiments will collect on the order\nof a few hundred pb$^{-1}$ of data during 2010.\nA higher center of mass energy of $14\\mbox{ TeV}$ is expected to be\nachieved after this run, once the necessary upgrades are completed.\nThe anticipated rate of data acquisition at $14 \\mbox{ TeV}$ in the early years of the LHC is expected to be\na few to 10~fb$^{-1}$\/year. Once the LHC acquires a few fb$^{-1}$ of data\nat $14\\mbox{ TeV}$, the Tevatron\nand LHC reaches for a light Higgs boson may be comparable and even\ncomplementary in some searches.\nEventually, once\nthe LHC experiments collect 10 to 30~fb$^{-1}$ of $14\\mbox{ TeV}$ data,\nthe LHC will probe the SM and MSSM Higgs sectors at high statistical\nsignificance, far superior to what is attainable at the Tevatron~\\cite{Aad:2009wy,Ball:2007zza}.\n\nIn this study, we analyze the Tevatron and LHC reach for the MSSM Higgs\nsector~\\cite{ERZ}--\\cite{mhiggsFD2} in the presence of explicit $CP$-violation~\\cite{Pilaftsis:1998dd}--\\cite{Ibrahim:2002zk}.\nAt the Tevatron we consider 10 fb$^{-1}$ in all channels\nand provide projections for a set of\npossible improvements in signal efficiencies. For the LHC, to study\nthe possible initial complementarity with the Tevatron results, we\nconsider the case of 3 fb$^{-1}$ at $14\\mbox{ TeV}$. To display the\nlong-term LHC\ncapabilities, we also show the results for\n30 fb$^{-1}$ at $14\\mbox{ TeV}$ in the most challenging scenario.\nLet us mention that\nthere are alternative\npossibilities for the LHC timeline where the collision energy is kept below $14\\mbox{ TeV}$ for several years, and the upgrade to $14\\mbox{ TeV}$ is\nonly completed later. We do not attempt an analysis of these scenarios\nbecause we base our study on the\nHiggs reach projections presented by the LHC collaborations, which\nare only fully complete for a center of mass energy of $14\\mbox{ TeV}$.\nFor reference, preliminary results indicate that to obtain the same Higgs reach, the\nluminosity at $10\\mbox{ TeV}$ should be twice as large as that\nwhich is required at 14~TeV~\\cite{Delmastro:2009wc}.\n\nOur work differs in three significant ways from previous analyses of\nHiggs searches in the MSSM with\nexplicit $CP$-violation\nperformed in\nRefs.~\\cite{Carena:2002bb,Accomando:2006ga}.\nFirst, for the Tevatron we use the 2009 limits from the CDF\nand D0 experiments given in\nRefs.~\\cite{CDFSM}--\\cite{cdfcharged}. For the LHC\nwe incorporate the projections presented by the experimental collaborations\nin the most recent technical design and expected physics performance\ndocuments~\\cite{Aad:2009wy,Ball:2007zza}. These projections show\nmarked differences from the earlier TDRs, and as a result the priority\nfor some channels has been reduced, while others have been elevated.\nSecondly, for both colliders we analyze the potential for the non-standard MSSM Higgs searches, and provide the combination with the SM-like Higgs reach. These two types of searches offer\nconsiderable complementarity and together can be used to cover most of the\nanalyzed parameter space. For the Tevatron, we also include the reach for the charged Higgs. Finally, we present the combination of the Tevatron discovery reach with the LHC reach at 3~fb$^{-1}$. At this low LHC integrated luminosity, the statistical significances offered by the two colliders may be comparable, and so it may be of interest to perform the combination. We present this analysis only for the SM-like Higgs search channels, which offer greater complementarity than the non-standard channels.\n\nFor the LHC reach we compute and combine discovery significances using Poisson statistics and the profile likelihood ratio, evaluated on data fixed to the expected values for the signals and backgrounds. For the Tevatron we work with 95\\% C.L. upper bounds on the signal presented by CDF and D0, combining them in inverse quadrature. This combination method is strictly valid only in the Gaussian limit; however, it was tested in Ref.~\\cite{Draper:2009fh} and found to match well with the combination derived from a full analysis performed by the collaborations. A further discussion of statistical methods is presented in Appendix A.\n\nIn the MSSM Higgs sector, $CP$-violation can occur via the incorporation of explicit phases in the supersymmetry breaking parameters.\n$CP$-violating phases can be removed from the tree-level Higgs\npotential by field redefinitions. However, phases in the soft trilinear couplings and the gaugino mass\nterms influence the effective Higgs Lagrangian through loop corrections~\\cite{Pilaftsis:1998pe,Pilaftsis:1999qt}.\nWe shall work in the CPX benchmark scenario, defined by the following\nparameter values at the soft scale~\\cite{Carena:2000yi}:\n\\begin{eqnarray*}\nM_S=500\\mbox{ GeV, }& &|A_t|=1\\mbox{ TeV,}\\nonumber\\\\\n\\mu=2\\mbox{ TeV, }& &M_{1,2}=200\\mbox{ GeV,}\\nonumber\\\\\nA_{b,\\tau}=A_t\\mbox{, }& &|M_{\\tilde{g}}|=1\\mbox{ TeV}.\n\\end{eqnarray*}\nIn the above, $A_f$ are the trilinear Higgs sfermion couplings,\n$M_S$ is the characteristic scale of soft supersymmetry breaking\nscalar masses, and $M_{\\tilde{g}}$ is the gluino mass.\nWe set the top quark mass to $m_t=173.1\\mbox{ GeV}$. We shall perform our\nanalysis scanning over the charged Higgs\nmass $M_{H^+}$ and $\\tan\\beta$ over the ranges\n$(100\\mbox{ GeV},400\\mbox{ GeV})$ and $(2,60)$, respectively, for a\nvariety of complex phases of $A_{t,b,\\tau}$ and $M_{\\tilde{g}}$\\footnote{If we began with phases for these parameters at a higher scale, phases for first and second generation trilinear parameters would be generated by RG running, which are highly constrained by EDM measurements~\\cite{Garisto:1996dj}. This can be avoided either by fixing these phases to zero at the weak scale, or by increasing the soft masses of the first and second generation sfermions. However, note that the first and second generation parameters do not have a significant influence on the results presented in this work.}. Masses, mixings, and branching ratios are computed with CPsuperH~\\cite{Lee:2007gn} and HDECAY~\\cite{Djouadi:1997yw}; SM Higgs cross sections are taken from Ref.~\\cite{maltoni} and rescaled to obtain cross sections in the MSSM. The dominant effects of the phases are twofold. First, they cause the neutral Higgs\nmass eigenstates to become admixtures of $CP$-even and $CP$-odd components, modifying\nthe couplings to gauge bosons relative to those in the case with no\n$CP$-violation. For example, the lightest neutral Higgs can now have a\nsignificant $CP$-odd component, strongly suppressing its couplings to the\n$W$ and $Z$ bosons. Secondly, the Yukawa couplings are altered, leading\nin particular to modifications of the neutral Higgs decay branching ratios to $b\\bar{b}$ and\n$\\tau^+\\tau^-$, and different production cross sections through the\nbottom quark fusion and gluon fusion mechanisms. We examine separately\nthe reach in those channels designed to search for a Higgs with SM-like\ngauge couplings (hereafter referred to as an ``SM-like Higgs\"), and in those\nchannels which probe either neutral scalars with negligible gauge couplings\n(hereafter, ``non-standard Higgs\") or charged Higgs states, in order to understand the\ncomplementarity of their coverages. Afterwards we combine the statistical significances of all channels to obtain an overall reach in the MSSM Higgs parameter space.\n\nOur presentation is organized as follows. In section 2 we review the\ncouplings of the MSSM effective Lagrangian that are of particular\nrelevance for understanding the CPX reach. In section 3 we present\nand analyze the results for the Tevatron and make conservative estimates\nof the improvements in signal efficiency necessary to cover large regions\nof parameter space. Section 4 contains the projections for the LHC, and we\noffer conclusions in section 5. Appendix A offers a brief review of statistical methods, followed by a discussion of the\napproximations used in the text to compute and combine exclusion limits\nand discovery significances for multiple channels at the Tevatron and LHC. Appendix B extends some of the discussion in the text to the case of $CP$-conserving benchmark scenarios.\n\n\\section{Effective Yukawa Couplings}\n\nThe radiative corrections to the Yukawa couplings of Higgs states to\ndown-type fermions~\\cite{Carena:1998gk}--\\cite{Carena:2001bg} play a significant role in inducing $CP$-violating\neffects in the Higgs sector and can strongly affect the SM-like Higgs search\nchannels at colliders.\nThe scalar and pseudoscalar neutral Higgs couplings to bottom quarks in the\neffective Lagrangian are given by~\\cite{Carena:2002bb}\n\\begin{equation}\n\\mathcal{L}=-g_fH_i\\bar{b}(g^S_{H_ib\\bar{b}}+\\imath\\gamma_5g^P_{H_ib\\bar{b}})b\n\\end{equation}\nwhere $g_f$ is the SM scalar coupling given by the bottom quark mass over the vacuum expectation value of the Higgs, and $g^{S,P}$ are given by\n\\begin{eqnarray}\ng^S_{H_ib\\bar{b}}&=&\\mbox{Re}\\left(\\frac{1}{1+\\kappa_b\\tan\\beta}\\right)\n\\frac{\\mathcal{O}_{1i}}{\\cos\\beta}+\\mbox{Re}\n\\left(\\frac{\\kappa_b}{1+\\kappa_b\\tan\\beta}\\right)\n\\frac{\\mathcal{O}_{2i}}{\\cos\\beta} \\nonumber\\\\\n&+&\\mbox{Im}\\left(\\frac{\\kappa_b(\\tan^2\\beta+1)}{1+\\kappa_b\\tan\\beta}\\right)\n\\mathcal{O}_{3i}\\nonumber\\\\\ng^P_{H_ib\\bar{b}}&=&-\\mbox{Re}\\left(\\frac{\\tan\\beta-\\kappa_b}\n{1+\\kappa_b\\tan\\beta}\\right)\\mathcal{O}_{3i}\n+\\mbox{Im}\\left(\\frac{\\kappa_b\\tan\\beta}{1+\\kappa_b\\tan\\beta}\\right)\n\\frac{\\mathcal{O}_{1i}}{\\cos\\beta} \\nonumber\\\\\n&-&\\mbox{Im}\\left(\\frac{\\kappa_b}{1+\\kappa_b\\tan\\beta}\\right)\n\\frac{\\mathcal{O}_{2i}}{\\cos\\beta}.\n\\label{effcoup}\n\\end{eqnarray}\nHere $\\mathcal{O}_{jk}$ is the neutral Higgs mixing matrix, where $j$\nis associated with the gauge eigenstate $\\{H^0_u,H^0_d,A\\}$ and\n$k$ runs over the\nmass states $\\{H_1,H_2,H_3\\}$ which are ordered so that $(M_{H_3}\\geq M_{H_2}\\geq M_{H_1})$, and $\\kappa_b$ parameterizes the\nradiative contributions from sbottom-gluino and stop-chargino loops,\n\\begin{eqnarray}\n\\kappa_b&=&\\frac{(\\Delta h_b\/h_b)}{1+(\\delta h_b\/h_b)}\\nonumber\\\\\n\\Delta h_b\/h_b&=&\\frac{2\\alpha_s}{3\\pi}\nM_{\\tilde{g}}^*\\mu^*I(m^2_{\\tilde{d}_1},m^2_{\\tilde{d}_2},|M_{\\tilde{g}}|^2)\n+\\frac{|h_u|^2}{16\\pi^2}A_u^*\\mu^*\nI(m^2_{\\tilde{u}_1},m^2_{\\tilde{u}_2},|M_{\\tilde{g}}|^2)\\nonumber\\\\\n\\delta h_b\/h_b&=&-\\frac{2\\alpha_s}{3\\pi}M_{\\tilde{g}}^*A_b\nI(m^2_{\\tilde{d}_1},m^2_{\\tilde{d}_2},|M_{\\tilde{g}}|^2)\n-\\frac{|h_u|^2}{16\\pi^2}|\\mu|^2\nI(m^2_{\\tilde{u}_1},m^2_{\\tilde{u}_2},|M_{\\tilde{g}}|^2)\n\\end{eqnarray}\nwhere $I(a,b,c)$ is a function that behaves as $1\/\\max(a^2,b^2,c^2)$~\\cite{deltamb1}--\\cite{deltamb2b}.\nThe size of the dominant loop corrections to the Higgs sector is controlled\nby $\\mu$, so a large value of $|\\mu|$ is taken in CPX to accentuate the\n$CP$-violating effects.\n\nFor illustration, let us consider the behavior of the effective couplings\nin the simplest case of vanishing phases. In this scenario $\\Delta h_b\/h_b$\nand $\\delta h_b\/h_b$ take the approximate numerical values $1\/20$ and $-1\/20$,\nrespectively, and $\\kappa_b\\approx\\Delta h_b\/h_b\\approx 1\/20$\\footnote{For\ncomparison, in the $CP$-conserving Maximal Mixing\nscenario $\\kappa_b\\simeq 1\/200$, and in the Minimal Mixing scenario\n$\\kappa_b\\simeq 1\/400$. In both of these scenarios, $|\\mu|= 200$~GeV.}. We denote the $CP$-even mass eigenstates by $h$ and $H$, where $M_h\\leq M_H$. We can always identify $h$ with $H_1$, but due to strong radiative corrections $H$ can either be $H_2$ or $H_3$ depending on $M_{H^+}$ and $\\tan\\beta$.\nThe mass states are related to the gauge eigenstates by a mixing angle $\\alpha$, which satisfies $(-\\sin\\alpha)=\\mathcal{O}_{1h}$ and $\\cos\\alpha=\\mathcal{O}_{1H}$. Furthermore, the pseudoscalar\neffective couplings for these states vanish, and the scalar couplings\nare rescaled relative to their tree level values. At tree level $g^S_{hb\\bar{b}}$ and $g^S_{Hb\\bar{b}}$ are given by $(-\\sin\\alpha\/\\cos\\beta)$ and $(\\cos\\alpha\/\\cos\\beta)$, respectively, and the rescaling factors are given by\n\\begin{eqnarray}\n\\frac{g^S_{hb\\bar{b}}}{-\\sin\\alpha\/\\cos\\beta}&=&\n\\frac{1-\\kappa_b\\cot\\alpha}{1+\\kappa_b\\tan\\beta}\\nonumber\\\\\n\\frac{g^S_{Hb\\bar{b}}}{\\cos\\alpha\/\\cos\\beta}&=&\n\\frac{1+\\kappa_b\\tan\\alpha}{1+\\kappa_b\\tan\\beta}.\n\\label{reltree}\n\\end{eqnarray}\nIn Fig.~\\ref{effhbb} we plot the squares\nof $g^S_{hb\\bar{b}}$ and $g^S_{Hb\\bar{b}}$.\nIn the large $M_{H^+}$ limit, $(-\\sin\\alpha)\\rightarrow\\cos\\beta$\nand $\\cos\\alpha\\rightarrow\\sin\\beta$, so the $h$ scalar coupling converges\nto the SM value. Similarly, in the small $M_{H^+}$ limit, the\n$H_3$ scalar coupling becomes that of the SM.\nFinally, in the non-standard Higgs limit (large $M_{H^+}$ for $H$, or\nsmall $M_{H^+}$ for $h$) and for large $\\tan\\beta$, the effective coupling\nof the non-standard Higgs approaches $\\tan\\beta\/(1+\\kappa_b\\tan\\beta)$, which\nshould be contrasted with the tree level behavior proportional to $\\tan\\beta$.\nAlthough the coupling is still enhanced relative to the SM value, it is\nsuppressed relative to the tree level MSSM value.\n\nIt is important to be precise with terminology: as defined in the\nIntroduction, an SM-like Higgs is one with significant couplings to vector\nbosons, not necessarily one with SM-like fermionic couplings. The\ndistinction is relevant, for example, at moderate values of $M_{H^+}$,\nwhere the first two terms of the $h$ scalar coupling are comparable because of the factor of $\\kappa_b$ in the\nsecond term. For these values of $M_{H^+}$, $h$ can be simultaneously\nSM-like and have an altered coupling to $b\\bar{b}$. Of course, this is true\neven for the tree level coupling, but the effective coupling is modified\nrelative to tree level by the factor in Eq.~\\ref{reltree} which can generate\na significant suppression when $(-\\cot\\alpha) < \\tan\\beta$.\n\nIt is also important to observe that the $\\tau^+\\tau^-H_i$ effective coupling must take a similar form to that of\n$b\\bar{b}H_i$, but with radiative terms proportional to $\\alpha_{1,2}$ instead of $\\alpha_s$. The effects of the threshold corrections are therefore much smaller for the $\\tau^+\\tau^-$ coupling and can be qualitatively neglected.\n\nWhen phases are introduced, $\\mathcal{O}_{3i}$ can become nonzero for all\nstates. Then the third term of $g^S_{H_ib\\bar{b}}$ in\nEq.~\\ref{effcoup} is nonzero and proportional to the imaginary part\nof $\\kappa_b$, which is dominated by the imaginary part of $\\Delta h_b\/h_b$.\nThis term is $\\tan\\beta$-enhanced, so significant phases for $A_t$\nand $M_{\\tilde{g}}$ can affect the conclusions drawn above for vanishing\nphases, potentially even countering the suppression effect in\n$g^S_{H_ib\\bar{b}}$.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.45\\textwidth]{.\/plots\/CPX_0000_mhptb_gSH1bb_nonorm.png} &\n\\includegraphics[width=0.45\\textwidth]{.\/plots\/CPX_0000_mhptb_gSH3bb_nonorm.png}\n\\end{tabular}\n\\caption{Effective bottom Yukawa couplings squared for the $CP$-even Higgs states $h$ and $H$ in the $CP$-conserving\nlimit of CPX.}\n\\label{effhbb}\n\\end{center}\n\\end{figure}\n\n\\section{Tevatron Results}\n\nNeutral Higgs states with SM-like couplings to gauge bosons are sought at\nthe Tevatron in associated production with a $W$ or $Z$ boson and\nin gluon fusion channels, with the Higgs decay to $b\\bar{b}$\nproviding the dominant decay channel in the former\ncase and Higgs decay to $W^+W^-$ providing the dominant channel in the\nlatter~\\cite{Abazov:2008eb}--\\cite{Aaltonen:2008ec}.\nWe compute projections for the expected upper limits on the signal from the combination of these channels at CDF and\nD0 with 10 fb$^{-1}$ per channel and 0\\%, 25\\%, and 50\\%\nimprovements in signal efficiencies. Neutral states with non-standard gauge\ncouplings are probed mainly in the inclusive $\\phi\\rightarrow\\tau^+\\tau^-$\nchannel~\\cite{:2008hu,Abulencia:2005kq}, but also exclusively in associated production with bottom quarks,\nwith the Higgs decaying into either bottom quark or $\\tau$ lepton pairs~\\cite{Abazov:2008zz,CDF3b}.\nFor these channels we consider 7 and 10 fb$^{-1}$, but without\nimprovement in efficiency. The extension of the reach by efficiency\nimprovements can be easily estimated: since the signal scales approximately\nwith $\\tan^2\\beta$, any eventual improvement in efficiency will produce a further\nextension of the reach in $\\tan\\beta$ by the square root of the efficiency\nimprovement.\nFurthermore, the 95\\% C.L. expected upper bound on the signal with\n25\\% improvements is essentially equivalent to the 90\\% limit with no\nimprovements\\footnote{This can be understood from the approximate formula for\nthe expected n$\\sigma$ upper limit on the signal,\n$s_{n\\sigma}\\approx n\\sqrt{b}\/(\\epsilon\\sqrt{L})$, where $L$ is the\nluminosity, $\\epsilon$ is the signal efficiency, and $b$ is the expected\nbackground.}. The charged Higgs is sought in decays of the top quark~\\cite{Collaboration:2009zh,Abulencia:2005jd}, $t\\rightarrow H^+b$, with $H^+\\rightarrow\\tau^+\\nu$ providing the dominant $H^+$ decay channel for $\\tan\\beta > 1$. We present the non-standard Higgs results in combination with those from the charged Higgs, since both particles can be classified together as strictly beyond-the-SM scalars. Finally, we combine the two classes of searches, SM-like and non-standard~+~charged, to derive\nthe strongest possible constraint on the MSSM Higgs parameter space. In all figures the shaded gray regions denote exclusion limits from\nLEP~\\cite{Schael:2006cr}, and solid black indicates theoretically\ndisallowed regions.\n\nNote that the expected 95\\% C.L. limits are obtained under the assumption that the data reflects only the average number of background events. If signal is also present and the data reflects the average value of signal+background for some point in the MSSM parameter space, the observed limit will be somewhat weaker, with an average value given by $R_{obs}\\approx R_{exp}+1$, where $R$ is the upper bound on the signal normalized to the expected signal in the MSSM\\footnote{Interestingly enough, the present combined Tevatron bounds on the SM Higgs given in Ref.~\\cite{Collaboration:2009je} show an observed bound that differs by about 1 from the expected bound in the low mass range. At $M_h=115\\mbox{ GeV}$, $R_{exp}=1.78$ and $R_{obs}=2.7$.}. For further discussion, see Appendix A.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_0000_mhptb_notau_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_9090_mhptb_notau_1.png} \\\\\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_0000_mhptb_onlytau_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_9090_mhptb_onlytau_1.png} \\\\\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_0000_mhptb_1.png} &\n\\includegraphics[width=0.40\\textwidth]{.\/plots\/CPX_9090_mhptb_1.png}\n\\end{tabular}\n\\caption{Projected Tevatron exclusion contours at 90\\% and 95\\% C.L. in the\nCPX scenario with $CP$-violating phases $\\arg(A_{t,b,\\tau})=0^{\\circ}$,\n$\\arg(M_{\\tilde{g}})=0^{\\circ}$ (\\textit{left}) and\n$\\arg(A_{t,b,\\tau})=90^{\\circ}$, $\\arg(M_{\\tilde{g}})=90^{\\circ}$\n(\\textit{right}). Row 1 gives results for SM-like Higgs searches,\nrow 2 includes only the non-standard Higgs searches, and row 3\ngives the combined constraints.}\n\\label{CPX0000}\n\\end{center}\n\\end{figure}\n\nIn this work we will consider three sets of phases for\n$A_{t,b,\\tau}$ and $M_{\\tilde{g}}$: $(0^{\\circ},0^{\\circ})$,\n$(90^{\\circ},90^{\\circ})$, and $(140^{\\circ},140^{\\circ})$. In the course\nof our study we examined other values of the $CP$-violating phases, including\nthe departure from setting common phases for $A_{t,b,\\tau}$ and\n$M_{\\tilde{g}}$. However, the results were qualitatively similar, and\nin particular, all unique features of interest also appeared in one or more\ncases discussed here.\n\n\nWe consider first the case without explicit $CP$-violation, in order\nto understand features which are independent of the phases. The results\nfrom SM-like searches, non-standard + charged Higgs search channels, and the combination are\ngiven in the first column of Fig.~\\ref{CPX0000}. The decoupling limit\nis probed at 95\\% C.L. with a 50\\% improvement in signal efficiency for the\nSM-like search channels. In this limit $M_{H_1}\\approx 121\\mbox{ GeV}$ with\nthe top mass set to $173.1\\mbox{ GeV}$. Smaller efficiency improvements more\nreadily probe the region of lower $\\tan\\beta$,\nwhere $M_{H_1}\\lesssim 121\\mbox{ GeV}$ and the SM Higgs constraint is stronger.\n\nHere a word about experimental mass resolution is in order. If the mass\ndifference between two Higgs states is under a certain finite threshold,\nthe detectors cannot resolve the particles. Consequently, their statistical\nsignificances should be added directly rather than in quadrature,\nleading to a stronger limit. We take a representative experimental mass\nresolution of $10\\mbox{ GeV}$. The main effect of this approximate treatment appears as a discontinuous\nspike in the range $M_{H^+}\\approx 135-155\\mbox{ GeV}$ of the non-standard\nHiggs search constraint. The reason for this behavior is that in this range of $M_{H^+}$ and for moderate\nto large $\\tan\\beta$, radiative corrections drive a $CP$-even Higgs mass to within $10\\mbox{ GeV}$ of the $CP$-odd Higgs mass. For slightly lower or higher values of $M_{H^+}$ the approximate degeneracy is lifted. In all subsequent figures of this work, we include the finite mass resolution effects.\n\n\nContrary to the $CP$-conserving cases analyzed in Ref.~\\cite{Draper:2009fh},\nthe SM-like Higgs search constraints become weaker with moderate,\ndecreasing $M_{H^+}$. At tree level, the increase in $\\mathcal{O}_{11}$\nsignificantly enhances the coupling of the SM-like Higgs to down-type fermions\nfor smaller $M_{H^+}$, typically leading to a large region just above the\nintense coupling regime where the constraint is stronger than in the\ndecoupling limit (see, for example, the Maximal Mixing scenario examined in\nRef.~\\cite{Draper:2009fh}). This feature does not appear in the CPX scenario\nin the absence of phases. The reason is that, as discussed before and\nshown in Fig.~\\ref{effhbb},\nthe $\\kappa_b$ threshold corrections in Eq.~\\ref{effcoup} are significant due\nto the large value of $\\mu$ taken in CPX. These corrections suppress the\n$b\\bar{b}H_1$ effective coupling relative to the tree level value, and therefore $Br(H_1\\rightarrow b\\bar{b})$ is\ndecreased while $Br(H_1\\rightarrow \\tau^+\\tau^-)$ is increased in this region.\nThis can be seen from the upper left plot of Fig.~\\ref{h1tth1bb}, where we\nshow the ratio of the $\\tau^+\\tau^-$ to $b\\bar{b}$ branching ratios, each\nnormalized to their SM values.\n\n\\begin{figure}[!htbp]\n\\begin{center}\n\\begin{tabular}{cc}\n\\includegraphics[width=0.40\\textwidth]\n{.\/plots\/CPX_0000_mhptb_BrH1ttH1bb.png} &\n\\includegraphics[width=0.40\\textwidth]\n{.\/plots\/CPX_9090_mhptb_BrH1ttH1bb.png} \\\\\n\\end{tabular}\n\\includegraphics[width=0.40\\textwidth]\n{.\/plots\/CPX_140140_mhptb_BrH1ttH1bb.png} \\\\\n\\caption{The ratio of $H_1$ branching ratios into $\\tau^+\\tau^-$\nand $b\\bar{b}$, normalized to their SM values, for vanishing\nphases (\\textit{upper left}), for $\\arg(A_{t,b,\\tau})=90^{\\circ}$,\n$\\arg(M_{\\tilde{g}})=90^{\\circ}$ (\\textit{upper right}), and for $\\arg(A_{t,b,\\tau})=140^{\\circ}$,\n$\\arg(M_{\\tilde{g}})=140^{\\circ}$ (\\textit{bottom}).}\n\\label{h1tth1bb}\n\\end{center}\n\\end{figure}\n\nThe constraint from the non-standard Higgs search in the $\\tau^+\\tau^-$\ninclusive channel is similar to what is obtained in other benchmark scenarios\ngeared towards the $CP$-conserving MSSM. At tree level the non-standard Higgs\nhas a $\\tan\\beta$ enhanced coupling to $\\tau^+\\tau^-$ that is not subject to\nlarge radiative corrections, and it is light enough to be produced for low to\nmoderate $M_{H^+}$. As a result the constraint is significant in this region.\nThe radiative increase in the $\\tau^+\\tau^-$ branching fraction due\nto the suppression of $b\\bar{b}$ is mostly compensated by the threshold\nsuppression of the non-standard Higgs coupling to bottom quarks, which enters\nin both of the dominant production mechanisms of $b\\bar{b}$ fusion and\ngluon fusion through a bottom loop~\\cite{Carena:2005ek}. $H_1$ becomes highly non-standard for moderate to large $\\tan\\beta$ and $M_{H^+}\\lesssim 150\\mbox{ GeV}$.\nTherefore the limit comes mostly from $H_1$ and the $CP$-odd Higgs for $M_{H^+}\\lesssim 150\\mbox{ GeV}$, and\nfrom the $CP$-odd and heavy $CP$-even Higgs for larger $M_{H^+}$. The non-standard constraint\nhas the virtue of mostly filling the dip in the LEP constraint\nat $M_{H^+}\\approx 140\\mbox{ GeV}$, which was due to a marginal excess\nin the LEP data around $M_{H_1}\\approx 90\\mbox{ GeV}$. The charged Higgs searches from top decays become also\nrelevant in this region of parameters, although in the absence of phases they do not independently reach 95\\% C.L. due to the large bottom coupling suppression in this scenario~\\cite{Carena:1999py}.\n\nWe stress the fact that neither the SM-like nor the non-standard~+~charged Higgs\nsearch is sufficient to reach the entire plane even with significant improvement.\nHowever, each search is most effective in the region where the other is\nweakest, providing excellent complementarity and strongly motivating a\nstatistical combination. In the bottom left plot of Fig.~\\ref{CPX0000},\nwe demonstrate that the combination of SM-like searches with 50\\%\nimprovements in signal efficiency and non-standard~+~charged search channels is\nsufficient to cover the entire region previously unprobed by LEP.\n\n\nNow we consider the effects of $CP$-violation, setting\n$\\arg(A_{t,b,\\tau})=\\arg(M_{\\tilde{g}})=90^{\\circ}$. The results are\npresented in the right-hand column of Fig.~\\ref{CPX0000}. In the\ndecoupling limit, $H_1$ is still SM-like; however,\nfor $M_{H^+}\\lesssim 160\\mbox{ GeV}$, $H_1$ becomes mostly $CP$-odd.\n$H_2$ is $CP$-odd in the decoupling limit, but transitions rapidly to become\nSM-like around $M_{H^+}\\approx 150\\mbox{ GeV}$, and\nfinally acquires non-standard couplings to gauge bosons for\n$M_{H^+}\\lesssim 135\\mbox{ GeV}$.\nThe fast transitions create a region of increased sensitivity compared to the $CP$-conserving case centered around\n$M_{H^+}\\approx 150\\mbox{ GeV}$ and stretching from low to moderate\n$\\tan\\beta$. Although $g_{ZZH_2}$ increases with $\\tan\\beta$, the $H_2\\rightarrow b\\bar{b}$ branching ratio suppression limits the height of the region. The region ends sharply at $M_{H^+}\\approx 130\\mbox{ GeV}$, where $H_3$ has become SM-like but the opening of the $H_3\\rightarrow H_1H_1$ channel heavily reduces the $H_3\\rightarrow b\\bar{b}$ branching ratio. As before the LEP constraint from $e^+e^-\\rightarrow H_1H_2\\rightarrow4b,2b2\\tau$ takes over for lower values of $M_{H^+}$.\n\nThe phase for $M_{\\tilde{g}}$ influences the Higgs masses more mildly than phases\nfor the trilinear couplings because it enters the mass matrix only at\nthe 2-loop level. Nonetheless, it strengthens the SM-like constraint around\n$M_{H^+}\\approx 200\\mbox{ GeV}$, primarily by counteracting the threshold\nsuppression of $g^S_{H_ib\\bar{b}}$ as discussed in Section 2 and leading to\na $g^S_{H_ib\\bar{b}}$ that is enhanced over the tree level value in this\nregion. Correspondingly the feature familiar from the Maximal Mixing scenario,\nwhich we noted earlier was absent in CPX with vanishing phases, has begun to\nreemerge in the small $M_{H^+}$, moderate $\\tan\\beta$ region. For comparison with the case of vanishing phases,\nin the upper right plot of Fig.~\\ref{h1tth1bb} we again present the ratio of\nthe $b\\bar{b}$ to $\\tau^+\\tau^-$ branching ratios, each normalized to their\nSM values, now in the presence of the $90^{\\circ}$ phases.\n\nThe non-standard~+~charged Higgs reach is similar to the case without phases. The two\nspikes, most visible on the combined plot, are the result of the mass\nresolution prescription discussed above. Below\n$M_{H^+}=150\\mbox{ GeV}$, $H_3$ and $H_2$ are within $10\\mbox{ GeV}$;\nabove, $H_2$ and $H_1$ share this property. From the final figure, it is\nagain evident that the combination of channels is essential to cover nearly the entire plane at 95\\% C.L.\n\nWe note that with $90^{\\circ}$ phases there appears a small hole in the\nLEP coverage at low $M_{H^+}$ and low $\\tan\\beta$ adjacent to the\ntheoretically disallowed region, which is unprobed by any of the\nTevatron search channels.\nThe presence of this hole was\nfirst discussed in Ref.~\\cite{Carena:2002bb} and is generated by the possible\ndecay of the SM-like Higgs boson into a pair of $H_1$'s, which acquire a\nsignificant $CP$-odd Higgs component. The hole is discussed in detail in Ref.~\\cite{Williams:2007dc}\nand channels which may help to cover it at hadron colliders are studied in\nRef.~\\cite{Kaplan:2009qt}--\\cite{Chang:2005ht}.\nOur results are qualitatively consistent with the LEP experiment plots\nfrom Ref.~\\cite{Schael:2006cr}\\footnote{In our plots the hole is somewhat\nsmaller than in Ref.~\\cite{Schael:2006cr} due to the finite grid size in our scan and\nthe approximations we used to implement the LEP constraints.}.\nIn the $M_{H^+}$ coordinates this hole appears as a very small region;\nhowever, it covers a significant portion of the range\n$M_{H_1}\\lesssim 45\\mbox{ GeV}$, and indicates that a light Higgs scenario\nphenomenologically similar to what has been proposed in the context of the\nNMSSM~\\cite{Dermisek:2005gg} has still not been fully ruled out in the\nMSSM. In this hole the decays $H_2\\rightarrow H_1H_1\\rightarrow 4b,4\\tau$\nconsidered by LEP are significant; however, since $\\tan\\beta$ is small,\n$H_2$ is of mixed composition and its dominant production mechanisms are\nsuppressed. Recently, the authors of Ref.~\\cite{Cranmer} reanalyzed the ALEPH data, extending the reach of the $4\\tau$ channel to higher values of $M_{H_2}$. However, this channel still only covers a small subset of this hole, because it is only efficient for $2m_{\\tau} 1) \\left( V^{\\pi^*}_1(\\mu; r, P) - V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k) \\right) \\notag \\\\\n +& \\sum_{k = 1}^K \\mathbbm{1}(|\\Pi^k| >1) \\left( V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k) - V^{\\pi^k}_1(\\mu; r, P)\\right) .\n \\label{eqn:no_violation_decompose} \n\\end{align}\n\nTo bound the first term on the right-hand side (RHS) of (\\ref{eqn:no_violation_decompose}), we have the\nfollowing lemma which gives an upper bound on the number of episodes for exploration by policy $\\pi^0$: \n\\begin{lemma}\\label{lem:burn-in}\nWith probability at least $(1 - \\delta)$, $\\sum_{k=1}^K \\mathbbm{1}(|\\Pi^k| = 1) \\leq C'$, where $C' = \\tilde{\\mathcal{O}}(H^4 |\\mathcal{S}|^3 |\\mathcal{A}| \/ ((\\tau - c^0)^2 \\land (\\tau - c^0)))$.\n\\end{lemma}\n \nTurning to the second term on the RHS of (\\ref{eqn:no_violation_decompose}), we first note that $\\pi^*$ may not be in $\\Pi^k$. To ensure that the term $V^{\\pi^*}_1(\\mu; r, P) - V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k)$ is nevertheless non-positive even when $\\pi^* \\not\\in \\Pi^k$, we set $\\alpha_r$ to the large value shown in (\\ref{alphar}). This increases $\\bar{r}^k$, and hence also $V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k)$. In addition, we show that there is a policy $\\hat{\\pi}^k$ that attains the same reward as the (non-Markov) probabilistic mixed policy, $\\tilde{\\pi}^k := B_{\\gamma_k} \\pi^* + (1 - B_{\\gamma_k}) \\pi^0$, where $B_{\\gamma_k}$ is a Bernoulli distributed random variable with mean $\\gamma_k$ for $\\gamma_k \\in [0, 1]$. $\\gamma_k$ will be chosen as the largest coefficient such that $V_1^{\\tilde{\\pi}^k}(\\mu; \\underbar{c}^k, \\hat{P}^k) \\le \\tau$. This latter policy in turn has a larger reward than $\\pi^0$ since it is a mixture with $\\pi^*$, yielding the following:\n\\begin{lemma}\n With probability at least $(1 - \\delta)$,\n \\begin{align*}\n \\sum_{k = 1}^K \\mathbbm{1}(|\\Pi^k| > 1) \\left( V^{\\pi^*}_1(\\mu; r, P) - V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k)\\right) \\le 0.\n \\end{align*}\n \\label{lem1}\n\\end{lemma}\n\nFinally, concerning the third term\non the RHS of (\\ref{eqn:no_violation_decompose}), akin to the closed-loop identifiability property \\cite{borkar1979adaptive,kumar1982new},\nwhile $\\hat{P}^k$ may not converge to $P$, the difference in the rewards $V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k) - V^{\\pi^k}_1(\\mu; r, P)$ grows sublinearly in $k$\nsince the same policy $\\pi^k$ is used in both values:\n\\begin{lemma}\n With probability at least $(1 - \\delta)$,\n \\begin{align*}\n \\sum_{k = 1}^K \\mathbbm{1}(|\\Pi^k| > 1) \\left( V^{\\pi^k}_1(\\mu; \\bar{r}^k, \\hat{P}^k) - V^{\\pi^k}_1(\\mu; r, P)\\right) = \\tilde{\\mathcal{O}}\\left(\\frac{H^3}{\\tau - c^0} \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K} + \\frac{H^5 |\\mathcal{S}|^3 |\\mathcal{A}|}{\\tau-c^0} \\right).\n \\end{align*}\n \\label{lem2}\n\\end{lemma}\n\nCombining Lemmas \\ref{lem:burn-in}, \\ref{lem1}, and \\ref{lem2} yields Theorem \\ref{thm1}.\n\n\n\\subsection{Analysis of OptPess-PrimalDual (Algorithm \\ref{alg:dual})}\n\\label{bounded_sketch}\nIn this section, we outline the steps in the proof of Theorem \\ref{thm2} by analyzing regret and constraint violation of OptPess-PrimalDual respectively.\n\n\nRecall $\\epsilon_k = 5 H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}|} (\\log\\frac{k}{\\delta'} + 1) \/ \\sqrt{k \\log\\frac{k}{\\delta'}}$, where $\\delta' = \\delta \/ (16 |\\mathcal{S}|^2 |\\mathcal{A}| H)$. The existence of a feasible solution to (\\ref{pertubed_problem}) can be guaranteed if $\\epsilon_k \\le \\tau - c^0$. Let $C''$ be the smallest value such that $\\forall k \\geq C''$, $\\epsilon_k \\le (\\tau - c^0)\/2$. Then the perturbed optimization problem (\\ref{pertubed_problem}) has at least one feasible solution for any $k \\ge C''$. By simple calculation, one can verify that $C'' = \\mathcal{O} (\\frac{H^4 |\\mathcal{S}|^3 |\\mathcal{A}|}{(\\tau - c^0)^2} \\log\\frac{H^4 |\\mathcal{S}|^3 |\\mathcal{A}|}{(\\tau - c^0)^2\\delta'})$. Notice that $\\epsilon_k$ is a function not depending on $K$, and so is the coefficient $C''$.\n\n\\paragraph{Constraint violation analysis}\nThe bounded constraint violation of OptPess-PrimalDual is established as follows. We first decompose the constraint violation as\n\\begin{align}\n Reg^{\\textbf{OPPD}}(K; c) &= \\left(\\sum_{k=1}^K \\left(V_1^{\\pi^k}(\\mu; c, P) - \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{c}^k, \\hat{P}^k)\\right) + \\sum_{k=1}^K \\left(\\hat{V}_1^{\\pi^k}(\\mu; \\tilde{c}^k, \\hat{P}^k) - \\tau\\right)\\right)_{+} \\notag\\\\\n & \\le \\left(\\sum_{k=1}^K \\left(V_1^{\\pi^k}(\\mu; c, P) - \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{c}^k, \\hat{P}^k)\\right) + \\lambda^{K+1} - \\sum_{k=1}^K \\epsilon_{k} \\right)_{+}.\n \\label{decompose_bounded_violation}\n\\end{align}\nThe first summation term and $\\lambda^{K+1}$ in (\\ref{decompose_bounded_violation}) can be bounded as follows:\n\\begin{lemma}\n Recall $\\delta' = \\delta \/ (16 |\\mathcal{S}|^2 |\\mathcal{A}| H)$, with probability at least $(1 - \\delta)$,\n \\begin{align*}\n \\sum_{k=1}^K \\left(V_1^{\\pi^k}(\\mu;c, P) - \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{c}^k, \\hat{P}^k)\\right) \\le 8 H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K \\log \\frac{K}{\\delta'}} + \\mathcal{O}(PolyLog(K)).\n \\end{align*}\n \\label{lem:violation_first_term}\n\\end{lemma}\n\n\\begin{lemma}\n For any $k \\ge C''$, with probability at least $(1 - \\delta)$,\n \\begin{align*}\n \\lambda^k \\le \\frac{1}{\\zeta} \\ln \\frac{11 \\nu_{\\max}^2}{3 \\rho^2} + C''(H-\\tau) + \\sum_{u=1}^{C''} \\epsilon_u + H + \\frac{4(H^2 + \\epsilon_k^2 + \\eta^k H)}{\\tau - c^0},\n \\end{align*}\n where $\\rho = (\\tau - c^0)\/4$, $\\nu_{\\max} = H$, $\\zeta = \\rho \/ (\\nu_{\\max }^{2}+\\nu_{\\max} \\rho \/ 3)$.\n \\label{lem7}\n\\end{lemma}\nTo guarantee bounded violation, we ensure that $\\sum_{k=1}^K \\epsilon_k$ in (\\ref{decompose_bounded_violation}) can cancel the dominant terms in the two lemmas above. According to Lemmas \\ref{lem:violation_first_term} and \\ref{lem7}, with probability at least $(1 - \\delta)$, the violation is bounded as\n\\begin{align*}\n Reg^{\\textbf{OPPD}}(K; c) = \\mathcal{O}\\left(C''H + H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| C'' \\log{(C'' \/ \\delta')}} \\right). \n\\end{align*}\n\n\n\\paragraph{Regret of reward analysis}\nFor episode $k$ with $k \\geq C''$, let $\\pi^{\\epsilon_k, *}$ be the optimal policy for (\\ref{pertubed_problem}), which is well-defined by the definition of $C''$. We decompose the reward regret as\n\\begin{align}\n Reg^{\\textbf{OPPD}}&(K; r) = \\sum_{k=1}^{C''} \\left(V_1^{\\pi^*}(\\mu; r, P) - V_1^{\\pi^k}(\\mu; r, P)\\right) \\label{bound_regret_decompose}\\\\\n + &\\sum_{k=C''}^K \\left(V_1^{\\pi^*}(\\mu; r, P) - V_1^{\\pi^{\\epsilon_k, *}}(\\mu; r, P)\\right) + \\sum_{k=C''}^K \\left(V_1^{\\pi^{\\epsilon_k, *}}(\\mu; r, P) - \\hat{V}_1^{\\pi^{\\epsilon_k, *}}(\\mu; \\tilde{r}^k, \\hat{P}^k)\\right) \\notag\\\\\n + &\\sum_{k=C''}^K \\left(\\hat{V}_1^{\\pi^{\\epsilon_k, *}}(\\mu; \\tilde{r}^k, \\hat{P}^k) - \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{r}^k, \\hat{P}^k)\\right) + \\sum_{k=C''}^K \\left(\\hat{V}_1^{\\pi^k}(\\mu; \\tilde{r}^k, \\hat{P}^k) - V_1^{\\pi^k}(\\mu; r, P)\\right). \\notag\n\\end{align}\n\nWe upper bound each term on the RHS of (\\ref{bound_regret_decompose}). Since $V_1^{\\pi}(\\mu;r, P) \\in [0, H]$ for any policy $\\pi$, the first term is upper bounded by $HC''$. The second and third terms can be bounded by the following two lemmas:\n\\begin{lemma}\n With probability at least $(1 - \\delta)$,\n \\begin{align*}\n \\sum_{k=C''}^K \\left(V_1^{\\pi^*}(\\mu; r, P) - V_1^{\\pi^{\\epsilon_k, *}}(\\mu; r, P)\\right) \\le \\sum_{k=C''}^K \\frac{\\epsilon_k H}{\\tau - c^0} = \\tilde{\\mathcal{O}}\\left(\\frac{H^3}{\\tau - c^0} \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K}\\right).\n \\end{align*}\n \\label{lem3}\n\\end{lemma}\n\n\\begin{lemma}\n With probability at least $(1 - \\delta)$,\n $\\sum_{k=C''}^K \\left(V_1^{\\pi^{\\epsilon_k, *}}(\\mu; r, P) - \\hat{V}_1^{\\pi^{\\epsilon_k, *}}(\\mu; \\tilde{r}^k, \\hat{P}^k)\\right) \\le 0$.\n \\label{lem4}\n\\end{lemma}\n\nThe pivotal step is to leverage optimism of $\\pi^k$ to further decompose the fourth term on the RHS of (\\ref{bound_regret_decompose}), and utilize the projected dual update to transfer it into the form of $\\lambda^k (\\lambda^k - \\lambda^{k+1})$. The following lemmas provide high probability bounds for the remaining two terms on the RHS of (\\ref{bound_regret_decompose}):\n\n\\begin{lemma}\n With probability at least $(1 - \\delta)$,\n \\begin{align*}\n \\sum_{k=C''}^K \\left(\\hat{V}_1^{\\pi^{\\epsilon_k, *}}(\\mu; \\tilde{r}^k, \\hat{P}^k) - \\hat{V}_1^{\\pi^k}(\\mu; \\tilde{r}^k, \\hat{P}^k)\\right) = \\tilde{\\mathcal{O}}\\left( \\frac{H}{\\tau - c^0} \\sqrt{K}\\right).\n \\end{align*}\n \\label{lem5}\n\\end{lemma}\n\n\\begin{lemma}\n With probability at least $(1 - \\delta)$, \n \\begin{align*}\n \\sum_{k=C''}^K \\left(\\hat{V}_1^{\\pi^k}(\\mu; \\tilde{r}^k, \\hat{P}^k) - V_1^{\\pi^k}(\\mu; r, P)\\right) = \\tilde{\\mathcal{O}}\\left(H^2 \\sqrt{|\\mathcal{S}|^3 |\\mathcal{A}| K} + H^4 |\\mathcal{S}|^3 |\\mathcal{A}|\\right).\n \\end{align*}\n \\label{lem6}\n\\end{lemma} \n\nApplying Lemmas \\ref{lem3}, \\ref{lem4}, \\ref{lem5}, and \\ref{lem6} yields Theorem \\ref{thm2}. \n\n\n\n\\section{Concluding remarks}\n\\label{conclusion}\nWe present two optimistic pessimism-based algorithms that maintain stringent safety constraints (either zero or bounded safety constraint violation) for unknown CMDPs with high probability, while still attaining an $\\tilde{\\mathcal{O}}(\\sqrt{K})$ regret of reward. The algorithms employ, respectively, a pessimistically safe policy set $\\Pi^k$ or an additional pessimistic term $\\epsilon_k$ into the safety constraint. \nThe first algorithm, OptPess-LP, guarantees zero violation with high probability by solving a linear programming with $\\Theta(|\\mathcal{S}||\\mathcal{A}|H)$ decision variables, while the second algorithm, OptPess-PrimalDual, is as efficient as policy-gradient-based algorithms in the tabular case, but violates constraints during initial episodes. A possible future direction for exploration is the application of the above OPFU principle in model-free algorithms. \n\n\n\n\\section*{Acknowledgement}\nP. R. Kumar's work is partially supported by US National Science Foundation under CMMI-2038625, HDR Tripods CCF-1934904; US Office of Naval Research under N00014-21-1-2385; US ARO under W911NF1810331, W911NF2120064; and U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) under the Solar Energy Technologies Office Award Number DE-EE0009031. The views expressed herein and conclusions contained in this document are those of the authors and should not be interpreted as representing the views or official policies, either expressed or implied, of the U.S. NSF, ONR, ARO, Department of Energy or the United States Government. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.\n\nDileep Kalathil gratefully acknowledges funding from the U.S. National Science Foundation (NSF) grants NSF-CRII- CPS-1850206 and NSF-CAREER-EPCN-2045783.\n\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\IEEEPARstart{A}s has been known, in conventional wireless communications, sending one information symbol from the transmitter to the receiver constitutes a transmission process over air interface. Apart from the desired receiver, the transmitted symbol can be interference to the other receivers, except for broadcasting services. \n\nThis work proposes a new method that enables one information symbol to serve, over air interface, two users located in geometrically orthogonal directions as shown in Figure~1. This scenario provides our research motivation to double transmission efficiency in the orthogonal geometry with the parallel- and perpendicular channel. The physical mechanism, allowing one symbol to appear with two different phases in the two directions, can be essentially attributed to the use of the spatial anisotropy of Doppler effect, which has been categorized into several sorts in general \\cite{Jackson1998}[2][3]. \n\nOur research starts from considering the Doppler effect on its emitted electromagnetic waves in the plane wave model and the observations from different angles with respect to the moving direction of the emission source. \n\nWhen a harmonic signal is emitted, the Doppler effect can be expressed mathematically by \n\\begin{eqnarray}\n\t\\begin{array}{l}\\label{1}\n\t\\Delta f(\\theta) = f^{'}-f=\\frac{v_x}{\\lambda}\\cos\\theta, \\ \\ \\\n\t\\end{array}\n\\end{eqnarray}\nwhere $\\Delta f$ is the Doppler frequency shift, $f^{'}$ is the frequency observed, $f$ is the frequency of the emission, $v_x$ is the speed of the source, $\\theta$ is the angle between directions of the observation and the source's moving, $\\lambda$ is the wavelength. \n\nIn a two-dimensional Cartesian coordinate model, we use $\\Delta f_x$ and $\\Delta f_y$ to express $\\Delta f(0)$ and $\\Delta f(\\pi\/2)$ for showing that $\\Delta{f}_y = 0 $ holds no matter what value $\\Delta{f}_x $ is. This indicates that any effects resulted from $\\Delta f_x$ will be nullified in the direction of y-axis. \n\nNow, we work on $\\Delta f_x$ with Maxwell equation\\cite{Jackson1998} by\n\\begin{eqnarray}\n\\begin{array}{l}\\label{2}\nR_x(t)= Ae^{j2\\pi f t-j(kx_0-kv_xt)}, \\\\ \n\\end{array}\n\\end{eqnarray}\nwhere $j=\\sqrt{-1}$, $R_x(t)$ is the received wave of a receiver located at x-axis for $x>0$, $A$ is the received amplitude, $x_0$ is the initial position of the receiver, and $k=2\\pi\/\\lambda$ is a constant. \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.35\\textwidth]{Fig1}\n\t\\caption{ Multiple antennas at the crossroads working through the parallel- and perpendicular channel.}\n\t\\label{Application Scenario}\n\\end{figure}\n\nThe Doppler frequency shift is found at $v_xk\/2\\pi = v_x\/\\lambda$ again in \\eqref{2}. However, because $v_xt$ is actually the position the emission source arrives at time $t$, the frequency shift can be regarded as a continuous phase modulation with respect to the position of the emission source. This can be more clearly seen at the baseband level by removing the carrier frequency $f$, i.e.,\n\\begin{eqnarray}\n\\begin{array}{l}\\label{3}\n\\hat{R}_x(x)=Ae^{jkx(t)-j\\phi_0}, \\ \\ \\ \n\\end{array}\n\\end{eqnarray}\nwhere $\\hat{R}_x(x)$ is the received baseband signal, $x(t)=v_x t$ denotes the position of the emission source at time $t$, $\\phi_0= kx_0$ is the initial phase. \n\nIt is noted that $kx(t)$ is a modulated phase that does not appear in direction of y-axis. \n \n\\section{Signal Modulation}\nIn this section, we propose a method in which the transmitter consists of a signal source and multiple antennas to transmit information symbols to the two receivers $\\hat{R}_x$ and $\\hat{R}_y$ located in two geometrically orthogonal directions, where the parallel and perpendicular channels are defined respectively, as shown in Fig.2. \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.35\\textwidth]{Fig2.pdf}\n\t\\caption{The transmitter architecture of the QD method and the ideal geometric model.}\n\t\\label{PhD}\n\\end{figure}\n\n\\subsection{Quasi-Doppler Effect}\nLet us use a harmonic signal, in the plane wave model, to explain the mechanism of generating Doppler effect by using multiple antennas to replace the moving of the emission source. These antennas are uniformly distributed along x-axis at $x_0 = 0,~x_1 = d,\\cdots,x_q=qd,\\cdots$, till $x_{Q-1} = (Q-1)d$ as shown in Fig.2, where $d$ is the distance between two adjacent antennas and $Qd= \\lambda$ is required for limiting the geometric range of the multi antennas. \n\nTo emulate the harmonic signal transmission of the Doppler source moving at speed $v_x$, we switch the signal source sequentially onto each antenna $x_0,~x_1,\\cdots,x_q,\\cdots, x_{Q-1}$, where the time of connecting the signal source and each antenna is set to $T=d\/v_x$ and the switching time, i.e., the time of hopping between any two antennas, is assumed to be zero. The received signal through the parallel channel can be written as\n\\begin{eqnarray}\n\\begin{array}{l}\\label{4}\n\\hat{R}_x(t)= h_xe^{jkv_x T \\lfloor t\/T\\rfloor+j\\phi_0} + n_0, \\ \\ \\\n\\end{array}\n\\end{eqnarray}\nwhere $\\hat{R}_x(t)$ is the received baseband signal, $v_x = d\/T$ is the speed of emulated moving, $\\lfloor \\cdot \\rfloor$ is the floor operator, $h_x$ is a complex channel response, $\\phi_0$ is an initial phase and $n_0$ is the Gaussian noise.\n\nThis equation shows the Doppler frequency shift at $kv_x\/2\\pi$, which is referred to as the Quasi Doppler (QD) effect, because the time variable behaves in a step manner with $\\lfloor t\/T \\rfloor$. \n\nSimilar with the transformation from \\eqref{2} to \\eqref{3}, assuming $\\phi_0=0$, we can transfer the QD frequency shift into desiccate phase modulations, according to the positions of those antennas, as expressed by\n\\begin{eqnarray}\n\\begin{array}{l}\\label{5}\n\\hat{R}_x(x_q)=h_xe^{jk x_q+j\\phi_0} + n_0, \n\\end{array}\n\\end{eqnarray}\nwhere $x_q$ represents the position of $q^{th}$ antenna, $q = 0,1,\\cdots,Q-1$. \n\nIn the following discussions, ``Antenna q'' stands for the antenna's position at position $x=x_q$. It is found that the QD effect yields the discrete phase modulation instead of the continuous one in \\eqref{3}. \n\nFurther, switching the signal source from antennas $q'$ to antenna $q$ can provide a phase modulation at $\\Delta {\\phi} = k(x_q-x_{q'}) = 2\\pi( x_q-x_{q'})\/\\lambda$. Thus, by assuming $\\phi_0=0$, the transmission at antenna $q$ can be regarded as a QD phase of \n\\begin{eqnarray}\n\\begin{array}{l}\\label{6}\n\\phi_q =2\\pi x_q \/\\lambda, \n\\end{array}\n\\end{eqnarray}\nwhere $\\phi_q$ is defined as the QD phase, and $x_q$ is the position of the transmit antenna. \n\nFor preparing a signal constellation for the transmission, $x_q$ should be pre-designed by\n\\begin{eqnarray}\n\\begin{array}{l}\\label{6-05}\nx_q = \\phi_q \\lambda\/2\\pi \n\\end{array}\n\\end{eqnarray}\nfor $q=1,2,....,Q-1$, according to the required $\\phi_q$ in the signal constellation. \n\nIn the communication scheme, the assumption of $\\phi_0 = 0$ is reasonable, because the channel characterization permits. \n\n\\subsection{Joint Symbol Modulation}\nThe transmission issue arises with the use of one information symbol to provide two desired phases that appear through the parallel channel and the perpendicular channel. \nTo be specific, $\\phi_x$ and $\\phi_y$ denote these two desired phases. The joint signal modulation is explained in the following two steps. \n\nAt the first step, the QD transmitter uses the conventional modulation at its signal source to produce an information symbol as \n\\begin{eqnarray}\n\\begin{array}{l}\\label{7}\nS_y = \\sqrt{E_s}e^{j\\phi_y}, \n\\end{array}\n\\end{eqnarray}\nwhere $S_y$ , $E_s$ and $\\phi_y$ are the modulated symbol, the symbol energy and the desired phase for the perpendicular channel, respectively. \n\nSecondly, for providing the desired phase, i.e., $\\phi_x$, through the parallel channel, the produced symbol in \\eqref{7} should be switched onto an appropriate antenna to complete the symbol transmission. The QD phase adds its contribution by \n\\begin{eqnarray}\n\\begin{array}{l}\\label{8}\nS_x = \\sqrt{E_s}e^{j(\\phi_y+ \\phi_q)}, \n\\end{array}\n\\end{eqnarray}\nand letting $\\phi_y+ \\phi_q = \\phi_x$ leads to \n\\begin{eqnarray}\\label{9} \n\\phi_q = \\begin{cases}\n\\begin{split}\n&\\phi_x - \\phi_y, \\quad &\\phi_x \\ge \\phi_y,\\\\\n&\\phi_x +2\\pi - \\phi_y, \\quad &\\phi_x <\\phi_y,\n\\end{split}\n\\end{cases}\n\\end{eqnarray}\nwhere $\\phi_q$ is the QD phase. Further, by using \\eqref{6-05}, we find the transmit antenna $q$ for $x=x_q$.\n \nBecause the QD phase. i.e., $\\phi_q$, does not show up through the perpendicular channel, the symbol transmission can be summarized in form of \n\\begin{eqnarray}\n\\begin{array}{l}\\label{10}\n\\hat{R}_\\beta = h_\\beta S_\\beta + n_0 = h_\\beta\\sqrt{E_s}e^{j\\phi_\\beta} + n_0, \n\\end{array}\n\\end{eqnarray}\nwhere $\\hat{R}_\\beta$ and $h_\\beta$ are the received signal and the channel gain factor, respectively, with $\\beta=x$ and $y$ indicating the signal received through the parallel- and perpendicular channel, respectively. \n \nBased on the above derivations, we conclude that the QD method provides a possibility to double the transmission efficiency in the orthogonal geometry model, because one jointly modulated symbol can satisfy the needs of two receivers. \n\n\\section{On the Performance}\nThe purpose of this section is to explore the application issues in the scenarios of crossroads, whereat the ideally orthogonal- and deviated geometric model are taken into the theoretical analysis and the simulations as well. \n\n\\subsection {Theoretical Analysis}\nIn addition to the ideal geometric model, we define the deviated geometric model for the scenarios, where the orthogonality is slightly deviated, as shown in Fig. \\ref{fig3}, and work on the cases of the parallel- and the perpendicular channel separately for showing the discrepancies. \n\nAs can be expected, in comparison with the ideal geometry, the deviated angles can cause some phase offsets as analysed below. \n\nAssuming that there is a small angle deviation $\\theta_x$ between x-axis and the line of sight from the transmit antenna to the receiver $\\hat{R}_x$ as shown in Fig. 3(a), we can find that the received $\\phi_y$ remains unchanged in \\eqref{8} and, however, the QD phase can change to (see the Appendix) \n\\begin{equation}\\label{11}\n{\\phi}_{q}' = k x_q\\cos\\theta_x = 2\\pi x_q \\cos\\theta_x\/\\lambda,\n\\end{equation}\nwhere ${\\phi}_q'$ is the QD phase received at the receiver $\\hat{R}_x$. Consequently, the received signal can be expressed as \n\\begin{eqnarray}\n\\begin{array}{l}\\label{12}\n\\hat{R}_{\\theta_x} = h_{\\theta_x} e^{j(\\phi_y+\\phi_q')}\n\t + n_0 = h_{\\theta_x} e^{j\\phi_x+ j\\Delta{\\phi}_{x}} + n_0,\n\\end{array}\n\\end{eqnarray}\nwhere $h_{\\theta_x}$ and $\\Delta{\\phi_x} = 2\\pi x_q (\\cos\\theta_x-1)\/\\lambda$ are the channel gain factor and the phase-offset owing to the geometric deviation angle $\\theta_x$, respectively. \n\nSecondly, for the deviated perpendicular channel as shown in Fig.3(b), the received signal can be expressed as (see the Appendix)\n\\begin{eqnarray}\n\\begin{array}{l}\\label{13}\n\\hat{R}_{\\theta_y} = h_{\\theta_y} e^{j(\\phi_y+ kx_qsin\\theta_y)} + n_0 = h_{\\theta y}, e^{j(\\phi_y+\\Delta{\\phi_y})}+n_0\n\\end{array}\n\\end{eqnarray}\nwhere $h_{\\theta_y}$ and $\\Delta \\phi_y= kx_q\\sin\\theta_y \/ \\lambda$ are the channel gain factor and the phase-offset owing to $\\theta_y$, respectively. \n\nApplying the Tailor expansion to \\eqref{12} and \\eqref{13}, we can find that the phase offsets are limited by the geometry deviation as $|\\Delta \\phi_x|\\le \\frac{\\pi\\theta_x^2}{2} $ and $ |\\Delta \\phi_y|\\le \\pi \\theta_y $. By comparison, the perpendicular channel is more sensitive to the deviation angle than the parallel channel, because $|\\Delta \\phi_y|$ converges in a linear manner, while $|\\Delta \\phi_x|$ with square, when the deviation angles approach zero. \n \nAs to the transmission performance, the QD method does not degrade much in principle, when the maximum values of $|\\Delta \\phi_x| $ and $|\\Delta \\phi_y|$ are much smaller than the phase difference between two adjacent points in the signal constellation used in the communication. To solidify this part of the work, simulations will be performed in the next subsection.\n\nFinally, we note that the assumption of zero switching-time in the theoretical model can be removed, because the phase detection is done by comparing the RF oscillations between the arriving signal and the local reference, both of which experience the same switching-time. Thus, there is no relative phase change as compared to the case with zero switching-time. \n\\begin{figure}[!t]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\centering\n\t\t\\includegraphics[width=0.38\\textwidth]{Fig3a.pdf}\n\t\t\\label{fig3a}\n\t}\n\t\\subfigure[]{\n\t\t\\centering\n\t\t\\includegraphics[width=0.38\\textwidth]{Fig3b.pdf}\n\t\t\\label{fig3b}\n\t}\n\t\\caption{Deviated geometric model: an deviated angel with the parallel channel in (a) and that with perpendicular channel in (b). }\n\t\\label{fig3}\n\\end{figure}\n\n\\subsection{Simulations}\nIn this subsection, simulation results are reported for investigating the performances of one symbol through two directions in the ideally orthogonal- and deviated geometric model. \n\nFor simplicity, we use BPSK modulation at the signal source and two omni-directional antennas located at positions $x_0 = 0 $ and $x_1 =\\lambda\/2$ to construct the QD transmitter. In addition, considering the crossroad scenarios where line of sight must exist from either the cars on the road or from people along the roadside to the traffic light, we neglect the multipath signals, because our antennas can be placed near the position of the traffic light. Actually, we select the AWGN channel model to the simulations for testing the bit-error-rate (BER) performances on various SNR-values, \n\nThe random phases $\\phi_x$ and $\\phi_y$ are generated at the QD transmitter by using Matlab 2019, and the Monte-Carlo method is used to simulate the receiver with perfect channel catheterization, where the noise power is set to zero in the pilots. Then, hard decision is used in the symbol demodulation. \n\nTo confirm the QD method, the ideal geometric model is taken to the simulations and the results are plotted in Fig. 4(a)(b) for the parallel- and perpendicular channel, respectively. Then, by setting a tolerable SNR loss at 0.8dB with $BER=10^{-8}$ in comparison with the ideal geometric model, we make a large number of simulations and found the corresponding deviation angles to $\\theta_x = 30^o$ and $\\theta_y = 8^o$, for which the BERs are also shown in Fig. \\ref{BER_Performance}(a)(b), respectively. The relativity of the two deviation angles, i.e., $\\theta_y < \\theta_x$, agrees with the theoretical conclusion in the analysis, i.e. the perpendicular channel is more sensitive compared to the parallel channel. \n\n\\begin{figure}[!t]\n\t\\centering\n\t\\subfigure[]{\n\t\t\\centering\n\t\t\\includegraphics[width=0.42\\textwidth]{BER_Performance_thetax.pdf}\n\t\t\\label{fig4a}\n\t}\n\t\\subfigure[]{\n\t\t\\centering\n\t\t\\includegraphics[width=0.42\\textwidth]{BER_Performance_thetay.pdf}\n\t\t\\label{fig4b}\n\t}\n\t\\caption{The BER performance of the ideal channel and the deviated channel: (a) the parallel channel and (b) the perpendicular channel. }\n\t\\label{BER_Performance}\n\\end{figure}\n\nFinally, the author would get rid of any possible confusion between the QD method and the spatial modulation (SM) by the following short statement: the QD method can double the transmission efficiency and work in the one-to-two system, while the SM can increase the transmission efficiency slightly \\cite{Yang2008} in the point-to-point manner \\cite{Mesleh2008,Rajashekar2013,Wen2014}. \n\n\\section{Conclusion}\nThe concept of the QD effect has been created for realizing the discrete phase modulation. By combining the anisotropy of electromagnetic wave propagation, we developed the QD method that can double the transmission efficiency. The application issues are addressed in the scenarios of crossroads with the ideal and deviated channel models, wherein the feasibility is confirmed by simulation results. \n\n\\section*{Appendix}\nBy considering the plane wave model, we use Fig. A to show the angle deviations between the ideal- and deviated geometric model by $\\theta_x$ and $\\theta_y$. \n\nExamining the angle deviation from the parallel channel, one can find that the wave propagation path-difference $x_q -x_0$ in the ideal geometric model should be projected onto the line in the direction of $\\hat{R}_\\theta$. The projected result is equal to $(x_q-x_0) cos\\theta_x$ that leads to the QD phase $kx_q cos\\theta_x$ at the receiver. \n \nFor the deviated perpendicular channel, the propagation path-difference is described by the deviation angle $\\theta_y$. Consequently, the QD phase can be changed to $kx_q sin\\theta_y$ because $ cos\\theta_x = sin\\theta_y$. \n\\renewcommand{\\thefigure}{\\Alph{figure}}\n\\setcounter{figure}{0}\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{appendix.pdf}\n\t\\caption{Illustration of the QD phase difference. }\n\t\\label{appendix}\n\\end{figure}\n\\section*{Acknowledgment}\n\nThe author expresses his thanks to Dongsheng Zheng, a Ph.D candidate, for his work on the simulations. \n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Algebra}\\label{subsection algebra}\n\nIn the following, $\\mathbf{F}$ denotes the field $\\mathbf{Z} \/ 2 \\mathbf{Z}$. Vector spaces are always over $\\mathbf{F}$.\n\n\\begin{defin}\\label{definition category}\n\t\n\tAn $A_{\\infty}$-category $\\mcA$ is the data of\n\t\\begin{enumerate}\n\t\t\n\t\t\\item a collection of objects $\\mathrm{ob} \\, \\mcA$,\n\t\t\n\t\t\\item for every objects $X,Y$, a graded vector space of morphisms $\\mcA \\left( X,Y \\right)$,\n\t\t\n\t\t\\item a family of degree $\\left( 2-d \\right)$ linear maps\n\t\t\\[\\mu^d : \\mcA \\left( X_0, X_1 \\right) \\otimes \\dots \\otimes \\mcA \\left( X_{d-1}, X_d \\right) \\to \\mcA \\left( X_0, X_d \\right) \\]\n\t\tindexed by the sequences of objects $\\left( X_0, \\dots, X_d \\right)$, $d \\geq 1$, such that\n\t\t\\[\\sum\\limits_{0 \\leq i < j \\leq d} \\mu^{d-(j-i)+1} \\circ \\left( \\mathbf{1}^i \\otimes \\mu^{j-i} \\otimes \\mathbf{1}^{d-j} \\right) = 0, \\]\n\t\tfor all $d \\geq 1$.\n\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition coalgebra}\n\t\n\tAn $A_{\\infty}$-cocategory $\\mcC$ is the data of \n\t\t\\begin{enumerate}\n\t\t\n\t\t\\item a collection of objects $\\mathrm{ob} \\, \\mcC$,\n\t\t\n\t\t\\item for every objects $X,Y$, a graded vector space of morphisms $\\mcC \\left( X,Y \\right)$,\n\t\t\n\t\t\\item a family of degree $\\left( 2-d \\right)$ linear maps\n\t\t\\[\\delta^d : \\mcC \\left( X_0, X_d \\right) \\to \\bigoplus \\limits_{d \\geq 1} \\bigoplus_{X_1, \\dots, X_{d-1}} \\mcC (X_0, X_1) \\otimes \\dots \\otimes \\mcC (X_{d-1}, X_d) \\]\n\t\tindexed by the sequences of objects $\\left( X_0, \\dots, X_d \\right)$, $d \\geq 1$, such that\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item for all $d \\geq 1$,\n\t\t\t\\[\\sum\\limits_{0 \\leq i < j \\leq d} \\left( \\mathbf{1}^i \\otimes \\delta^{j-i} \\otimes \\mathbf{1}^{d-j} \\right) \\circ \\delta^{d-(j-i)+1} = 0, \\]\n\t\t\t\n\t\t\t\\item the map \n\t\t\t\\[C \\to \\prod_{d \\geq 1} C^{\\otimes d}, \\quad x \\mapsto \\left( \\delta^d \\left( x \\right) \\right)_{d \\geq 1} \\]\n\t\t\tfactors through the inclusion $\\bigoplus\\limits_{d \\geq 1} C^{\\otimes d} \\to \\prod\\limits_{d \\geq 1} C^{\\otimes d}$.\n\t\t\t\n\t\t\\end{itemize}\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{rmks}\n\t\n\t\\begin{enumerate}\n\t\t\n\t\t\\item If $\\mcE$ is some set, denote by $\\mathbf{F}_{\\mcE}$ the semi-simple algebra over $\\mathbf{F}$ generated by elements $e_X$, $X \\in \\mcE$, such that\n\t\t\\[e_X \\cdot e_Y = \n\t\t\\left\\{ \n\t\t\\begin{array}{ll}\n\t\t\te_X & \\text{if } X = Y \\\\\n\t\t\t0 & \\text{if } X \\ne Y.\n\t\t\\end{array}\n\t\t\\right. \\]\n\t\tTo any $A_{\\infty}$-category $\\mcA$ with $\\mathrm{ob} (\\mcA) = \\mcE$, we can associate an $A_{\\infty}$-algebra over $\\mathbf{F}_{\\mcE}$ where \n\t\t\\begin{itemize}\n\n\t\t\t\\item the underlying graded vector space is $\\bigoplus_{X,Y \\in \\mcE} \\mcA (X, Y)$,\n\t\t\t\n\t\t\t\\item given $x \\in \\mcA (X_0, Y_0)$, \n\t\t\t\\[e_X \\cdot x = \n\t\t\t\\left\\{ \n\t\t\t\\begin{array}{ll}\n\t\t\t\tx & \\text{if } X = X_0 \\\\\n\t\t\t\t0 & \\text{if } X \\ne X_0\n\t\t\t\\end{array}\n\t\t\t\\right. \\text{ and } \n\t\t\tx \\cdot e_Y = \n\t\t\t\\left\\{ \n\t\t\t\\begin{array}{ll}\n\t\t\t\tx & \\text{if } Y = Y_0 \\\\\n\t\t\t\t0 & \\text{if } Y \\ne Y_0,\n\t\t\t\\end{array}\n\t\t\t\\right. \\]\t\n\t\t\t\n\t\t\t\\item operations are the same as on $\\mcA$.\n\n\t\t\\end{itemize}\n\t\tConversely, to any $A_{\\infty}$-algebra over $\\mathbf{F}_{\\mcE}$, one can associate an $A_{\\infty}$-category with $\\mathrm{ob} (\\mcA) = \\mcE$. \n\t\tNote that the above discussion also applies to $A_{\\infty}$-cocategories. As a result, the theory of $A_{\\infty}$-(co)categories with $\\mcE$ as set of objects is equivalent to the theory of $A_{\\infty}$-(co)algebras over $\\mathbf{F}_{\\mcE}$.\n\t\n\t\t\\item In this paper, we will appeal to several standard notions in the theory of $A_{\\infty}$-(co)categories that we choose not to recall: instead, we list them and give corresponding references.\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item For $A_{\\infty}$-(co)maps, (co)augmentations and (co)bar, graded dual, Koszul dual constructions, see \\cite[section 2]{EL21} (where everything is written in the language of $A_{\\infty}$-(co)algebras over $\\mathbf{F}_{\\mcE}$).\n\t\t\t\n\t\t\t\\item For general definitions and results about $A_{\\infty}$-categories (in particular about homotopy between $A_{\\infty}$-functors, homological perturbation theory, directed (sub)categories and twisted complexes), see \\cite[chapter 1]{Sei08}.\n\t\t\t\n\t\t\t\\item For quotient of $A_{\\infty}$-categories, see \\cite{LO06}, and for localization of $A_{\\infty}$-categories, see \\cite[section 3.1.3]{GPS20}.\n\n\t\t\\end{itemize}\n\t\n\t\t\\item An Adams-graded vector space is a $\\mathbf{Z} \\times \\mathbf{Z}$-graded vector space: if $x$ is an element in the $(i,j)$ component, we say that $i$ is the cohomological degree of $x$, and $j$ is the Adams degree of $x$.\n\t\tAn Adams-graded $A_{\\infty}$-(co)category is an $A_{\\infty}$-(co)category enriched over Adams-graded vector spaces, where the operations are required to be of degree $0$ with respect to the Adams grading. \n\t\tSee \\cite{LPWZ08} for a treatment of Koszul duality in the context of Adams-graded $A_{\\infty}$-algebras.\n\t\t\n\t\\end{enumerate}\n\n\\end{rmks}\n\n\\subsection{Modules over $A_{\\infty}$-categories}\n\nLet $\\mcC, \\mcD$ be two $A_\\infty$-categories, and let $\\mcA, \\mcB$ be two full subcategories of $\\mcC, \\mcD$ respectively.\n\n\\begin{defin}\\label{definition bimodule}\n\t\n\tA $\\left( \\mcC, \\mcD \\right)$-bimodule $\\mcM$ consists of the following data: \n\t\\begin{enumerate}\n\t\t\n\t\t\\item for every pair $\\left( X, Y \\right) \\in \\mathrm{ob} \\left( \\mcC \\right) \\times \\mathrm{ob} \\left( \\mcD \\right)$, a vector space $\\mcM \\left( X, Y \\right)$, \n\t\t\n\t\t\\item a family of degree $\\left( 1-p-q \\right)$ linear maps \n\t\t\\begin{align*}\n\t\t\\mu_{\\mcM} : \\mcC \\left( X_0, X_1 \\right) \\otimes & \\dots \\otimes \\mcC \\left( X_{p-1}, X_p \\right) \\otimes \\mcM \\left( X_p, Y_q \\right) \\\\ \n\t\t& \\otimes \\mcD \\left( Y_q, Y_{q-1} \\right) \\otimes \\dots \\otimes \\mcD \\left( Y_1, Y_0 \\right) \\to \\mcM \\left( X_0, Y_0 \\right)\n\t\t\\end{align*}\n\t\tindexed by the sequences \n\t\t\\[\\left( X_0, \\dots, X_p, Y_0, \\dots, Y_q \\right) \\in \\mathrm{ob} \\left( \\mcC \\right)^{p+1} \\times \\mathrm{ob} \\left( \\mcD \\right)^{q+1}, \\]\n\t\twhich satisfy the relations \n\t\t\\begin{align*}\n\t\t\\sum \\mu_{\\mcM} \\left( \\dots , \\mu_{\\mcC} \\left( \\dots \\right) , \\dots, u \\dots \\right) & + \\sum \\mu_{\\mcM} \\left( \\dots , \\mu_{\\mcM} \\left( \\dots, u, \\dots \\right), \\dots \\right) \\\\ \n\t\t& + \\sum \\mu_{\\mcM} \\left( \\dots , u, \\dots, \\mu_{\\mcD} \\left( \\dots \\right) , \\dots \\right) = 0 .\n\t\t\\end{align*}\n\t\t\n\t\\end{enumerate}\n\t\n\tA degree $s$ morphism $t : \\mcM_1 \\to \\mcM_2$ between two $\\left( \\mcC, \\mcD \\right)$-bimodules consists of a family of degree $\\left( s-p-q \\right)$ linear maps\n\t\\begin{align*}\n\tt : \\mcC \\left( X_0, X_1 \\right) \\otimes & \\dots \\otimes \\mcC \\left( X_{p-1}, X_p \\right) \\otimes \\mcM_1 \\left( X_p, Y_q \\right) \\\\ \n\t& \\otimes \\mcD \\left( Y_q, Y_{q-1} \\right) \\otimes \\dots \\otimes \\mcD \\left( Y_1, Y_0 \\right) \\to \\mcM_2 \\left( X_0, Y_0 \\right)\n\t\\end{align*}\n\tindexed by the sequences \n\t\\[\\left( X_0, \\dots, X_p, Y_0, \\dots, Y_q \\right) \\in \\mathrm{ob} \\left( \\mcC \\right)^{p+1} \\times \\mathrm{ob} \\left( \\mcD \\right)^{q+1}. \\]\n\tThe differential of such a morphism is defined by\n\t\\begin{align*}\n\t\\mu_{\\mathrm{Mod}_{\\mcC, \\mcD}}^1 & \\left( t \\right) \\left( \\dots, u, \\dots \\right) \\\\\n\t& = \\sum t \\left( \\dots , \\mu_{\\mcC} \\left( \\dots \\right) , \\dots, u, \\dots \\right) + \\sum t \\left( \\dots , \\mu_{\\mcM_1} \\left( \\dots, u, \\dots \\right), \\dots \\right) \\\\ \n\t& +\\sum t \\left( \\dots , u, \\dots, \\mu_{\\mcD} \\left( \\dots \\right), \\dots \\right) + \\sum \\mu_{\\mcM_2} \\left( \\dots , t \\left( \\dots, u, \\dots \\right), \\dots \\right) .\n\t\\end{align*} \n\tFinally, the composition of $t_1 : \\mcM_1 \\to \\mcM_2$ and $t_2 : \\mcM_2 \\to \\mcM_3$ is such that\n\t\\[\t\\mu_{\\mathrm{Mod}_{\\mcC}}^2 \\left( t_1 , t_2 \\right) \\left( \\dots, u, \\dots \\right) = \\sum t_2 \\left( \\dots , t_1 \\left( \\dots, u, \\dots \\right), \\dots \\right) .\\]\n\tWe denote by $\\mathrm{Mod}_{\\mcC, \\mcD}$ the DG-category of $\\left( \\mcC, \\mcD \\right)$-bimodules.\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition pullback bimodule}\n\t\n\tLet $\\Phi_1, \\Phi_2 : \\mcC \\to \\mcD$ be two $A_{\\infty}$-functors. Then there is a $\\mcC$-bimodule $\\mcD \\left( \\Phi_1 (-), \\Phi_2 (-) \\right)$ defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, it sends $(X_1, X_2)$ to $\\mcD \\left( \\Phi_1 X_1, \\Phi_2 X_2 \\right)$,\n\t\t\n\t\t\\item on morphisms, it sends a sequence $\\left( \\dots, y, \\dots \\right)$ in\n\t\t\\begin{align*}\n\t\t\\mcC \\left( X_0, X_1 \\right) \\times & \\dots \\times \\mcC \\left( X_{p-1}, X_p \\right) \\times \\mcD \\left( \\Phi_1 X_p, \\Phi_2 X_{p+1} \\right) \\\\ \n\t\t& \\times \\mcC \\left( X_{p+1}, X_{p+2} \\right) \\times \\dots \\times \\mcC \\left( X_{p+q}, X_{p+q+1} \\right)\n\t\t\\end{align*}\n\t\tto \n\t\t\\begin{align*}\n\t\t\\mu_{\\mcD \\left( \\Phi_1 (-), \\Phi_2 (-) \\right)} & \\left( \\dots, y, \\dots \\right) \\\\ \n\t\t& = \\sum \\mu_{\\mcD} \\left( \\Phi_1 \\left(\\dots \\right) , \\dots , \\Phi_1 \\left( \\dots \\right) , y, \\Phi_2 \\left( \\dots \\right) , \\dots , \\Phi_2 \\left( \\dots \\right) \\right) .\n\t\t\\end{align*}\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\nIn the following, we will focus on \\emph{left} $\\mcC$-modules, which correspond to $\\left( \\mcC, \\mathbf{F} \\right)$-bimodules.\nWe denote by $\\mathrm{Mod}_{\\mcC}$ the DG-category of (left) $\\mcC$-modules.\n\n\\begin{defin}\\label{definition quasi-isomorphism between modules}\n\t\n\tLet $t : \\mcM_1 \\to \\mcM_2$ be a degree $0$ closed $\\mcC$-module map. We say that $t$ is a quasi-isomorphism if the induced chain map $t : \\mcM_1 \\left( X \\right) \\to \\mcM_2 \\left( X \\right)$ is a quasi-isomorphism for every object $X$ in $\\mcC$. (See \\cite[section A.2]{GPS19} for a discussion on quasi-isomorphisms between $A_{\\infty}$-modules).\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition homotopy between morphisms of modules}\n\t\n\tLet $t, t' : \\mcM_1 \\to \\mcM_2$ be two degree $0$ closed morphisms of $\\mcC$-modules. A homotopy between $t$ and $t'$ is a $\\mcC$-module map $h : \\mcM_1 \\to \\mcM_2$ such that\n\t\\[ t + t' = \\mu^1_{\\mathrm{Mod}_{\\mcC}} \\left( h \\right) . \\]\n\t\n\\end{defin}\n\n\\begin{defin}[See \\cite{Sei08} paragraph (1l) and \\cite{GPS19} section A.1]\\label{definition Yoneda functor}\n\t \n\tThere is an $A_{\\infty}$-functor \n\t\\[ \\mcC \\to \\mathrm{Mod}_{\\mcC}, \\quad Y \\mapsto \\mcC \\left( - , Y \\right), \\]\n\tcalled the Yoneda $A_{\\infty}$-functor, defined as follows.\n\tFor every object $X$, \n\t\\[ \\mcC \\left( -,Y \\right) \\left( X \\right) = \\mcC \\left( X,Y \\right). \\]\n\tBesides, a sequence \n\t\\[ \\left( x_0 , \\dots , x_{d-1} \\right) \\in \\mcC \\left( X_0 , X_1 \\right) \\times \\dots \\times \\mcC \\left( X_{d-1} , X_d \\right) \\]\n\tacts on an element $u$ in $\\mcC \\left( X_d , Y \\right)$ via the operations \n\t\\[ \\mu_{\\mcC \\left( -,Y \\right)} \\left( x_0 , \\dots , x_{d-1} , u \\right) = \\mu_{\\mcC} \\left( x_0 , \\dots , x_{d-1} , u \\right) . \\]\n\tFinally, let \n\t\\[\\mathbf{y} = \\left( y_0 , \\dots , y_{p-1} \\right) \\in \\mcC \\left( Y_0 , Y_1 \\right) \\times \\dots \\times \\mcC \\left( Y_{p-1} , Y_p \\right) \\]\n\tbe a sequence of morphisms in $\\mcC$. Then the Yoneda functor gives a morphism of $\\mcC$-modules $t_{\\mathbf{y}} : \\mcC \\left( -,Y_0 \\right) \\to \\mcC \\left( -,Y_p \\right)$ which sends every sequence $\\left( x_0 , \\dots , x_{d-1} , u \\right)$ as above to \n\t\\[ \\mu_{\\mcC} \\left( x_0 , \\dots , x_{d-1} , u , y_0, \\dots, y_{p-1} \\right) \\in \\mcC \\left( X_0,Y_p \\right) . \\]\n\t\n\\end{defin}\n\nWe have the following important result.\n\n\\begin{prop}[Yoneda lemma]\\label{prop Yoneda lemma}\n\t\n\tThe Yoneda $A_{\\infty}$-functor \n\t\\[ \\mcC \\to \\mathrm{Mod}_{\\mcC}, \\quad Y \\mapsto \\mcC \\left( - , Y \\right) \\]\n\tis cohomologically full and faithful.\n\t\n\\end{prop}\n\n\\begin{proof}\n\t\n\tThis is Lemma 2.12 in \\cite{Sei08}, and also Lemma A.1 in \\cite{GPS19}.\n\t\n\\end{proof}\n\nThe Yoneda lemma has the following easy consequence. We state it for future reference.\n\n\\begin{coro}\\label{coro closed module map homotopic to Yoneda module map}\n\t\n\tEvery closed $\\mcC$-module map $f : \\mcC \\left( -, X \\right) \\to \\mcC \\left( -, Y \\right)$ is homotopic to the $\\mcC$-module map $t_{f \\left( e_X \\right)}$ induced by $f \\left( e_X \\right) \\in \\mcC \\left( X, Y \\right)$. (see Definition \\ref{definition Yoneda functor}).\n\t\n\\end{coro}\n\n\\begin{proof}\n\t\n\tAccording to the Yoneda lemma, $f$ is homotopic to $t_x$ for some closed $x$ in $\\mcC \\left( X , Y \\right)$. Thus, there exists a $\\mcC$-module map $h : \\mcC \\left( -, X \\right) \\to \\mcC \\left( -, Y \\right)$ such that \n\t\\[f = t_x + \\mu_{\\mathrm{Mod}_{\\mcC}}^1 \\left( h \\right) . \\]\n\tEvaluating the latter relation at the unit $e_X \\in \\mcC \\left( X, X \\right)$ gives \n\t\\[f \\left( e_X \\right) = x + \\mu_{\\mcC}^1 \\left( h e_X \\right) . \\]\n\tTherefore, $x$ is homotopic to $f \\left( e_X \\right)$, and this implies that $t_x$ is homotopic to $t_{f \\left( e_X \\right)}$ by the Yoneda lemma. Finally, we have that $f$ is homotopic to $t_{f \\left( e_X \\right)}$.\n\t\n\\end{proof}\n\n\\paragraph{Pullback of $A_{\\infty}$-modules.}\n\n\\begin{defin}[See \\cite{Sei08} paragraph (1k)]\\label{definition pullback functor}\n\t \n\tLet $\\Phi : \\mcC \\to \\mcD$ be an $A_{\\infty}$-functor. Then there is a DG-functor \n\t\\[\\Phi^* : \\mathrm{Mod}_{\\mcD} \\to \\mathrm{Mod}_{\\mcC}, \\quad \\mcN \\mapsto \\Phi^* \\mcN \\]\n\tdefined as follows.\n\tLet $\\mcN$ be a $\\mcD$-module. For every object $X$, \n\t\\[ \\Phi^* \\mcN \\left( X \\right) = \\mcN \\left( \\Phi X \\right) . \\]\n\tBesides, a sequence \n\t\\[ \\left( x_0 , \\dots , x_{d-1} \\right) \\in \\mcC \\left( X_0 , X_1 \\right) \\times \\dots \\times \\mcC \\left( X_{d-1} , X_d \\right) \\]\n\tacts on an element $u \\in \\Phi^* \\mcN \\left( X_d \\right)$ via the operations \n\t\\[ \\mu_{\\Phi^* \\mcN} \\left( x_0 , \\dots , x_{d-1} , u \\right) = \\sum \\mu_{\\mcN} \\left( \\Phi \\left( x_0 , \\dots , x_{i_1-1} \\right) , \\dots , \\Phi \\left( x_{d-i_r} , \\dots , x_{d-1} \\right) , u \\right) . \\]\n\tFinally, let $t : \\mcN_1 \\to \\mcN_2$ be a $\\mcD$-module map. Then the above functor gives a $\\mcC$-module map $\\Phi^* t : \\Phi^* \\mcN_1 \\to \\Phi^* \\mcN_2$ which sends every sequence $\\left( x_0 , \\dots , x_{d-1} , u \\right)$ as above to \n\t\\[ \\Phi^* t \\left( x_0 , \\dots , x_{d-1} , u \\right) = \\sum t \\left( \\Phi \\left( x_0 , \\dots , x_{i_1-1} \\right) , \\dots , \\Phi \\left( x_{d-i_r} , \\dots , x_{d-1} \\right) , u \\right) . \\]\n\t\n\\end{defin}\n\n\\begin{rmk}\\label{rmk composition of pullback functors}\n\t\n\tLet $\\Phi : \\mcC \\to \\mcD$ be an $A_{\\infty}$-functor, and let $\\Psi : \\mcD \\to \\mcE$ be another $A_{\\infty}$-functor towards a third $A_{\\infty}$-category $\\mcE$. Then $\\Phi^* \\circ \\Psi^* = \\left( \\Psi \\circ \\Phi \\right)^*$ as DG-functors.\n\t\n\t\n\\end{rmk}\n\n\\begin{defin}\\label{definition modules morphism induced by functor}\n\t\n\tLet $Y$ be an object of $\\mcC$, and let $\\Phi : \\mcC \\to \\mcD$ be an $A_{\\infty}$-functor.\n\tThen there is a degree $0$ closed $\\mcC$-module map $t_{\\Phi} : \\mcC \\left( - , Y \\right) \\to \\Phi^* \\mcD \\left( - , \\Phi \\left( Y \\right) \\right)$ which sends any sequence \n\t\\[ \\left( x_0 , \\dots , x_{d-1} , u \\right) \\in \\mcC \\left( X_0 , X_1 \\right) \\times \\dots \\times \\mcC \\left( X_{d-1} , X_d \\right) \\times \\mcC \\left( X_d , Y \\right) \\]\n\tto \n\t\\[t_{\\Phi} \\left( x_0 , \\dots , x_{d-1} , u \\right) = \\Phi \\left( x_0 , \\dots , x_{d-1} , u \\right) \\in \\mcD \\left( \\Phi X_0 , \\Phi Y \\right) . \\]\n\t\n\\end{defin}\n\n\\paragraph{Quotient of $A_{\\infty}$-modules.}\n\n\\begin{defin}[See \\cite{GPS20} section 3.1.3]\\label{definition localization of modules}\n\t \n\tThere is a DG-functor \n\t\\[ \\mathrm{Mod}_{\\mcC} \\to \\mathrm{Mod}_{\\mcC \/ \\mcA}, \\quad \\mcM \\, \\mapsto \\, _{\\mcA \\backslash} \\mcM \\]\n\tdefined as follows. Let $\\mcM$ be a $\\mcC$-module. For every object $X$, \n\t\\begin{align*}\n\t_{\\mcA \\backslash} \\mcM \\left( X \\right) & = \\\\\n\t& \\mcM \\left( X \\right) \\bigoplus \\left( \\bigoplus\\limits_{\\substack{ p \\geq 1 \\\\ A_1,\\dots,A_p \\in \\mcA}} \\mcC \\left( X , A_1 \\right) [1] \\otimes \\dots \\otimes \\mcC \\left( A_{p-1} , A_p \\right) [1] \\otimes \\mcM \\left( A_p \\right) \\right) .\n\t\\end{align*}\n\tBesides, a sequence\n\t\\[ \\mathbf{x}_i = \\left( x_i^0 , \\dots , x_i^{p_i-1} \\right) \\in \\mcC \/ \\mcA \\left( X_i , X_{i+1} \\right) \\quad \\left( 0 \\leq i \\leq d-1 \\right) \\]\n\tacts on an element\n\t\\[ \\mathbf{u} = \\left( x_d^0 , \\dots , x_d^{p_d-1} , u \\right) \\in \\, _{\\mcA \\backslash} \\mcM \\left( X_d \\right) \\]\n\tvia the operations \n\t\\begin{align*}\n\t\\mu_{_{\\mcA \\backslash} \\mcM} & \\left( \\mathbf{x}_0 , \\dots , \\mathbf{x}_{d-1} , \\mathbf{u} \\right) = \\\\ \n\t& \\sum\\limits_{ \\substack{ 0 \\leq i \\leq p_0, 1 \\leq j \\leq p_d \\\\ i 0. \\]\nWe denote by $\\mcU$ a small open neighborhood of $\\left( F_1, f_1 \\right)$ in $C^{\\infty} \\left( L \\right) \\times C^{\\infty} \\left( \\Lambda \\right)$ with the following property: for every pair $\\left( F, f \\right)$ in $\\mcU$, the functions $F$, $f$ are positive, Morse, and their critical points and gradient trajectories are in bijection with those of $F_1$, $f_1$ respectively. \nThen we choose a family $\\left( F_n, f_n \\right)_{n \\geq 2}$ of pairs in $\\mcU$ such that in the ends,\n\\[F_n \\left( t,q \\right) = a_n \\left( t + f_n \\left( q \\right) \\right) + b_n \\text{ for } 0 < a_n \\ll 1 \\text{ and } b_n > 0. \\]\nWe also choose a decreasing sequence $\\left( \\delta_k \\right)_{k \\geq 1}$ of small enough positive numbers so that the family\n\\[\\left( H_n = \\sum_{k=1}^{n} \\delta_k F_k, h_n = \\sum_{k=1}^{n} \\delta_k f_k \\right)_{n \\in \\mathbf{N}} \\]\nhas the following property: for every $ij$ ; the bigrading of an intersection point $c \\in \\mcO \\left( \\Lambda^i, \\Lambda^j \\right)$ corresponding to $q_0 \\in \\mathrm{Crit} \\, h_0$ is $\\left( 2 \\left( j-i \\right) + \\mathrm{ind} \\left( q_0 \\right) \\right)$, and \n\t\t\n\t\t\\item the operations are such that $e_{L_n} \\in \\mcO \\left( L_n, L_n \\right)$ is a strict unit, and for every sequence of integers $i_0 < \\dots < i_d$, for every sequence of intersection points \n\t\t\\[\\left( x_1, \\dots, x_d \\right) \\in \\left( L_{i_0} \\cap L_{i_1} \\right) \\times \\dots \\times \\left( L_{i_{d-1}} \\cap L_{i_d} \\right) \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcO} \\left( x_1, \\dots, x_d \\right) = \\sum \\limits_{x_0 \\in L_{i_0} \\cap L_{i_d}} \\# \\mcM_{x_0, x_d \\dots x_1} \\left( \\mathbf{L}, j \\right) x_0 \\]\n\t\t(see Definition \\ref{definition moduli spaces for Lagrangians}).\n\t\tBesides, let $W_L$ be the set made of the morphisms $u^n \\in \\mcO \\left( L_n, L_{n+1} \\right)$ corresponding to the minimum of $\\left( H_{n+1} - H_n \\right)$ for $n \\in \\mathbf{N}$.\n\t\tWe define\n\t\t\\[CF^* \\left( L \\right) := \\mathrm{End}_{\\mcO \\left[ W_L^{-1} \\right]} \\left( L \\right) . \\]\n\t\t\n\t\\end{enumerate} \t\n\t\n\\end{defin}\n\n\nThere are natural $A_{\\infty}$-functors $\\psi : \\mcO \\to \\mcA_{++}$ (see \\cite{EL21} section 5), $\\iota : \\mcA_{++} \\to \\mcA_+$ (the inclusion) and $\\pi : \\mcA_+ \\to \\mcA_0$ (the projection). \nThe composition $\\phi = \\pi \\circ \\iota \\circ \\psi : \\mcO \\to \\mcA_0$ sends $W_L$ to $W_0$, and thus it induces an $A_{\\infty}$-map $\\widetilde{\\phi} : CF^* \\left( L \\right) \\to LA_0^* \\left( \\Lambda \\right)$.\n\n\\begin{defin}\\label{definition relative cohomology} \n\t\n\tThe Floer complex of $L$ relative to its boundary $\\Lambda$ is the non unital $A_{\\infty}$-algebra\n\t\\[CF^* \\left( L, \\Lambda \\right) := \\mathrm{Cone} \\left( CF^* \\left( L \\right) \\xrightarrow{\\widetilde{\\phi}} LA_0^* \\left( \\Lambda \\right) \\right) \\]\n\t(see Definition \\ref{definition cone of an algebra map}). \n\t\n\\end{defin}\n\n\\section{Results}\n\nRecall that we are considering $A_{\\infty}$ functors $\\psi : \\mcO \\to \\mcA_{++}$ (as in \\cite{EL21} section 5) and $\\iota : \\mcA_{++} \\to \\mcA_+$ (the inclusion). Moreover, the composition $\\iota \\circ \\psi : \\mcO \\to \\mcA_+$ sends $W_L$ to $W_+$.\n\n\\begin{thm}\\label{thm relation Lagrangians Legendrians}\n\t(Part of Theorem 63 in \\cite{EL21})\n\t\n\tIf $HW^* \\left( L \\right) = 0$, then the $A_{\\infty}$-map\n\t\\[\\widetilde{\\iota \\circ \\psi} : CF^* \\left( L \\right) \\to LA_+^* \\left( \\Lambda \\right) \\]\n\tinduced by $\\iota \\circ \\psi$ is a quasi-isomorphism. \n\t\n\\end{thm}\n\n\\begin{proof}\n\t\n\tOn the objects, $\\psi \\left( L_n \\right) = \\Lambda_n$ and, on the morphisms, $\\psi$ is defined using the exact same count of pseudo-holomorphic curves than in \\cite{EL21} Theorem 63 (the corresponding $A_{\\infty}$-map is denoted $\\mathfrak{e}$ there). In \\cite{EL21} Theorem 63, the authors prove that for every $i < j$,\n\t\\[\\mathrm{Cone} \\left( \\mcO \\left( L_i, L_j \\right) \\xrightarrow{\\psi} \\mcA_{++} \\left( \\Lambda_i, \\Lambda_j \\right) \\right)^{\\#} \\simeq CW^* \\left( L_i, \\widehat{L_j} \\right) \\]\n\twhere $\\widehat{L_j}$ is a negative wrapping of $L_j$ across $L_i$.\n\tBy hypothesis, $CW^* \\left( L \\right)$ is acyclic, which implies that $\\psi^1$ is a quasi-isomorphism. Thus, $\\psi$ is a quasi-equivalence, and since it sends $W_L$ into $W_+^{dir}$, the induced functor $\\widetilde{\\psi} : \\mcO \\left[ W_L^{-1} \\right] \\to \\mcA_{++} \\left[ \\left( W_+^{dir} \\right)^{-1} \\right]$ is also a quasi-equivalence (see Lemma 2.4 in \\cite{GPS19}). This concludes the proof according to Lemma \\ref{lemma directed and non-directed positive augmentation categories}.\n\n\n\n\n\n\t\n\\end{proof}\n\n\\begin{thm}\n\t(Part of Theorem 63 in \\cite{EL21})\n\t\n\tIf $HW^* \\left( L \\right) = 0$, then there is a quasi-isomorphism of $A_{\\infty}$-algebras\n\t\\[\\mathbf{F} \\oplus CF^* \\left( L, \\Lambda \\right) \\xrightarrow{\\sim} LA^* \\left( \\Lambda \\right) . \\] \n\t\n\\end{thm}\n\n\\begin{proof}\n\t\n\tFirst recall that the projection to the space of short Reeb chords induces an $A_{\\infty}$-functor $\\pi : \\mcA_+ \\to \\mcA_0$ which sends $W_+$ to $W_0$, and Proposition \\ref{prop Lagrangian algebra as a cone} gives\n\t\\[\\overline{LA^*} \\left( \\Lambda \\right) \\simeq \\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right) . \\] \n\tThen, recall that the composition $\\phi = \\pi \\circ \\iota \\circ \\psi : \\mcO \\to \\mcA_0$ sends $W_L$ to $W_0$, and thus it induces an $A_{\\infty}$-map $\\widetilde{\\phi} : CF^* \\left( L \\right) \\to LA_0^* \\left( \\Lambda \\right)$. \n\tNow observe that we have the following commutative diagram\n\t\\[\\begin{tikzcd}\n\tCF^* \\left( L \\right) \\ar[r, \"\\widetilde{\\iota \\circ \\psi}\"] \\ar[d, \"\\widetilde{\\phi}\"] & LA_+^* \\left( \\Lambda, \\varepsilon_L \\right) \\ar[d, \"\\widetilde{\\pi}\"] \\\\\n\tLA_0^* \\left( \\Lambda, \\varepsilon_L \\right) \\ar[r, equal] & LA_0^* \\left( \\Lambda, \\varepsilon_L \\right) .\n\t\\end{tikzcd} \\]\n\tThe top horizontal arrow is a quasi-isomorphism by Theorem \\ref{thm relation Lagrangians Legendrians}. According to Proposition \\ref{prop invariance cone of an algebra map}, we get a quasi-isomorphism of non unital $A_{\\infty}$-algebras \n\t\\[CF^* \\left( L, \\Lambda \\right) \\xrightarrow{\\sim} \\overline{LA^*} \\left( \\Lambda \\right). \\]\n\tThis concludes the proof because \n\t\\[LA^* \\left( \\Lambda \\right) = \\mathbf{F} \\oplus \\overline{LA^*} \\left( \\Lambda \\right). \\]\n\t\n\\end{proof}\n\n\n\\begin{thm}\\label{thm dga with coefficients in chains of based loop space}\n\t(Theorem 51 in \\cite{EL21})\n\t\n\tIf $\\Lambda$ is simply connected, then \n\t\\[E \\left( CE_{-*}^+ \\left( \\Lambda \\right) \\right) \\simeq LA_+^* \\left( \\Lambda \\right) . \\]\n\t\n\\end{thm}\n\n\\begin{proof}[Idea of proof]\n\t\n\tAssume that we can choose the copies to be parallel. In \\cite{EL21} section 3.5, the authors define a quasi-isomorphism $\\phi : CE_{-*}^+ \\left( \\Lambda\\right) \\to CE_{-*}^{\\parallel} \\left( \\Lambda \\right)$. Applying first bar and then graded dual, we get a quasi-isomorphism $B \\left( CE_{-*}^{\\parallel} \\left( \\Lambda \\right) \\right)^{\\#} \\to B \\left( CE_{-*}^+ \\left( \\Lambda \\right) \\right)^{\\#}$. Besides, recall that \n\t\\[CE_{-*}^{\\parallel} \\left( \\Lambda \\right) = \\Omega LC_*^{\\parallel} \\left( \\Lambda \\right). \\]\n\tThere is a natural quasi-isomorphism $B \\left( \\Omega LC_*^{\\parallel} \\left( \\Lambda \\right) \\right) \\to LC_*^{\\parallel} \\left( \\Lambda \\right)$ (see section 2.2.2 in \\cite{EL21}), which induces a quasi-isomorphism $LA^*_{\\parallel} \\left( \\Lambda \\right) \\to B \\left( \\Omega LC_*^{\\parallel} \\left( \\Lambda \\right) \\right)^{\\#}$. As a result, we get a quasi-isomorphism \n\t\\[LA^*_{\\parallel} \\left( \\Lambda \\right) \\to B \\left( CE_{-*} \\left( \\Lambda \\right) \\right)^{\\#} . \\] \n\t\n\tNow going back to the case where the copies are not parallel, we should still be able to define an $A_{\\infty}$-functor $\\Phi : \\mcA_+ \\to B \\left( CE_{-*} \\left( \\Lambda, C_{-*} \\left( \\Omega \\Lambda \\right) \\right) \\right)^{\\#}$ which mimic the map we just described. This functor should look like this. Take sequences \n\t\\[\\left\\{\n\t\\begin{array}{ccl}\n\t\\left( c_0, \\dots, c_{d-1} \\right) & \\in & \\mcR \\left( \\Lambda_{n_0}, \\Lambda_{n_1} \\right) \\times \\dots \\times \\mcR \\left( \\Lambda_{n_{d-1}}, \\Lambda_{n_d} \\right) \\\\\n\t\\left( x_0, \\dots, x_{r-1} \\right) & \\in & CE_{-*}^+ \\left( \\Lambda \\right)^{r}\n\t\\end{array}\n\t\\right. \\]\n\tsuch that none of the Reeb chords $c_i$ correspond to the minimum of the Morse function $f_1$, and each $x_i$ is a word in elements of $C_{-*} \\left( \\Omega \\Lambda \\right) \\cup \\mcR \\left( \\Lambda \\right)$.\n\tIf $r$ is positive, then \n\t\\[\\langle \\Phi \\left( c_0, \\dots, c_{d-1} \\right), s x_0 \\otimes \\dots \\otimes s x_{r-1} \\rangle = 0 . \\]\n\tAssume now that $r=0$. Let $\\left( \\mathbf{y}_1, \\dots, \\mathbf{y}_p \\right)$ be the family of short Reeb chords appearing in $c_0 \\cdots c_{d-1}$.\n\tThen \n\t\\[\t\\langle \\Phi \\left( c_0, \\dots, c_{d-1} \\right), s x_0 \\rangle = \\prod_{i=1}^{p} \\vert \\mcT \\left( \\sigma_i, \\mathbf{y}_i \\right) \\vert \\]\n\t(the moduli spaces $\\mcT \\left( \\sigma_i, \\mathbf{y}_i \\right)$ are defined in \\cite{EL21} section 3.5) if $x_0$ is obtained from $c_0 \\cdots c_{d-1}$ by replacing each $\\mathbf{y}_i$ by $\\sigma_i$, and $0$ otherwise. \n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\t\n\tIt remains to show that $\\Phi$ sends $W_+$ to the unit (this should follow from Lemma 38 in \\cite{EL21}), and that the induced functor $\\widetilde{\\Phi} : \\mcA_+ \\left[ W_+^{-1} \\right] \\to B \\left( CE_{-*}^+ \\left( \\Lambda \\right) \\right)^{\\#} \\left[ \\mathbf{1}^{-1} \\right]$ is a quasi-equivalence. \n\t\n\\end{proof}\n\n\n\n\\section{Extended Legendrian $A_{\\infty}$-algebra}\\label{subsection positive augmentation category}\n\nIn this appendix, we study the endomorphism $A_{\\infty}$-algebra $LA_+^* \\left( \\Lambda \\right)$ of an augmentation $\\varepsilon : CE_{-*} \\left( \\Lambda \\right) \\to \\mathbf{F}$ in the positive augmentation category $\\mcA ug_+ \\left( \\Lambda \\right)$ defined by Chantraine in \\cite{Cha19} (it was first introduced by Ng, Rutherford, Shende, Sivek and Zaslow in \\cite{NRS+20} for Legendrians in $\\mathbf{R}^3$).\nThe goal is to prove Proposition \\ref{prop Lagrangian algebra as a cone}, which shows that there is a quasi-isomorphism of non-unital $A_{\\infty}$-algebras between $\\overline{LA^* \\left( \\Lambda \\right)}$ (which is the endomorphism algebra of $\\varepsilon$ in the augmentation category $\\mcA ug_- \\left( \\Lambda \\right)$ defined in \\cite{BC14}) and the cone of an $A_{\\infty}$-map from $LA_+^* \\left( \\Lambda \\right)$ to some $A_{\\infty}$-algebra $LA_0^* \\left( \\Lambda \\right)$ generated by Morse type generators.\n\n\\subsection{Setting}\n\nLet $(V, \\xi)$ be a contact manifold of dimension $(2n+1)$ equipped with a hypertight contact form $\\alpha$, and let $\\Lambda$ be a connected compact Legendrian submanifold in $\\left( V, \\xi \\right)$ which is chord generic with respect to $\\alpha$ (see section \\ref{subsection Legendrian invariants}).\nIn the following we fix an augmentation $\\varepsilon : CE_{-*} \\left( \\Lambda \\right) \\to \\mathbf{F}$. \n\nChoose a positive Morse function $f_1 : \\Lambda \\to \\mathbf{R}$ with a unique minimum.\nWe denote by $\\mcU$ a small open neighborhood of $f_1$ in $C^{\\infty} \\left( \\Lambda \\right)$ with the following property: for every $f$ in $\\mcU$, $f$ is positive, Morse, and its critical points and gradient trajectories are in bijection with those of $f_1$. \nThen we choose a family $\\left( f_n \\right)_{n \\geq 2}$ of functions in $\\mcU$.\nWe also choose a decreasing sequence $\\left( \\delta_k \\right)_{k \\geq 1}$ of small enough positive numbers so that the family\n\\[h_n = \\sum_{k=1}^{n} \\delta_k f_k, \\quad n \\in \\mathbf{N}, \\]\nhas the following property: for every $i j$. \n\t\t\n\t\t\\item the operations are such that $e_{\\Lambda_n} \\in \\mcA_{\\triangle} \\left( \\Lambda_n, \\Lambda_n \\right)$ is a strict unit, and for every sequence of generators \n\t\t\\[\\left( x_1, \\dots, x_d \\right) \\in \\mcA_{\\triangle} \\left( \\Lambda_{i_0}, \\Lambda_{i_1} \\right) \\times \\dots \\times \\mcA_{\\triangle} \\left( \\Lambda_{i_{d-1}}, \\Lambda_{i_d} \\right), \\]\n\t\twe have \n\t\t\\begin{align*}\n\t\t\\mu_{\\mcA_{\\triangle}} & \\left( x_1, \\dots, x_d \\right) \\\\ \n\t\t& = \\sum \\limits_{\\substack{ a, \\boldsymbol{\\gamma_0}, \\dots, \\boldsymbol{\\gamma_d} }} \\# \\mcM_{a, \\boldsymbol{\\gamma_d} x_d \\boldsymbol{\\gamma_{d-1}} x_{d-1} \\dots x_1 \\boldsymbol{\\gamma_0}} \\left( \\mathbf{R} \\times \\mathbf{\\Lambda}, J \\right) \\varepsilon_{i_0} \\left( \\boldsymbol{\\gamma_0} \\right) \\dots \\varepsilon_{i_d} \\left( \\boldsymbol{\\gamma_d} \\right) a ,\n\t\t\\end{align*}\n\t\twhere in the sum, $a$ is a generator of $\\mcA_{\\triangle} \\left( \\Lambda_{i_0}, \\Lambda_{i_d} \\right)$ and $\\boldsymbol{\\gamma_j}$ is a word of Reeb chords in $\\mcR \\left( \\Lambda_{i_j}, \\Lambda_{i_j} \\right)$\n\t\t(see Definition \\ref{definition moduli spaces} for the moduli spaces).\n\t\t\n\t\\end{enumerate}\n\tMoreover, we let $W_{\\triangle}$ be the set made of the generators $c_n \\in \\mcA_{\\triangle} \\left( \\Lambda_n, \\Lambda_{n+1} \\right)$ corresponding to the minimum of $\\left( h_{n+1} - h_n \\right)$ for $n \\in \\mathbf{N}$.\n\tWe define\n\t\\[LA_{\\triangle}^* \\left( \\Lambda \\right) := \\mathrm{End}_{\\mcA_{\\triangle} \\left[ W_{\\triangle}^{-1} \\right]} \\left( \\Lambda \\right) . \\]\n\t\n\\end{defin}\n\n\\begin{lemma}\\label{lemma multiplication by the minimum in the different versions}\n\t\n\tFor every integer $j$, for every $\\triangle \\in \\{ ++, +, 0 \\}$ the morphism $c_j \\in \\mcA_{\\triangle} \\left( \\Lambda_j, \\Lambda_{j+1} \\right) \\cap W_{\\triangle}$ satisfies the following. For every $i < j < k$, the chain maps\n\t\\[\\left\\{\n\t\\begin{array}{ccccl}\n\t\\mu_{\\mcA_{\\triangle}}^2 \\left( - , c_j \\right) & : & \\mcA_{\\triangle} \\left( \\Lambda_i , \\Lambda_j \\right) & \\to & \\mcA_{\\triangle} \\left( \\Lambda_i , \\Lambda_{j+1} \\right) \\\\\n\t\\mu_{\\mcA_{\\triangle}}^2 \\left( c_j , - \\right) & : & \\mcA_{\\triangle} \\left( \\Lambda_{j+1}, \\Lambda_{k+1} \\right) & \\to & \\mcA_{\\triangle} \\left( \\Lambda_j, \\Lambda_{k+1} \\right)\n\t\\end{array}\n\t\\right. \\]\n\tare quasi-isomorphisms. \n\tMoreover, if we denote by $\\mcA_{long} \\left( \\Lambda_m, \\Lambda_n \\right)$ the subcomplex of $\\mcA_+ \\left( \\Lambda_m, \\Lambda_n \\right)$ generated by $\\mcR_{long} \\left( \\Lambda_m, \\Lambda_n \\right)$, then the induced chain maps \n\t\\[\\left\\{\n\t\\begin{array}{ccccl}\n\t\\mu_{\\mcA_+}^2 \\left( - , c_j \\right) & : & \\mcA_{long} \\left( \\Lambda_i , \\Lambda_j \\right) & \\to & \\mcA_{long} \\left( \\Lambda_i , \\Lambda_{j+1} \\right) \\\\\n\t\\mu_{\\mcA_+}^2 \\left( c_j , - \\right) & : & \\mcA_{long} \\left( \\Lambda_{j+1}, \\Lambda_{k+1} \\right) & \\to & \\mcA_{long} \\left( \\Lambda_j, \\Lambda_{k+1} \\right)\n\t\\end{array}\n\t\\right. \\]\n\talso are quasi-isomorphisms. \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWhen the contact manifold is $V = \\mathbf{R} \\times P$, the result follows from the main analytic theorem of \\cite{EES09} (Theorem 3.6). Indeed, this result implies that (when the perturbation functions $h_n$ are small enough) the 3-punctured pseudo-holomorphic discs with boundary on $\\mathbf{R} \\times \\bigsqcup_n \\Lambda_n$, asymptotic to $c_j$ at one of their punctures, correspond to trivial cylinders with boundary on $\\mathbf{R} \\times \\Lambda$.\n\t\n\tThe general case is part of a work in progress of Chantraine, Dimitroglou-Rizell and Ghiggini. \n\t\n\\end{proof}\n\n\n\\subsection{Relations between $A_{\\infty}$-categories and $A_{\\infty}$-algebras}\n\nObserve that the inclusion $\\iota : \\mcA_{++} \\to \\mcA_+$ defines a strict $A_{\\infty}$-functor which sends $W_{++}$ to $W_+$.\n\n\n\\begin{lemma}\\label{lemma directed and non-directed positive augmentation categories}\n\t\n\tThe $A_{\\infty}$-functor\n\t\\[\\widetilde{\\iota} : \\mcA_{++} \\left[ W_{++}^{-1} \\right] \\xrightarrow{\\sim} \\mcA_+ \\left[ W_+^{-1} \\right] \\]\n\tinduced by $\\iota$ (see Definition \\ref{definition induced functor between quotients}) is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tFix an object $Y = \\Lambda_m$ in $\\mcA_{++}$. We want to prove that the map \n\t\\[\\widetilde{\\iota} : \\mcA_{++} \\left[ W_{++}^{-1} \\right] \\left( X, Y \\right) \\to \\mcA_+ \\left[ W_+ ^{-1} \\right] \\left( \\widetilde{\\iota} X, \\widetilde{\\iota} Y \\right) \\]\n\tis a quasi-isomorphism for every object $X$. \n\tOur strategy is to apply Proposition \\ref{prop quasi-isomorphism between localizations}. We consider the modules $\\mcM_{\\mcA_+}$, $\\mcM_{\\mcA_{++}}$ and module maps $t_{\\mcA_{++}} : \\mcA_{++} \\left( -, \\Lambda_m \\right) \\to \\mcM_{\\mcA_{++}}$, $t_{\\mcA_+} : \\mcA_+ \\left( -, \\Lambda_m \\right) \\to \\mcM_{\\mcA_+}$ of Definition \\ref{definition specific module}. \n\tObserve that the diagram of $\\mcA_{++}$-modules \n\t\\[\\begin{tikzcd}\n\t\\bigoplus \\limits_{n \\geq 0} \\mcA_{++} \\left( -, \\Lambda_n \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[r, \"\\bigoplus \\limits_{n \\geq 0} t_{c_n}\"] & \\bigoplus \\limits_{n \\geq 0} \\mcA_{++} \\left( -, \\Lambda_{n+1} \\right) \\ar[d, \"t_{\\iota}\"] \\\\\n\t\\bigoplus \\limits_{n \\geq 0} \\mcA_{++} \\left( -, \\Lambda_n \\right) \\ar[r, \"t_{\\iota}\"] & \\iota^* \\mcM_{\\mcA_+}\n\t\\end{tikzcd} \\]\n\tcommutes up to the homotopy $h : \\bigoplus \\limits_{n \\geq 0} \\mcA_{++} \\left( -, \\Lambda_n \\right) \\to \\iota^* \\mcM_{\\mcA_+}$ which is the strict module map sending $x \\in \\mcA_{++} \\left( \\Lambda_k, \\Lambda_n \\right)$ to \n\t\\[h(x) = sx \\in \\mcA_+ \\left( \\Lambda_k, \\Lambda_n \\right) \\left[ 1 \\right] \\subset \\iota^* \\mcM_{\\mcA_+} \\left( \\Lambda_k \\right) \\]\n\t($s : \\mcA_+ \\left( \\Lambda_k, \\Lambda_n \\right) \\to \\mcA_+ \\left( \\Lambda_k, \\Lambda_n \\right) \\left[ 1 \\right]$ is the usual degree $(-1)$ map).\n\tAccording to Proposition \\ref{prop morphism induced by homotopy}, this induces a closed $\\mcA_{++}$-module map $t_0 : \\mcM_{\\mcA_{++}} \\to \\iota^* \\mcM_{\\mcA_+}$. Moreover, the following diagram of $\\mcA_{++}$-modules is commutative\n\t\\[\\begin{tikzcd}\n\t\\mcA_{++} \\left( - , Y \\right) \\ar[d, \"t_{\\mcA_{++}}\"] \\ar[r, \"t_{\\iota}\"] & \\iota^* \\mcA_+ \\left( - , \\iota Y \\right) \\ar[d, \"\\iota^* t_{\\mcA_+}\"] \\\\\n\t\\mcM_{\\mcA_{++}} \\ar[r, \"t_0\"] & \\iota^* \\mcM_{\\mcA_+} .\n\t\\end{tikzcd} \\] \n\t\n\tIt remains to check the three items of Proposition \\ref{prop quasi-isomorphism between localizations}. The first two items are satisfied according to Lemmas \\ref{lemma multiplication by the minimum in the different versions} and \\ref{lemma second result for specific module}. We now have to prove that $t_0 : \\mcM_{\\mcA_{++}} \\to \\iota^* \\mcM_{\\mcA_+}$ is a quasi-isomorphism. Let $X = \\Lambda_k$ be some object in $\\mcA_{++}$. Observe that the following diagram of chain complexes is commutative \n\t\\[\\begin{tikzcd}\n\t\\mcM_{\\mcA_{++}} \\left( \\Lambda_k \\right) \\ar[r, \"t_0\"] & \\iota^* \\mcM_{\\mcA_+} \\left( \\Lambda_k \\right) = \\mcM_{\\mcA_+} \\left( \\Lambda_k \\right) \\\\\n\t\\mcA_{++} \\left( \\Lambda_k, \\Lambda_{k+1} \\right) \\ar[u, hook] \\ar[r, \"\\iota\"] & \\mcA_+ \\left( \\Lambda_k, \\Lambda_{k+1} \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tEach vertical map is a quasi-isomorphism according to Lemma \\ref{lemma first result for specific module}, and the bottom horizontal map is the identity map. This implies that $t_0$ is a quasi-isomorphism, which is what we needed to prove.\n\t\n\\end{proof}\n\nObserve that the projection $\\pi : \\mcA_+ \\to \\mcA_0$ defines a strict $A_{\\infty}$-functor which sends $W_+$ to $W_0$. Thus it induces a strict $A_{\\infty}$-functor $\\widetilde{\\pi} : \\mcA_+ \\left[ W_+^{-1} \\right] \\to \\mcA_0 \\left[ W_0^{-1} \\right]$ (see Definition \\ref{definition induced functor between quotients}). \nBesides, let $\\overline{LA^*} \\left( \\Lambda \\right)$ be the augmented ideal of \n\\[LA^* \\left( \\Lambda \\right) = \\mathrm{End}_{\\mcA_+} \\left( \\Lambda \\right) \\]\n($\\overline{LA^*} \\left( \\Lambda \\right)$ is the endomorphism algebra of $\\varepsilon$ in the $A_{\\infty}$-category $\\mcA ug_- \\left( \\Lambda \\right)$ defined in \\cite{BC14}, see subsection \\ref{subsection Chekanov-Eliashberg algebra}). The inclusion $\\overline{LA^*} \\left( \\Lambda \\right) \\hookrightarrow \\mcA_+ \\left( \\Lambda, \\Lambda \\right)$ induces a strict $A_{\\infty}$-map $\\overline{LA^*} \\left( \\Lambda \\right) \\to LA_+^* \\left( \\Lambda \\right)$ whose image is contained in $\\ker \\left( \\widetilde{\\pi} \\right)$. Since $\\ker \\left( \\widetilde{\\pi} \\right)$ is an $A_{\\infty}$-subalgebra of $\\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right)$ (see Definition \\ref{definition cone of an algebra map}), we get a strict $A_{\\infty}$-map \n\\[\\phi : \\overline{LA^*} \\left( \\Lambda \\right) \\to \\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right) . \\]\n\n\\begin{prop}\\label{prop Lagrangian algebra as a cone}\n\t\n\tThe $A_{\\infty}$-map\n\t\\[\\phi : \\overline{LA^*} \\left( \\Lambda \\right) \\to \\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right) \\]\n\tis a quasi-isomorphism. \n\t\n\\end{prop}\n\n\\begin{proof}\n\t\n\tWe will apply Proposition \\ref{prop quasi-isomorphism between localizations} to the $A_{\\infty}$-functor $\\pi : \\mcA_+ \\to \\mcA_0$ and the object $\\Lambda_0 = \\Lambda$ of $\\mcA_+$. \n\tWe consider the modules $\\mcM_{\\mcA_0}$, $\\mcM_{\\mcA_+}$ and module maps $t_{\\mcA_+} : \\mcA_+ \\left( -, \\Lambda \\right) \\to \\mcM_{\\mcA_+}$, $t_{\\mcA_0} : \\mcA_0 \\left( -, \\Lambda \\right) \\to \\mcM_{\\mcA_0}$ of Definition \\ref{definition specific module}. \n\tObserve that the diagram of $\\mcA_+$-modules \n\t\\[\\begin{tikzcd}\n\t\\bigoplus \\limits_{n \\geq 0} \\mcA_+ \\left( -, \\Lambda_n \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[r, \"\\bigoplus \\limits_{n \\geq 0} t_{c_n}\"] & \\bigoplus \\limits_{n \\geq 0} \\mcA_+ \\left( -, \\Lambda_{n+1} \\right) \\ar[d, \"t_{\\pi}\"] \\\\\n\t\\bigoplus \\limits_{n \\geq 0} \\mcA_+ \\left( -, \\Lambda_n \\right) \\ar[r, \"t_{\\pi}\"] & \\pi^* \\mcM_{\\mcA_0}\n\t\\end{tikzcd} \\]\n\tcommutes up to the homotopy $h : \\bigoplus \\limits_{n \\geq 0} \\mcA_+ \\left( -, \\Lambda_n \\right) \\to \\pi^* \\mcM_{\\mcA_0}$ which is the strict module map sending $x \\in \\mcA_+ \\left( \\Lambda_k, \\Lambda_n \\right)$ to \n\t\\[h(x) = s \\pi \\left( x \\right) \\in \\mcA_0 \\left( \\Lambda_k, \\Lambda_n \\right) \\left[ 1 \\right] \\subset \\pi^* \\mcM_{\\mcA_0} \\left( \\Lambda_k \\right) \\]\n\t($s : \\mcA_0 \\left( \\Lambda_k, \\Lambda_n \\right) \\to \\mcA_0 \\left( \\Lambda_k, \\Lambda_n \\right) \\left[ 1 \\right]$ is the usual degree $(-1)$ map).\n\tAccording to Proposition \\ref{prop morphism induced by homotopy}, this induces a closed $\\mcA_+$-module map $t_0 : \\mcM_{\\mcA_+} \\to \\pi^* \\mcM_{\\mcA_0}$. Moreover, the following diagram of $\\mcA_+$-modules is commutative\n\t\\[\\begin{tikzcd}\n\t\\mcA_+ \\left( - , \\Lambda \\right) \\ar[d, \"t_{\\mcA_+}\"] \\ar[r, \"t_{\\pi}\"] & \\pi^* \\mcA_0 \\left( - , \\Lambda \\right) \\ar[d, \"\\pi^* t_{\\mcA_0}\"] \\\\\n\t\\mcM_{\\mcA_+} \\ar[r, \"t_0\"] & \\pi^* \\mcM_{\\mcA_0} .\n\t\\end{tikzcd} \\] \n\t\n\tNow according to Proposition \\ref{prop quasi-isomorphism between localizations}, there is a chain map $u : \\, _{W_+^{-1}} \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\to \\, _{W_0^{-1}} \\mcM_{\\mcA_0} \\left( \\Lambda \\right)$ such that the following diagram of chain complexes commutes\n\t\\[\\begin{tikzcd}\n\t\\mcA_+ \\left[ W_+^{-1} \\right] \\left( \\Lambda , \\Lambda \\right) \\ar[d, \"_{W_+^{-1}} t_{\\mcA_+}\"] \\ar[r, \"\\widetilde{\\pi}\"] & \\mcA_0 \\left[ W_0^{-1} \\right] \\left( \\Lambda , \\Lambda \\right) \\ar[d, \"_{W_0^{-1}} t_{\\mcA_0}\"] \\\\\n\t_{W_+^{-1}} \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\ar[r, \"u\"] & _{W_0^{-1}} \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\\\\n\t\\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\ar[u, hook] \\ar[r, \"t_0\"] & \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\ar[u, hook]\n\t\\end{tikzcd} \\]\n\t(the two vertical maps on the bottom are the inclusions). \n\tMoreover, the inclusions \n\t\\[\\overline{LA^*} \\left( \\Lambda \\right) \\hookrightarrow LA_+^* \\left( \\Lambda \\right), \\, \\overline{LA^*} \\left( \\Lambda \\right) \\hookrightarrow \\, _{W_+^{-1}} \\mcM_{\\mcA_+} \\left( \\Lambda \\right), \\, \\overline{LA^*} \\left( \\Lambda \\right) \\hookrightarrow \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\]\n\tmake the following diagram commutes \n\t\\[\\begin{tikzcd}\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r, hook] \\ar[d, equal] & LA_+^* \\left( \\Lambda \\right) \\ar[d, \"_{W_+^{-1}} t_{\\mcA_+}\"] \\ar[r, \"\\widetilde{\\pi}\"] & LA_0^* \\left( \\Lambda \\right) \\ar[d, \"_{W_0^{-1}} t_{\\mcA_0}\"] \\\\\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r, hook] \\ar[d, equal] & _{W_+^{-1}} \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\ar[r, \"u\"] & _{W_0^{-1}} \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\\\\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r, hook] & \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\ar[u, hook] \\ar[r, \"t_0\"] & \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tAccording to Lemma \\ref{lemma multiplication by the minimum in the different versions}, $\\mcA_+$ and $\\mcA_0$ satisfy the hypotheses of Lemma \\ref{lemma second result for specific module}. Thus, using Lemma \\ref{lemma second result for specific module} and Proposition \\ref{prop quasi-iso between a module and its localization}, we know that the vertical maps are quasi-isomorphisms. Therefore, we get the following commutative diagram of chain complexes\n\t\\[\\begin{tikzcd}\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r, \"\\phi\"] \\ar[d, equal] & \\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right) \\ar[d, \"\\sim\"] \\\\\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r] \\ar[d, equal] & \\mathrm{Cone} \\left( _{W_+^{-1}} \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\xrightarrow{u} \\, _{W_0^{-1}} \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\right) \\\\\n\t\\overline{LA^*} \\left( \\Lambda \\right) \\ar[r, \"\\psi\"] & \\mathrm{Cone} \\left( \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\xrightarrow{t_0} \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\right) \\ar[u, \"\\sim\" right] .\n\t\\end{tikzcd} \\]\n\tTherefore, in order to show that \n\t\\[\\phi : \\overline{LA^*} \\left( \\Lambda \\right) \\to \\mathrm{Cone} \\left( LA_+^* \\left( \\Lambda \\right) \\xrightarrow{\\widetilde{\\pi}} LA_0^* \\left( \\Lambda \\right) \\right) \\]\n\tis a quasi-isomorphism, it suffices to show that \n\t\\[\\psi : \\overline{LA^*} \\left( \\Lambda \\right) \\to \\mathrm{Cone} \\left( \\mcM_{\\mcA_+} \\left( \\Lambda \\right) \\xrightarrow{t_0} \\mcM_{\\mcA_0} \\left( \\Lambda \\right) \\right) \\]\n\tis a quasi-isomorphism. Observe that the inclusion $\\ker \\left( t_0 \\right) \\hookrightarrow \\mathrm{Cone} (t_0)$ is a quasi-isomorphism because $t_0$ is surjective. Thus it suffices to show that the inclusion $\\overline{LA^*} \\left( \\Lambda \\right) \\hookrightarrow \\ker \\left( t_0 \\right)$ is a quasi-isomorphism.\n\tRecall that we denote by $\\mcA_{long} \\left( \\Lambda_m, \\Lambda_n \\right)$ the subcomplex of $\\mcA_+ \\left( \\Lambda_m, \\Lambda_n \\right)$ generated by $\\mcR_{long} \\left( \\Lambda_m, \\Lambda_n \\right)$. Then observe that $\\overline{LA^*} \\left( \\Lambda \\right) = \\mcA_{long} \\left( \\Lambda, \\Lambda \\right)$ and \n\t\\[\\ker \\left( t_0 \\right) = \\left[\n\t\\begin{tikzcd}\n\t\\mcA_{long} \\left( \\Lambda , \\Lambda_0 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_0}\"] & \\mcA_{long} \\left( \\Lambda , \\Lambda_1 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_1}\"] & \\dots \\\\\n\t\\mcA_{long} \\left( \\Lambda , \\Lambda_0 \\right) & \\mcA_{long} \\left( \\Lambda , \\Lambda_1 \\right) & \\dots\n\t\\end{tikzcd} \n\t\\right] . \\]\n\tThus the result follows from Lemmas \\ref{lemma multiplication by the minimum in the different versions} and \\ref{lemma first result for specific module}. \n\t\n\n\\end{proof}\n\\section*{Introduction}\n\\addcontentsline{toc}{section}{Introduction} \n\nLegendrian contact homology was introduced by Chekanov \\cite{Che02} and Eliashberg \\cite{Eli98}, and it fits into the Symplectic Field Theory introduced in \\cite{EGH00}. It has been rigorously defined in the contactization of a Liouville manifold in \\cite{EES07}, following \\cite{EES05}. The importance of Legendrian contact homology goes beyond its applications to the Legendrian isotopy problem: for example, it was used by Bourgeois, Ekholm and Eliashberg in \\cite{BEE12} to compute symplectic invariants of Weinstein manifolds, and in a different way by Chantraine, Dimitroglou Rizell, Ghiggini and Golovko in \\cite{CDGG17} to prove a generation result for the wrapped Fukaya category of Weinstein manifolds. \n\nThe motivation for this paper is the study of Legendrian contact homology in subcritically fillable and Boothby-Wang contact manifolds, the latter being named after \\cite{BW58}. This has been done combinatorially in dimension three by Ekholm and Ng in \\cite{EN15} for the subcritically fillable case, and by Sabloff in \\cite{Sab03} for the Boothby-Wang case. The importance of the first kind of manifolds comes from the fact that every Weinstein manifold is obtained from a subcritical Weinstein manifold (of the form $\\mathbf{C} \\times P$ for some Weinstein manifold $P$) by attaching handles along Legendrian submanifolds in its boundary at infinity. The importance of the second kind of manifolds comes from a theorem of Donaldson in \\cite{Don96}, which states that any integral symplectic manifold $\\left( X, \\omega \\right)$ admits a symplectic submanifold $D \\subset X$ of codimension $2$, such that $X \\setminus D$ is a Liouville manifold whose boundary at infinity is a Boothby-Wang contact manifold. \nThe first step before attacking both cases presented above is to study Legendrian contact homology in the circular contactization of a Liouville manifold. In fact, both subcritically fillable and Boothby-Wang contact manifolds can be seen as compactifications of such spaces. This paper links together the Fukaya $A_{\\infty}$-category of a family of connected compact exact Lagrangians in a Liouville manifold $(P, \\lambda)$, and the Chekanov-Eliashberg DG-category of Legendrian lifts in the circular contactization $\\left( S^1 \\times P, \\ker ( d \\theta - \\lambda ) \\right)$.\n\nThe strategy we follow is to lift the situation to the usual contactization $\\mathbf{R} \\times P$ which has been much more studied. This naturally leads to consider an $A_{\\infty}$-category whose objects are the lifts in $\\mathbf{R} \\times P$ of our starting Legendrians, and morphisms spaces are generated by Reeb chords. Moreover, the deck transformations of the cover $\\mathbf{R} \\to S^1$ induce an $A_{\\infty}$-autoequivalence of this category. \nThe rest of the proof has two main ingredients: \n\\begin{enumerate}\n\t\n\t\\item Functorial properties of the Legendrian invariants, which are used to bring us in a situation where we can apply the correspondence result of \\cite{DR16} between discs in the symplectization $\\mathbf{R} \\times \\mathbf{R} \\times P$ and polygons in $P$.\n\t\n\t\\item Two algebraic results of independent interest about mapping tori of $A_{\\infty}$-autoequivalences, that allow us to bridge the gaps between the algebraic invariants we are interested in. \n\t\n\\end{enumerate} \nWe now proceed to describe the organization of the paper and state our main results.\n\n\\paragraph{Algebra.} \n\nIn section \\ref{subsection algebra}, we briefly recall the definitions of $A_{\\infty}$-(co)categories and give references for standard notions that we do not recall, such as (co)bar, graded dual and Koszul dual constructions.\nOn the other hand, we discuss in some details the notions of modules over $A_{\\infty}$-categories, as well as Grothendieck construction and homotopy pushout associated to a diagram of $A_{\\infty}$-categories following \\cite[section A.4]{GPS19}.\nWe use it to introduce the notion of ``cylinder object for an $A_{\\infty}$-category'', which is supposed to mimic the corresponding notion in homotopy theory. \n\n\\paragraph{Mapping torus of an $A_{\\infty}$-autoequivalence.} \n\nIn section \\ref{subsection mapping torus}\\footnote{In this section, $A_{\\infty}$-categories are always assumed to be \\emph{strictly unital} (see \\cite[paragraph (2a)]{Sei08}).}, we define the mapping torus associated to a quasi-autoequivalence $\\tau$ of an $A_{\\infty}$-category $\\mcA$ as the $A_{\\infty}$-category\n\\[\\mathrm{MT} (\\tau) := \\mathrm{hocolim} \\left( \n\\begin{tikzcd}\n\t\\mcA \\sqcup \\mcA \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\mathrm{id} \\sqcup \\mathrm{id}\" left] & \\mcA \\\\\n\t\\mcA .\n\\end{tikzcd} \\right). \\]\nObserve that this terminology was also used in \\cite{Kar21}, but we do not know if the two notions coincide. When considering an $A_{\\infty}$-autoequivalence $\\tau : \\mcA \\to \\mcA$, we always assume that $\\mcA$ is equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} (\\mcA)$ compatible with $\\tau$, which is a bijection \n\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA \\right), \\quad \\left( n, E \\right) \\mapsto X^n \\left( E \\right) \\]\nsuch that $\\tau \\left( X^n \\left( E \\right) \\right) = X^{n+1} \\left( E \\right)$ for every $n \\in \\mathbf{Z}$ and $E \\in \\mcE$ (see Definition \\ref{definition group-action}).\nThis naturally turns $\\mcA$ into an Adams-graded $A_{\\infty}$-category, where the Adams-degree of a morphism in $\\mcA (X^i (E), X^j (E'))$ is defined to be $(j-i)$.\nIt then follows that the mapping torus of $\\tau$ is also Adams-graded.\n\nSection \\ref{subsection mapping torus} contains two results about mapping tori of $A_{\\infty}$-autoequivalences: we choose to state only the most important in this introduction.\nWe denote by $\\mathbf{F} \\left[ t_m \\right]$ the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $(m, 1)$.\nObserve that if $\\mcC$ is a subcategory of an $A_{\\infty}$-category $\\mcD$ with $\\mathrm{ob} (\\mcC) = \\mathrm{ob} (\\mcD)$, then $\\mcC \\oplus (t_m \\mathbf{F} [t_m] \\otimes \\mcD)$ is naturally an Adams-graded $A_{\\infty}$-category, where the Adams degree of $t_m^k \\otimes x$ equals $k$.\nBesides, if $\\mcC$ is an $A_{\\infty}$-category equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcC \\right)$, we denote by $\\mcC^0$ the full $A_{\\infty}$-subcategory of $\\mcC$ whose set of objects corresponds to $\\{ 0 \\} \\times \\mcE$.\nFinally, we use the functor $\\mcC \\mapsto \\mcC_m$ of Definition \\ref{definition forgetful functor}. \n\n\\begin{thmintro}\\label{thm mapping torus in weak situation introduction}\n\t\n\tLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$, weakly directed with respect to some compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\n\tAssume that there exists a closed degree $0$ bimodule map $f : \\mcA_m \\left( -, - \\right) \\to \\mcA_m \\left( -, \\tau(-) \\right)$ such that $f : \\mcA_m \\left( X^i(E) , X^j(E') \\right) \\to \\mcA_m \\left( X^i(E), X^{j+1} (E') \\right)$ is a quasi-isomorphism for every $i < j$ and $E, E' \\in \\mcE$.\n\tThen there is a quasi-equivalence of Adams-graded $A_{\\infty}$-categories \n\t\\[\\mathrm{MT} (\\tau) \\simeq \\mcA_m^0 \\oplus \\left( t_m \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ f \\left( \\mathrm{units} \\right)^{-1} \\right]^0 \\right) . \\]\n\t\n\\end{thmintro}\n\n\\begin{rmksintro}\n\t\n\t\\begin{enumerate}\n\t\t\n\t\t\\item In \\cite{Gan13}, the chain complex of $\\mcA$-bimodule maps from the diagonal bimodule $\\mcA \\left( -, - \\right)$ to some $\\mcA$-bimodule $\\mcB$ is called the two-pointed complex for Hochschild cohomology of $\\mcA$ with coefficients in $\\mcB$. According to \\cite[Proposition 2.5]{Gan13}, this complex is quasi-isomorphic to the (ordinary) Hochschild cochain complex of $\\mcA$ with coefficients in $\\mcB$. In particular, the bimodule map $f$ in Theorem \\ref{thm mapping torus in weak situation introduction} defines a class in the Hochschild cohomology of $\\mcA_m$ with coefficients in $\\mcA_m \\left( -, \\tau (-) \\right)$.\n\t\t\n\t\t\\item The $A_{\\infty}$-category which computes the mapping torus in Theorem \\ref{thm mapping torus in weak situation introduction} is very similar to the categories studied in \\cite{Sei08bis}, with main difference the presence of curvature in Seidel's setting.\n\t\t\n\t\t\\item The use of the functor $\\mcC \\mapsto \\mcC_m$ in Theorem \\ref{thm mapping torus in weak situation introduction} is not of any deep importance. It was convenient for us to introduce it here for our application to Legendrian contact homology (see Theorem \\ref{thm mainthm} below).\n\t\t\n\t\\end{enumerate}\n\t\n\\end{rmksintro}\n\n\\paragraph{Chekanov-Eliashberg DG-algebra.} \n\nIn section \\ref{section Legendrian invariants}, we recall the definition and functorial properties of the Chekanov-Eliashberg DG-category associated to a family of Legendrians in a hypertight contact manifold.\n\n\\paragraph{Legendrian lifts of exact Lagrangians in the circular contactization.} \n\nIn section \\ref{subsection proof of main thm}, we start with a family \n\\[\\mathbf{L} = \\left( L(E) \\right)_{E \\in \\mcE}, \\quad \\mcE = \\left\\{ 1, \\dots, N \\right\\}, \\]\nof mutually transverse compact connected exact Lagrangian submanifolds in a Liouville manifold $\\left( P, \\lambda \\right)$, and we study a Legendrian lift of $\\mathbf{L}$ in the circular contactization $\\left( S^1 \\times P, \\ker (d \\theta - \\lambda) \\right)$.\nMore precisely, we assume\\footnote{This can always be achieved by applying the Liouville flow in backwards time.} that there are primitives $f_E : L(E) \\to \\mathbf{R}$ of $\\lambda_{| L(E)}$ such that $0 \\leq f_1 < \\dots < f_N \\leq 1\/2$,\nand we consider the family of Legendrians\n\\[\\mathbf{\\Lambda^{\\circ}} := (\\Lambda^{\\circ}(E))_{E \\in \\mcE}, \\text{ where } \\Lambda^{\\circ}(E) = \\left\\{ (f_E(x), x) \\in (\\mathbf{R} \/ \\mathbf{Z}) \\times P \\mid x \\in L(E) \\right\\} . \\]\nWe denote by $CE (\\mathbf{\\Lambda^{\\circ}})$ the Chekanov-Eliashberg category of $\\mathbf{\\Lambda^{\\circ}}$, by $\\mcF uk (\\mathbf{L})$ the Fukaya category generated by the Lagrangians $L(E)$ (see for example \\cite[chapter 2]{Sei08}), and by $\\overrightarrow{\\mcF uk} (\\mathbf{L})$ its directed subcategory (see \\cite[paragraph (5n)]{Sei08}).\n\nIn order for the latter algebraic objects to be $\\mathbf{Z}$-graded, we assume that $H_1(P)$ is free, that the first Chern class of $P$ (equipped with any almost complex structure compatible with $(- d \\lambda)$) is 2-torsion, and that the Maslov class of the Lagrangians $L(E)$ vanish. \nAs explained in section \\ref{subsection Conley-Zehnder index}, the grading on $CE (\\mathbf{\\Lambda^{\\circ}})$ depends on the choice of a symplectic trivialization of the contact structure along a fiber $h_0 = S^1 \\times \\{ a_0 \\}$.\nWe denote by $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ the Chekanov-Eliashberg DG-category of $\\mathbf{\\Lambda^{\\circ}}$ with grading induced by the trivialization\n\\[\\left( \\xi^{\\circ}_{| h_0}, d \\alpha^{\\circ} \\right) \\xrightarrow{\\sim} \\left( h_0 \\times \\mathbf{C}^n, dx \\wedge dy \\right), \\quad \\left( \\left( \\theta, a_0 \\right), \\left( \\lambda_{a_0} (v), v \\right) \\right) \\mapsto \\left( \\left( \\theta, a_0 \\right) , e^{2 i \\pi r \\theta} \\psi (v) \\right), \\]\nwhere $\\psi : \\left( T_{a_0} P, - d \\lambda_{a_0} \\right) \\xrightarrow{\\sim} \\left( \\mathbf{C}^n, dx \\wedge dy \\right)$ is a symplectic isomorphism.\n\nIn this setting, $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ is augmented (with the trivial augmentation) and Adams-graded (by the number of times a Reeb chord winds around the fiber).\nAs above, we denote by $\\mathbf{F} \\left[ t_m \\right]$ the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $(m, 1)$.\nMoreover, we denote by $E(-) = B(-)^{\\#}$ (graded dual of bar construction) the Koszul dual functor (see \\cite[section 2]{LPWZ08} or \\cite[section 2.3]{EL21}).\n\n\\begin{thmintro}\\label{thm mainthm}\n\t\n\tKoszul duality holds for $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$, and there is a quasi-equivalence of augmented Adams-graded $A_{\\infty}$-categories\n\t\\[E \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right) \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}) \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcF uk (\\mathbf{L}) \\right). \\]\n\t\n\\end{thmintro}\n\n\\begin{rmkintro}\n\t\t\n\tKoszul duality has many important consequences, see for example \\cite{LPWZ08} or \\cite{EL21}. In particular, by definition of Koszul duality (see \\cite[Theorem 2.4]{LPWZ08} or \\cite[Definition 17]{EL21}), Theorem \\ref{thm mainthm} implies that there is a quasi-equivalence of augmented Adams-graded DG-categories\n\t\\[CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\simeq E \\left( \\overrightarrow{\\mcF uk} (\\mathbf{L}) \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcF uk (\\mathbf{L}) \\right) \\right). \\]\n\tObserve that this formula is closely related to Conjecture 6.3 in \\cite{Sei08bis}, which was also discussed by Ganatra and Maydanskiy in the appendix of \\cite{BEE12}.\n\t\n\\end{rmkintro}\n\nWe now give a corollary of the latter result.\nIf $B$ is a (unpointed) space, we consider its one-point compactification $B^*$ and view it as a pointed space (with base point the point at infinity). If moreover $X$ is a pointed space, we consider the half-smash product of $B$ and $X$,\n\\[X \\rtimes B := X \\wedge B^* \\]\n(where $\\wedge$ denotes the smash product of pointed spaces). \nFinally, if $Y$ is a pointed space, we denote by $\\Omega Y$ its loop space.\n\n\\begin{corointro}\n\t\n\tIf $L$ is a connected compact exact Lagrangian and $\\Lambda^{\\circ}$ is a Legendrian lift of $L$ in the circular contactization, then there is a quasi-equivalence of augmented DG-algebras \n\t\\[ CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right) \\simeq C_{-*} \\left( \\Omega \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right). \\]\n\n\t\n\\end{corointro}\n\n\\paragraph{Acknowledgments.}\n\nThis work is part of my PhD thesis that I did at Nantes Universit\u00e9 under the supervision of Paolo Ghiggini and Vincent Colin, who I thank for their guidance and support. I also thank Baptiste Chantraine, Georgios Dimitroglou Rizell and Tobias Ekholm for helpful discussions.\n\\section{Legendrian lifts of exact Lagrangians in the circular contactization}\\label{subsection proof of main thm}\n\nIn this section, we start with a family $\\mathbf{L}$ of mutually transverse compact connected exact Lagrangian submanifolds in a Liouville manifold, and we study a Legendrian lift $\\mathbf{\\Lambda^{\\circ}}$ of $\\mathbf{L}$ in the circular contactization. \nFor the standard contact form, each point on a Legendrian gives rise to a (countable) infinite set of Reeb chords, and thus the situation is degenerate. In section \\ref{subsection setting and Legendrian invariants}, we explain how we perturb the contact form and we state our central result, which relates the Chekanov-Eliashberg DG-category of $\\mathbf{\\Lambda^{\\circ}}$ and the Fukaya $A_{\\infty}$-category of $\\mathbf{L}$. \n\n\\subsection{Setting}\\label{subsection setting and Legendrian invariants}\n\nLet $\\left( P , \\lambda \\right)$ be a Liouville manifold, and let \n\\[\\mathbf{L} = \\left( L(E) \\right)_{E \\in \\mcE}, \\quad \\mcE = \\left\\{ 1, \\dots, N \\right\\}, \\]\nbe a family of mutually transverse compact connected exact Lagrangian submanifolds in $(P, \\lambda)$ such that there are primitives $f_E : L(E) \\to \\mathbf{R}$ of $\\lambda_{| L(E)}$ satisfying $0 \\leq f_1 < \\dots < f_N \\leq 1\/2$.\nWe consider the contact manifold \n\\[ \\left( V^{\\circ}, \\xi^{\\circ} \\right) = \\left( S^1 \\times P, \\mathrm{ker} \\, \\alpha^{\\circ} \\right), \\text{ where } S^1 = \\mathbf{R}_{\\theta} \/ \\mathbf{Z}, \\, \\alpha^{\\circ} = d \\theta - \\lambda, \\]\nand the family of Legendrian submanifolds \n\\[\\mathbf{\\Lambda^{\\circ}} := (\\Lambda^{\\circ}(E))_{E \\in \\mcE}, \\text{ where } \\Lambda^{\\circ}(E) = \\left\\{ (f_E(x), x) \\in (\\mathbf{R} \/ \\mathbf{Z}) \\times P \\mid x \\in L(E) \\right\\} . \\]\nIn order for the Chekanov-Eliashberg category of $\\mathbf{\\Lambda^{\\circ}}$ and the Fukaya category of $\\mathbf{L}$ to be $\\mathbf{Z}$-graded, we assume that $H_1(P)$ is free, that the first Chern class of $P$ (equipped with any almost complex structure compatible with $(- d \\lambda)$) is 2-torsion, and that the Maslov classes of the Lagrangians $L(E)$ vanish.\n\n\\subsubsection{Reeb chords}\\label{subsection perturbed Reeb chords}\n\nObserve that $\\Lambda^{\\circ} = \\bigcup_{E \\in \\mcE} \\Lambda (E)$ is not chord generic for $\\alpha^{\\circ}$ (see Definition \\ref{definition chord generic}). We will choose a compactly supported function $H : P \\to \\mathbf{R}$, and consider the perturbed contact form \n\\[ \\alpha_H^{\\circ} = e^H \\alpha^{\\circ} . \\]\nThe Reeb vector field of $\\alpha_H^{\\circ}$ is then \n\\[ R_{\\alpha_H^{\\circ}} = e^{- H} \\left( \n\\begin{array}{c}\n1 + \\lambda \\left( X_H \\right) \\\\\nX_H\n\\end{array} \\right), \\]\nwhere $X_H$ is the unique vector field on $P$ satisfying $\\iota_{X_H} d \\lambda = -d H$.\n\nWe fix a compact neighborhood $K$ of $L$ which is contained in a Weinstein neighborhood of $L$ in $P$. \nIt is not hard to see that for every positive integer $N$, the space of smooth functions $H$ on $P$ supported in $K$, such that the $R_{\\alpha_H^{\\circ}}$-chords of $\\Lambda^{\\circ}$ with action less than $N$ are generic, is open and dense in $C_K^{\\infty} \\left( P \\right)$. Therefore, the space of functions $H \\in C_K^{\\infty} \\left( P \\right)$ such that $\\Lambda^{\\circ}$ is chord generic with respect to $\\alpha_H^{\\circ}$ is a Baire subset of $C_K^{\\infty} \\left( P \\right)$. In particular, the latter is dense in $C_K^{\\infty} \\left( P \\right)$. In the following, we choose $H \\in C_K^{\\infty} \\left( P \\right)$ such that \n\\begin{enumerate}\n\t\n\t\\item $\\Lambda^{\\circ}$ is chord generic with respect to $\\alpha_H^{\\circ}$,\n\t\n\t\\item $H$ is sufficiently close to $0$ so that\n\t\\[d \\theta \\left( R_{\\alpha_H^{\\circ}} \\right) = e^{-H} \\left( 1 + \\lambda \\left( X_H \\right) \\right) \\geq 1\/2 . \\]\n\t\n\\end{enumerate}\n\n\\begin{exa}\\label{example cotangent case}\n\t\n\tAssume that we are in the case\n\t\\[\\left( P, \\lambda \\right) = \\left( T^* M, p dq \\right), \\, L = 0_M, \\text{ and } H \\left( q, p \\right) = h(q), \\]\n\twhere $h : M \\to \\mathbf{R}$ is a Morse function (we present this example in order to see what happens, even if $H$ is not compactly supported in $T^*M$).\n\tThe Reeb vector field of $\\alpha_H^{\\circ}$ is \n\t\\[ R_{\\alpha_H^{\\circ}} = e^{- h} \\left( \n\t\\begin{array}{c}\n\t1 \\\\\n\t0 \\\\\n\t-d h\n\t\\end{array} \\right), \\]\n\tand therefore the Reeb flow satisfies\n\t\\[ \\varphi_{R_{\\alpha_H^{\\circ}}}^t \\left( \\theta , \\left( q,p \\right) \\right) = \\left( \\theta + t e^{- h (q)}, \\left( q , p - t e^{-h (q)} dh \\left( q \\right) \\right) \\right) . \\]\n\tThus, the $R_{\\alpha_H^{\\circ}}$-chords of $\\Lambda^{\\circ}$ are the paths $c : \\left[ 0,T \\right] \\to S^1 \\times T^* M$ of the form \n\t\\[ c(t) = \\left( t e^{- h (q_0)} , \\left( q_0 , 0 \\right) \\right), \\text{ with } T e^{- h (q_0)} \\in \\mathbf{Z}_{\\geq 1} \\text{ and } q_0 \\in \\mathrm{Crit} \\, h. \\]\n\tObserve that these Reeb chords are transverse but lie on top of each others. See Figure \\ref{figure perturbed Reeb chords}, where we illustrate this perturbation when $M = S^1$.\n\t\n\\end{exa}\n\n\\begin{figure}\n\t\\def1\\textwidth{1.2\\textwidth}\n\t\\input{Invariants_lift\/Figures\/dessin10.pdf_tex}\t\t\n\t\\caption{Reeb chords (in blue) of $\\Lambda^{\\circ} = \\{ 0 \\} \\times 0_{S^1}$ for $\\alpha^{\\circ}$ (on the left) and for $\\alpha_H^{\\circ}$ (on the right)}\t\n\t\\label{figure perturbed Reeb chords}\n\\end{figure}\n\n\\subsubsection{Conley-Zehnder index}\n\nIn order to define the Conley-Zehnder index (see section \\ref{subsection Conley-Zehnder index}), we need to choose a family $(h_0, h_1, \\dots, h_s)$ of embedded circles in $V^{\\circ} = S^1 \\times P$ which represent a basis of $H_1 ( V^{\\circ} )$, and a symplectic trivialization of $\\xi^{\\circ}$ over each $h_i$. We let $h_0 = S^1 \\times \\{ a_0 \\}$ be some fiber of $S^1 \\times P \\to P$, and we fix $\\left( h_1, \\dots, h_s \\right)$ to be any family of embedded circles in $P$ which represent a basis of $H_1 ( P )$. \nWe choose a symplectic isomorphism $\\psi : \\left( T_{a_0} P, - d \\lambda_{a_0} \\right) \\xrightarrow{\\sim} \\left( \\mathbf{C}^n, dx \\wedge dy \\right)$, and then we choose the symplectic trivialization\n\\[\\left( \\xi^{\\circ}_{| h_0}, d \\alpha^{\\circ} \\right) \\xrightarrow{\\sim} \\left( h_0 \\times \\mathbf{C}^n, dx \\wedge dy \\right), \\quad \\left( \\left( \\theta, a_0 \\right), \\left( \\lambda_{a_0} (v), v \\right) \\right) \\mapsto \\left( \\left( \\theta, a_0 \\right) , e^{2 i \\pi r \\theta} \\psi (v) \\right) \\]\nwhere $r$ is some integer, that we call \\emph{$r$-trivialization of $\\xi^{\\circ}$ over the fiber}.\nFinally, we choose some trivialization of $\\xi^{\\circ}$ over each $h_i$, $1 \\leq i \\leq s$.\n\n\\begin{exa}\\label{example CZ index in cotangent case}\n\t\n\tWe compute the Conley-Zehnder index of a Reeb chord in the case of Example \\ref{example cotangent case}, i.e. when \n\t\\[\\left( P, \\lambda \\right) = \\left( T^* M, p dq \\right), \\, L = 0_M, \\text{ and } H \\left( q, p \\right) = h(q), \\]\n\twhere $h : M \\to \\mathbf{R}$ is a Morse function .\n\tIn this case, the Reeb flow is given by\n\t\\[ \\varphi_{R_{\\alpha_H^{\\circ}}}^t \\left( \\theta , \\left( q,p \\right) \\right) = \\left( \\theta + t e^{- h (q)}, \\left( q , p - t e^{- h (q)} dh \\left( q \\right) \\right) \\right) . \\]\n\tLet $c : \\left[ 0,T \\right] \\to V^{\\circ}$ be a Reeb chord of $\\Lambda^{\\circ}$. Then there exists a positive integer $k$ and a critical point $q_0$ of $h$ such that \n\t\\[c(t) = \\left( t e^{- h (q_0)} , \\left( q_0 , 0 \\right) \\right) \\text{ and } T e^{- h (q_0)} = k. \\]\n\tObserve that $c(0) = c(T)$, and thus there is no need to choose a capping path for $c$. \n\tBesides, for every $u$ in $T_{q_0} M$, we have \n\t\\[D \\varphi_{R_{\\alpha_H^{\\circ}}}^t \\left( c(0) \\right) \\left( 0, u, 0 \\right) = \\left( 0, u, - t e^{- h (q_0)} D^2 h \\left( q_0 \\right) u \\right) . \\]\n\tIn order to compute the index of $c$, we first choose coordinates $\\left( x_1, \\dots, x_n \\right)$ around $q_0 \\in M$ in which \n\t\\[h = h \\left( q_0 \\right) + \\frac{1}{2} \\sum_{j=1}^{\\mathrm{dim}(M)} \\sigma_j x_j^2, \\text{ where } \\sigma_j = \\pm 1, \\]\n\tand we extend it to symplectic coordinates $\\left( x_1, \\dots, x_n, y_1, \\dots, y_n \\right)$ around $\\left( q_0, 0 \\right) \\in T^*M$ by setting \n\t\\[y_j \\left( q,p \\right) = \\langle p, \\frac{\\partial}{\\partial x_j} (q) \\rangle . \\] \n\tOur choice of trivialization for a fiber of $S^1 \\times \n\tP \\to P$ induces the trivialization \n\t\\[e^{2 i \\pi r k t \/ T} \\left( dx + i dy \\right) : c^{-1} \\xi^{\\circ} \\xrightarrow{\\sim} \\left( \\mathbf{R} \/ T \\mathbf{Z} \\right) \\times \\mathbf{C}^n \\]\n\t(observe that $\\xi^{\\circ}_{c(t)} = \\{ 0 \\} \\times T_{\\left( q_0, 0 \\right)} \\left( T^*M \\right)$).\n\tAccordingly, the path $t \\mapsto D \\varphi_{R_{\\alpha_H^{\\circ}}}^t \\left( T_{c(0)} \\Lambda^{\\circ} \\right)$ induces a path of Lagrangians \n\t\\[\\Gamma_c : t \\in \\left[ 0, T \\right] \\mapsto \\left\\{ \\left( e^{2 i \\pi m k t \/ T} \\left( u_j - i t e^{- h(q_0)} \\sigma_j u_j \\right) \\right)_{1 \\leq j \\leq n} \\mid u \\in \\mathbf{R}^n \\right\\} \\subset \\mathbf{C}^n . \\]\n\tWe close this path using a counterclockwise rotation $\\Gamma$, and call the resulting loop $\\overline{\\Gamma_c}$. In order to compute the Conley-Zehnder index of $c$, we have to look at how $\\overline{\\Gamma_c}$ intersects the Lagrangian $i \\mathbf{R}^n$ (as explained in \\cite[section 2.2]{EES05}).\n\tObserve that $\\Gamma_c$ intersects $i \\mathbf{R}^n$ positively $2rk$ times, so that $\\Gamma_c$ contributes $2rk$ to the Conley-Zehnder index of $c$. Moreover, since $\\Gamma$ is a counterclockwise rotation bringing\n\t\\[\\left\\{ \\left( u_j - i T e^{- h(q_0)} \\sigma_j u_j \\right)_{1 \\leq j \\leq n} \\mid u \\in \\mathbf{R}^n \\right\\} \\text{ to } \\mathbf{R}^n, \\]\n\tthe contributions to the intersection between $\\Gamma$ and $i \\mathbf{R}^n$ come from the negative eigenvalues $\\sigma_j$. The computation done in \\cite[Lemma 3.4]{EES05bisbis} implies that $\\Gamma$ contributes $\\mathrm{ind} (q_0)$ to the Conley-Zehnder index of $c$. We conclude that the Conley-Zehnder index of $c$ is\n\t\\[CZ \\left( c \\right) = \\mu \\left( \\overline{\\Gamma_c} \\right) = 2rk + \\mathrm{ind} (q_0) . \\] \n\n\\end{exa}\n\n\\subsubsection{Main result and strategy of proof}\\label{subsection invariants and statement of the result}\n\nLet $j$ be an almost complex structure on $P$ compatible with $\\left( -d\\lambda \\right)$, and let $J^{\\circ}$ be its lift to a complex structure on $\\xi^{\\circ}$.\nRecall from section \\ref{subsection Chekanov-Eliashberg algebra} the definition of the Chekanov-Eliashberg DG-category of a family of Legendrians. In our situation, $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) = CE_{-*} \\left( \\mathbf{\\Lambda^{\\circ}}, J^{\\circ}, \\alpha_{H}^{\\circ} \\right)$ (with grading induced by the $r$-trivialization of $\\xi^{\\circ}$ over the fiber) is an Adams-graded DG-algebra, where the Adams-degree of a Reeb chord $c$ is the number of times $c$ winds around the fiber. Besides, the map $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\to \\mathbf{F}$ which sends every Reeb chord to zero (and preserves units) defines an augmentation of $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$. \n\n\\begin{rmk}\n\t\n\tIn the case of Example \\ref{example cotangent case}, the cohomological degree of a Reeb chord $c$ in $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ corresponding to a positive integer $k$ and a critical point $q_0$ is \n\t\\[1 - CZ (c) = 1 -2rk - \\mathrm{ind} (q_0) \\]\n\t(see Example \\ref{example CZ index in cotangent case}).\n\t\n\\end{rmk}\n\nBesides, we denote by $\\mcF uk (\\mathbf{L})$ the Fukaya category generated by the Lagrangians $L(E)$ (see for example \\cite[chapter 2]{Sei08}), and by $\\overrightarrow{\\mcF uk} (\\mathbf{L})$ its directed subcategory:\n\\[\\hom_{\\overrightarrow{\\mcF uk} (\\mathbf{L})} (L(E), L(E')) =\n\\left\\{\n\\begin{array}{cl}\n\t\\langle L(E) \\cap L(E') \\rangle & \\text{if } E < E' \\\\\n\t\\mathbf{F} & \\text{if } E = E' \\\\\n\t0 & \\text{if } E > E' \\\\\n\\end{array}\n\\right. \\]\n(see \\cite[paragraph (5n)]{Sei08}). \n\nLet $\\mathbf{F} \\left[ t_m \\right]$ be the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $\\left( m,1 \\right)$. \nObserve that if $\\mcC$ is a subcategory of an $A_{\\infty}$-category $\\mcD$ with $\\mathrm{ob} (\\mcC) = \\mathrm{ob} (\\mcD)$, then $\\mcC \\oplus (t_m \\mathbf{F} [t_m] \\otimes \\mcD)$ is naturally an Adams-graded $A_{\\infty}$-category, where the Adams degree of $t_m^k \\otimes x$ equals $k$.\nMoreover, we denote by $E(-) = B(-)^{\\#}$ (graded dual of bar construction) the Koszul dual functor (see \\cite[section 2.3]{EL21} or \\cite[section 2]{LPWZ08}).\n\n\\begin{thm}[Theorem \\ref{thm mainthm} in the Introduction]\\label{thm mainthm restatement}\n\t\n\tKoszul duality holds for $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$, and there is a quasi-equivalence of augmented Adams-graded $A_{\\infty}$-categories\n\t\\[E \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right) \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}) \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcF uk (\\mathbf{L}) \\right). \\]\n\t\n\\end{thm}\n\t\n\\begin{coro}\\label{coro topological expression}\n\t\n\tIf $L$ is a connected compact exact Lagrangian and $\\Lambda^{\\circ}$ is a Legendrian lift of $L$ in the circular contactization, then there is a quasi-equivalence of augmented DG-algebras \n\t\\[ CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right) \\simeq C_{-*} \\left( \\Omega \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right). \\]\n\t\n\\end{coro}\n\n\\begin{proof}\n\t\n\tLet $x_0$ be the base point of $\\mathbf{CP}^{\\infty}$, and set $P := \\mathbf{CP}^{\\infty} \\setminus \\{ x_0 \\}$. \n\tObserve that \n\t\\[\\left( P \\times L \\right)^* = P^* \\wedge L^* = \\mathbf{CP}^{\\infty} \\wedge L^* = \\mathbf{CP}^{\\infty} \\rtimes L . \\]\n\tWe have \n\t\\[\\mathbf{F} \\oplus \\left( t_2 \\mathbf{F} \\left[ t_2 \\right] \\otimes CF^* \\left( L \\right) \\right) \\simeq \\mathbf{F} \\oplus \\left( t_2 \\mathbf{F} \\left[ t_2 \\right] \\otimes C^* \\left( L \\right) \\right) \\simeq C^* \\left( \\left( P \\times L \\right)^* \\right) \\simeq C^* \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right). \\]\n\tThus, it follows from Theorem \\ref{thm mainthm restatement} that \n\t\\[E \\left( CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right) \\right) \\simeq C^* \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) . \\]\n\tSince Koszul duality holds for $CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right)$, \n\t\\[CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right) \\simeq E \\left( C^* \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right) . \\]\n\tObserve that the graded algebra $H^*\\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right)$ is locally finite (i.e. each degree component is finitely generated) and simply connected (i.e. its augmentation ideal is concentrated in components of degree strictly greater than 1). Thus, according to the homological perturbation lemma (see \\cite[Proposition 1.12]{Sei08}), we can assume that $C^*\\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right)$ is a locally finite and simply connected $A_{\\infty}$ model for the DG-algebra of cochains on $\\mathbf{CP}^{\\infty} \\rtimes L$. \n\tTherefore, \\cite[Lemma 10]{EL21} implies that\n\t\\[CE_{-*}^1 \\left( \\Lambda^{\\circ} \\right) \\simeq \\Omega \\left( C_{-*} \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right). \\]\n\tNow, since $\\mathbf{CP}^{\\infty} \\rtimes L$ is simply connected, Adams result (see \\cite{Ada56}, \\cite{AH56} and also \\cite{EL21}) yields\n\t\\[\\Omega \\left( C_{-*} \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right) \\simeq C_{-*} \\left( \\Omega \\left( \\mathbf{CP}^{\\infty} \\rtimes L \\right) \\right) . \\]\n\tThis concludes the proof.\n\t\n\\end{proof}\n\n\\paragraph{Strategy of proof.}\n\nWe explain the startegy to compute $E ( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) )$.\nRecall from the last paragraph of section \\ref{subsection Chekanov-Eliashberg algebra} that there is a coaugmented $A_{\\infty}$-cocategory $LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ such that \n\\[CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) = \\Omega ( LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) ). \\]\n$LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ inherits an Adams-grading from $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ (the same), and we denote by $LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ its graded dual (see \\cite[section 2.1.3]{EL21}). In our situation, $LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ is an augmented Adams-graded $A_{\\infty}$-category whose augmentation ideal is generated by the Reeb chords of $\\mathbf{\\Lambda^{\\circ}}$ (and the Adams-degree of a Reeb chord $c$ is the number of times $c$ winds around the fiber).\nSince there is a quasi-isomorphism $B \\left( \\Omega C \\right) \\simeq C$ for every $A_{\\infty}$-cocategory $C$ (see \\cite[section 2.2.2]{EL21}), it follows that \n\\[E \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right) = B \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right)^{\\#} \\simeq LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)^{\\#} = LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\]\n(graded dual preserves quasi-isomorphisms).\n\n\\begin{rmk}\n\t\n\tIn the case of Example \\ref{example cotangent case}, the cohomological degree of a Reeb chord $c$ in $LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ corresponding to a positive integer $k$ and a critical point $q_0$ is \n\t\\[CZ (c) = 2rk + \\mathrm{ind} (q_0) \\]\n\t(see Example \\ref{example CZ index in cotangent case}).\n\t\n\\end{rmk}\n\nIn order to compute $LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$, we apply a sequence of geometric modifications to the situation. Each of the section numbered from \\ref{subsection lift} to \\ref{subsection projection on P} explains one of these modifications and describes how the algebraic invariants change accordingly. \nThe main ingredients are respectively: Theorem \\ref{thm mapping torus in strict situation} in section \\ref{subsection lift}, Lemma \\ref{lemma rectify contact form} in section \\ref{subsection rectification of the contact form}, Theorem \\ref{thm invariance} in section \\ref{subsection back to the original acs}, and \\cite[Theorem 2.1]{DR16} in section \\ref{subsection projection on P}.\nFinally, we end the proof in section \\ref{subsection proof of the main result} using Theorem \\ref{thm mapping torus in weak situation} (Theorem \\ref{thm mapping torus in weak situation introduction} in the introduction).\n\n\\subsection{Lift to $\\mathbf{R} \\times P$}\\label{subsection lift}\n\nIn the following we will consider the contact manifold \n\\[ \\left( V, \\xi \\right) = \\left( \\mathbf{R}_{\\theta} \\times P, \\ker \\, \\alpha \\right) \\text{ where } \\alpha = d \\theta - \\lambda, \\]\nand the family of Legendrian submanifolds\n\\[\\mathbf{\\Lambda} := (\\Lambda^n(E))_{(n, E) \\in \\mathbf{Z} \\times \\mcE}, \\text{ where } \\Lambda^{\\theta}(E) = \\left\\{ (f_E(x) + \\theta, x) \\in \\mathbf{R} \\times P \\mid x \\in L(E) \\right\\} . \\]\nMoreover we set $\\Lambda^n := \\bigcup_{E \\in \\mcE} \\Lambda^n (E)$ and $\\Lambda := \\bigcup_{n \\in \\mathbf{Z}} \\Lambda^n$.\n\nRecall from section \\ref{subsection perturbed Reeb chords} that we chose a compactly supported function $H : P \\to \\mathbf{R}$ such that \n\\begin{enumerate}\n\t\n\t\\item $\\Lambda^{\\circ}$ is chord generic with respect to $\\alpha_H^{\\circ}$,\n\t\n\t\\item $H$ is sufficiently close to $0$ so that\n\t\\[d \\theta \\left( R_{\\alpha_H^{\\circ}} \\right) = e^{-H} \\left( 1 + \\lambda \\left( X_H \\right) \\right) \\geq 1\/2 . \\]\n\t\n\\end{enumerate}\nWe consider the contact form \n\\[ \\alpha_{H} = e^{ H} \\alpha, \\]\nwith Reeb vector field \n\\[ R_{\\alpha_H} = e^{- H} \\left( \n\\begin{array}{c}\n1 + \\lambda \\left( X_H \\right) \\\\\n X_H\n\\end{array} \\right). \\]\nMoreover, we denote by $J$ the lift of $J^{\\circ}$ to an almost complex structure on $\\xi$. \n\n\\subsubsection{The $A_{\\infty}$-category $\\mcA$}\n\n\\begin{defin}\\label{definition category A}\n\t\n\tWe consider the $A_{\\infty}$-category $\\mcA$ defined as follows\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\mcA$ are the Legendrians $\\Lambda^n (E)$, $(n, E) \\in \\mathbf{Z} \\times \\mcE$,\n\t\t\n\t\t\\item the space of morphisms from $\\Lambda^i (E)$ to $\\Lambda^j (E')$ is either generated by the $R_{\\alpha_{H}}$-chords from $\\Lambda^i (E)$ to $\\Lambda^j (E')$ if $(i,E)<(j,E')$, or $\\mathbf{F}$ if $(i,E)=(j,E')$, or $0$ otherwise,\n\t\t\n\t\t\\item the operations are such that $1 \\in \\mcA \\left( \\Lambda^n (E), \\Lambda^n (E) \\right)$ is a strict unit, and for every sequence $(i_0, E_0) < \\dots < (i_d, E_d)$, for every sequence of Reeb chords \n\t\t\\[\\left( c_1, \\dots, c_d \\right) \\in \\mcR \\left( \\Lambda^{i_0} (E_0), \\Lambda^{i_1} (E_1) \\right) \\times \\dots \\times \\mcR \\left( \\Lambda^{i_{d-1}} (E_{d-1}), \\Lambda^{i_d} (E_d) \\right), \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcA} \\left( c_1, \\dots, c_d \\right) = \\sum \\limits_{c_0 \\in \\mcR \\left( \\Lambda^{i_0} (E_0), \\Lambda^{i_d} (E_d) \\right)} \\# \\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha_{H} \\right) c_0 \\]\n\t\t(see Definition \\ref{definition moduli spaces} for the moduli spaces).\n\t\t\n\t\\end{enumerate} \n\t\n\\end{defin}\n\n\\subsubsection{The quasi-autoequivalence $\\tau$} \n\nWe introduce an $A_{\\infty}$-endofunctor of $\\mcA$ that will be important in the following.\n\n\\begin{defin}\\label{definition functor tau}\n\t\n\tWe denote by $\\tau : \\mcA \\to \\mcA$ the $A_{\\infty}$-functor defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\tau$ sends $\\Lambda^n (E)$ to $\\Lambda^{n+1} (E)$,\n\t\t\n\t\t\\item on morphisms, the map \n\t\t\\[\\tau : \\mcA \\left( \\Lambda^{i_0} (E_0), \\Lambda^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcA \\left( \\Lambda^{i_{d-1}} (E_{d-1}), \\Lambda^{i_d} (E_d) \\right) \\to \\mcA \\left( \\Lambda^{i_0+1} (E_0), \\Lambda^{i_d+1} (E_d) \\right) \\]\n\t\tis obtained by dualizing the components of the DG-isomorphism\n\t\t\\[CE_{-*} \\left( \\left( \\Lambda^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha_{H} \\right) \\to CE_{-*} \\left( \\left( \\Lambda^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha_{H} \\right) \\]\n\t\tinduced by the path $ \\left( \\left( \\Lambda^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J \\right)_{0 \\leq t \\leq 1}$ (see Theorem \\ref{thm invariance}). \n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{lemma}\\label{lemma tau is a quasi-equivalence}\n\t\n\tThe $A_{\\infty}$-functor $\\tau : \\mcA \\to \\mcA$ is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tConsider the $A_{\\infty}$-functor $\\overline{\\tau} : \\mcA \\to \\mcA$ defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\overline{\\tau}$ sends $\\Lambda^n (E)$ to $\\Lambda^{n-1} (E)$,\n\n\t\t\\item on morphisms, the map \n\t\t\\[\\overline{\\tau} : \\mcA \\left( \\Lambda^{i_0} (E_0), \\Lambda^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcA \\left( \\Lambda^{i_{d-1}} (E_{d-1}), \\Lambda^{i_d} (E_d) \\right) \\to \\mcA \\left( \\Lambda^{i_0-1} (E_0), \\Lambda^{i_d-1} (E_d) \\right) \\]\n\t\tis obtained by dualizing the components of the \\emph{inverse} of the DG-isomorphism\n\t\t\\[CE_{-*} \\left( \\left( \\Lambda^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha_{H} \\right) \\to CE_{-*} \\left( \\left( \\Lambda^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha_{H} \\right) \\]\n\t\tinduced by the path $ \\left( \\left( \\Lambda^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J \\right)_{0 \\leq t \\leq 1}$. \n\t\t\n\t\\end{enumerate} \n\tThen $\\tau \\circ \\overline{\\tau} = \\overline{\\tau} \\circ \\tau = \\mathrm{id}_{\\mcA}$. \n\t\n\\end{proof}\n\nHere the $\\mathbf{Z}$-splitting \n\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA \\right), \\quad (n, E) \\mapsto \\Lambda^n (E) \\]\nis compatible with the quasi-autoequivalence $\\tau$ in the sense of Definition \\ref{definition group-action}. As explained there, this turns $\\mcA$ into an Adams-graded $A_{\\infty}$-category: the Adams-degree of a morphism $c \\in \\mcA \\left( \\Lambda^i (E), \\Lambda^j (E') \\right)$ is $\\left( j-i \\right)$. \n\n\\subsubsection{Relation between $LA^* \\left( \\Lambda^{\\circ} \\right)$ and $\\left( \\mcA, \\tau \\right)$}\n\nWe now explain how $LA^* \\left( \\Lambda^{\\circ} \\right)$ and $\\left( \\mcA, \\tau \\right)$ are related.\nSee Figure \\ref{figure relation usual and circular contactization}, where we illustrate the action of the projection $\\Pi_{S^1 \\times P}$ in the case \n\\[\\left( P, \\lambda \\right) = \\left( T^* S^1, p dq \\right), \\, L = 0_{S^1}, \\text{ and } H \\left( q, p \\right) = h(q), \\]\nwhere $h : S^1 \\to \\mathbf{R}$ is a Morse function. \n\n\\begin{figure}\n\t\\def1\\textwidth{1.3\\textwidth}\n\t\\input{Invariants_lift\/Figures\/dessin12.pdf_tex}\t\t\n\t\\caption{Action of the projection $\\Pi_{S^1 \\times T^* S^1}$}\t\n\t\\label{figure relation usual and circular contactization}\n\\end{figure}\n\n\\begin{lemma}\\label{lemma tau is strict}\n\t\n\tThe $A_{\\infty}$-functor $\\tau$ is strict, and it sends a Reeb chord $t \\mapsto \\left( \\theta \\left( t \\right) , x \\left( t \\right) \\right)$ in $\\mcA \\left( \\Lambda^i (E) , \\Lambda^j (E') \\right)$ to the Reeb chord $t \\mapsto \\left( \\theta \\left( t \\right) + 1 , x \\left( t \\right) \\right)$ in $\\mcA \\left( \\Lambda^{i+1} (E) , \\Lambda^{j+1} (E') \\right)$.\n\tIn particular, $\\tau$ acts bijectively on hom-sets. \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tRecall that $\\alpha_H = e^H \\alpha$, with $H$ a function defined on the base manifold $P$.\n\tIn particular, the flow $\\varphi_{\\partial_{\\theta}}^t$ of $\\partial_{\\theta}$ is a strict contactomorphism of $\\left( V, \\alpha_{H} \\right)$.\n\tMoreover, since $J$ is the lift of an almost complex structure $j$ on $P$, we have\n\t\\[\\left( \\left( \\Lambda^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J \\right) = \\left( \\left( (\\varphi_{\\partial_{\\theta}}^t)^{-1} \\Lambda^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, (\\varphi_{\\partial_{\\theta}}^t)^* J \\right) . \\]\n\tThe result follows from Theorem \\ref{thm invariance}.\n\t\n\\end{proof}\n\nWe denote by $\\mcA_{\\tau}$ the Adams-graded $A_{\\infty}$-category associated to $\\tau$ as in Definition \\ref{definition category of coinvariants}. \n\n\\begin{lemma}\\label{lemma A^rond = A_Z}\n\t\n\tThere is a quasi-isomorphism of Adams-graded $A_{\\infty}$-categories\n\t\\[LA^* \\left( \\Lambda^{\\circ} \\right) \\simeq \\mcA_{\\tau} . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tConsider the map which sends a Reeb chord $c \\in \\mcR \\left( \\Lambda^i (E), \\Lambda^j (E') \\right)$ to the corresponding chord $\\Pi_{S^1 \\times P} \\left( c \\right) \\in \\mcR \\left( \\Lambda^{\\circ} (E), \\Lambda^{\\circ} (E') \\right)$ (where $\\Pi_{S^1 \\times P} : \\mathbf{R} \\times P \\to S^1 \\times P$ is the projection). According to Lemma \\ref{lemma tau is strict}, $\\Pi_{S^1 \\times P} \\left( \\tau c \\right) = \\Pi_{S^1 \\times P} \\left( c \\right)$, and thus the map $c \\mapsto \\Pi_{S^1 \\times P} \\left( c \\right)$ induces a map $\\psi : \\mcA_{\\tau} \\to LA^* \\left( \\Lambda^{\\circ} \\right)$. Moreover, observe that $\\psi$ is a bijection on hom-spaces. It remains to prove that $\\psi$ is an $A_{\\infty}$-map. This follows from the fact that the map\n\t\\[ u = \\left( \\sigma , v \\right) \\mapsto \\left( \\sigma , \\Pi_{S^1 \\times P} \\circ v \\right) \\]\n\tinduces a bijection\n\t\\[\\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha_{H} \\right) \\xrightarrow{\\sim} \\mcM_{\\psi (c_d), \\dots, \\psi (c_1), \\psi (c_0)} \\left( \\mathbf{R} \\times \\Lambda^{\\circ}, J^{\\circ}, \\alpha_{H}^{\\circ} \\right) . \\] \n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma A^rond = mapping torus of tau}\n\t\n\tThe Adams-graded $A_{\\infty}$-category $LA^* \\left( \\Lambda^{\\circ} \\right)$ is quasi-equivalent to the mapping torus of $\\tau : \\mcA \\to \\mcA$ (see Definition \\ref{definition mapping torus}). \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows directly from Theorem \\ref{thm mapping torus in strict situation} using Lemmas \\ref{lemma tau is strict} and \\ref{lemma A^rond = A_Z}. \n\t\n\\end{proof}\n\n\\subsection{Rectification of the contact form}\\label{subsection rectification of the contact form}\n\nNow that we are in the usual contactization, we have the following result. \n\n\\begin{lemma}\\label{lemma rectify contact form} \n\t\n\tThere exists a contactomorphism $\\phi_H$ of $\\left( V,\\xi \\right)$ such that \n\t\\[ \\phi_H^* \\alpha_{H} = \\alpha . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tRecall that $\\alpha_{H} = e^{ H} \\alpha$, with $H$ a compactly supported function on the base manifold $P$ such that $e^{-H} \\left( 1 + \\lambda \\left( X_H \\right) \\right) \\geq 1\/2$.\n\t\n\tAssume that there is a contact isotopy $\\left( \\phi_{t} \\right)_{0 \\leq t \\leq 1}$ such that $\\phi_0 = \\mathrm{id}$ and \n\t\\begin{equation}\\label{eq1}\n\t\\phi_{t}^* \\alpha_{t H} = \\alpha\n\t\\end{equation}\n\tfor every $t$. Let $\\left( F_{t} \\right)_{t}$ be the family of functions on $V$ such that \n\t\\[ \\frac{d}{d t} \\phi_{t} = Y_{F_{t}} \\circ \\phi_{t}, \\]\n\twhere $Y_F$ is the contact vector field on $V$ satisfying \n\t\\begin{align*}\n\t\\left\\{\n\t\\begin{array}{ccl}\n\t\\alpha \\left( Y_F \\right) & = & F \\\\\n\t\\iota_{Y_F} d \\alpha & = & d F \\left( R_{\\alpha} \\right) \\alpha - d F . \n\t\\end{array}\n\t\\right.\n\t\\end{align*}\n\tTaking the derivative of equation \\eqref{eq1} with respect to $t$, we get \n\t\\begin{equation}\\label{eq2}\n\tH + d \\left( e^{t H} F_{t} \\right) \\left( R_{\\alpha_{t H}} \\right) = 0\n\t\\end{equation}\n\tbecause $Y_F$ satisfies \n\t\\begin{align*}\n\t\\left\\{\n\t\\begin{array}{ccl}\n\t\\alpha_{t H} \\left( Y_F \\right) & = & e^{t H} F \\\\\n\t\\iota_{Y_F} d \\alpha_{t H} & = & d \\left( e^{t H} F \\right) \\left( R_{\\alpha_{t H}} \\right) \\alpha_{t H} - d \\left( e^{t H} F \\right) .\n\t\\end{array}\n\t\\right.\n\t\\end{align*}\n\tBesides, we deduce from \n\t\\[ R_{\\alpha_{t H}} = e^{- t H} \\left( \n\t\\begin{array}{c}\n\t1 + t \\lambda \\left( X_H \\right) \\\\\n\tt X_H\n\t\\end{array} \\right), \\quad \\iota_{X_H} d \\lambda = - dH, \\]\n\tthat \n\t\\[ d H \\left( R_{\\alpha_{t H}} \\right) = 0 . \\]\n\tThen equation \\eqref{eq2} gives \n\t\\begin{equation}\\label{eq3}\n\td F_{t} \\left( R_{\\alpha_{t H}} \\right) = - H e^{- t H} .\n\t\\end{equation} \n\t\n\tConversely, if $\\left( F_{t} \\right)_{t}$ is a family of functions on $V$ satisfying equation \\eqref{eq3}, then the contact isotopy $\\left( \\phi_{t} \\right)_{t}$ defined by \n\t\\[ \\phi_0 = \\mathrm{id} \\text{ and } \\frac{d}{d t} \\phi_{t} = Y_{F_{t}} \\circ \\phi_{t}, \\]\n\tsatisfies\n\t\\[ \\frac{d}{d t} \\left( \\phi_{t}^* \\alpha_{t H} \\right) = 0 , \\]\n\tand thus $\\phi_H := \\phi_1$ gives the desired result.\n\t\n\tTherefore, it remains to find a family $\\left( F_{t} \\right)_{t}$ satisfying equation \\eqref{eq3}. First recall that \n\t\\[ R_{\\alpha_{t H}} = e^{- t H} \\left( \n\t\\begin{array}{c}\n\t1 + t \\lambda \\left( X_H \\right) \\\\\n\tt X_H\n\t\\end{array} \\right) . \\]\n\tBy assumption on $H$, the function $d \\theta \\left( R_{\\alpha_{t H}} \\right)$ is greater than $1\/2$ for every $t \\in \\left[ 0,1 \\right]$. Thus, for every $t \\in \\left[ 0,1 \\right]$ and every $\\left( \\theta , x \\right)$ in $V$, there exists a unique real number $\\rho_{t} \\left( \\theta , x \\right)$ such that \n\t\\[ \\varphi_{R_{\\alpha_{t H}}}^{- \\rho_{t} \\left( \\theta , x \\right) } \\left( \\theta , x \\right) \\in \\{ 0 \\} \\times P . \\]\n\tThen we let \n\t\\[ F_{t}:= - \\rho_{t} H e^{- t H} . \\]\n\tFor every real number $t$, we have \n\t\\[ F_{t} \\circ \\varphi_{R_{\\alpha_{t H}}}^t = - \\left( \\rho_{t} \\circ \\varphi_{R_{\\alpha_{t H}}}^t \\right) H e^{- t H} \\text{ because } dH \\left( R_{\\alpha_{t H}} \\right) = 0 . \\]\n\tBut the map $\\varphi_{R_{\\alpha_{t H}}}^{- \\rho_{t} \\circ \\varphi_{R_{\\alpha_{t H}}}^t + t}$ takes its values in $\\{ 0 \\} \\times P$ by definition of $\\rho_{t}$, so by uniqueness we have\n\t\\[ \\rho_{t} \\circ \\varphi_{R_{\\alpha_{t H}}}^t = \\rho_{t} + t . \\] \n\tThen we have \n\t\\[ F_{t} \\circ \\varphi_{R_{\\alpha_{t H}}}^t = - \\left( \\rho_{t} + t \\right) H e^{- t H}, \\]\n\tand thus \n\t\\[ d F_{t} \\left( R_{\\alpha_{t H}} \\right) = - H e^{- t H} . \\]\n\tThis concludes the proof.\n\t\n\\end{proof}\n\n\\begin{exa}\\label{example rectify contact form in cotangent case}\n\t\n\tAssume that we are in the case \n\t\\[\\left( P, \\lambda \\right) = \\left( T^* M, p dq \\right), \\, L = 0_M, \\text{ and } H \\left( q, p \\right) = h(q), \\]\n\twhere $h : M \\to \\mathbf{R}$ is a Morse function. \n\tThen the diffeomorphism $\\phi_H$ defined by\n\t\\[\\phi_H^{-1} \\left( \\theta, \\left( q,p \\right) \\right) = \\left( \\theta e^{h(q)}, \\left( q, e^{h(q)} p + \\theta e^{h(q)} dh (q) \\right) \\right) \\]\n\tsatisfies $\\phi_H^* \\alpha_H = \\alpha$.\n\tWith this choice of $\\phi_H$, we have in particular\n\t\\[\\phi_H^{-1} \\left( \\{ \\theta \\} \\times 0_M \\right) = j^1 \\left( \\theta e^h \\right) \\subset \\mathbf{R} \\times T^*M . \\] \n\t\n\\end{exa}\n\n\\subsubsection{The $A_{\\infty}$-category $\\mcA_1$}\n\nIn the following, we fix a contactomorphism $\\phi_H$ as in Lemma \\ref{lemma rectify contact form}.\nWe define an $A_{\\infty}$-category $\\mcA_1$, which is roughly obtained by pulling back the data of $\\mcA$ by $\\phi_H$. \n\n\\begin{defin}\n\t\n\tLet \n\t\\[\\Lambda_H^{\\theta} (E) := \\phi_H^{-1} \\left( \\Lambda^{\\theta} (E) \\right), \\, \\Lambda_H^n := \\bigcup\\limits_{E \\in \\mcE} \\Lambda_H^n (E), \\, \\Lambda_H := \\bigcup\\limits_{n \\in \\mathbf{Z}} \\Lambda_H^n \\text{ and } J_H := \\phi_H^* J. \\]\n\tWe consider the $A_{\\infty}$-category $\\mcA_1$ defined as follows\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\mcA_1$ are the Legendrians $\\Lambda_H^n (E)$, $(n, E) \\in \\mathbf{Z} \\times E$,\n\t\t\n\t\t\\item the vector space $\\mcA_1 \\left( \\Lambda_H^i (E), \\Lambda_H^j (E') \\right)$ is either generated by the $R_{\\alpha}$-chords from $\\Lambda_H^i (E)$ to $\\Lambda_H^j (E')$ if $(i,E)<(j,E')$, or $\\mathbf{F}$ if $(i,E)=(j,E')$, or $0$ otherwise, and\n\t\t\n\t\t\\item the operations are such that $1 \\in \\mcA_1 \\left( \\Lambda_H^n (E), \\Lambda_H^n (E) \\right)$ is a strict unit, and for every sequence $(i_0, E_0) < \\dots < (i_d, E_d)$, for every sequence of Reeb chords \n\t\t\\[\\left( c_1, \\dots, c_d \\right) \\in \\mcR \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_1} (E_1) \\right) \\times \\dots \\times \\mcR \\left( \\Lambda_H^{i_{d-1}} (E_{d-1}), \\Lambda_H^{i_d} (E_d) \\right) \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcA_1} \\left( c_1, \\dots, c_d \\right) = \\sum \\limits_{c_0 \\in \\mcR \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_d} (E_d) \\right)} \\# \\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda_H, J_H, \\alpha \\right) c_0 \\]\n\t\t(see Definition \\ref{definition moduli spaces} for the moduli spaces).\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\subsubsection{The quasi-autoequivalence $\\tau_1$}\n\n\\begin{defin}\\label{definition functor tau_1}\n\t\n\tWe denote by $\\tau_1 : \\mcA_1 \\to \\mcA_1$ the $A_{\\infty}$-functor defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\tau_1 \\left( \\Lambda_H^n (E) \\right) = \\Lambda_H^{n+1} (E)$,\n\t\t\n\t\t\\item on morphisms, the map \n\t\t\\[\\tau_1^d : \\mcA_1 \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcA_1 \\left( \\Lambda_H^{i_{d-1}} (E_{d-1}), \\Lambda_H^{i_d} (E_d) \\right) \\to \\mcA_1 \\left( \\Lambda_H^{i_0+1} (E_0), \\Lambda_H^{i_d+1} (E_d) \\right) \\]\n\t\tis obtained by dualizing the components of the DG-isomorphism\n\t\t\\[CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\]\n\t\tinduced by the path $ \\left( \\left( \\Lambda_H^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J_H \\right)_{0 \\leq t \\leq 1}$ (see Theorem \\ref{thm invariance}). \n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{lemma}\\label{lemma tau_1 is a quasi-equivalence}\n\t\n\tThe $A_{\\infty}$-functor $\\tau_1 : \\mcA_1 \\to \\mcA_1$ is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows because $\\tau_1$ is defined by dualizing the components of a DG-isomorphism (see the proof of Lemma \\ref{lemma tau is a quasi-equivalence}). \n\t\n\\end{proof}\n\nHere the $\\mathbf{Z}$-splitting \n\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA_1 \\right), \\quad (n, E) \\mapsto \\Lambda_H^n (E), \\]\nis compatible with the quasi-autoequivalence $\\tau_1$ in the sense of Definition \\ref{definition group-action}. As explained there, this turns $\\mcA_1$ into an Adams-graded $A_{\\infty}$-category. \n\n\\subsubsection{Relation between $\\left( \\mcA, \\tau \\right)$ and $\\left( \\mcA_1, \\tau_1 \\right)$}\n\nWe now explain how the pairs $\\left( \\mcA, \\tau \\right)$ and $\\left( \\mcA_1, \\tau_1 \\right)$ are related.\nSee Figure \\ref{figure action of the contactomorphism} where we illustrate the action of the contactomorphism $\\phi_H^{-1}$ in the case \n\\[\\left( P, \\lambda \\right) = \\left( T^* S^1, p dq \\right), \\, L = 0_{S^1}, \\text{ and } H \\left( q, p \\right) = h(q), \\]\nwhere $h : S^1 \\to \\mathbf{R}$ is a Morse function.\n\n\\begin{figure}\n\t\\def1\\textwidth{1.4\\textwidth}\n\t\\input{Invariants_lift\/Figures\/dessin13.pdf_tex}\t\t\n\t\\caption{Action of the contactomorphism $\\phi_H^{-1}$}\t\n\t\\label{figure action of the contactomorphism}\n\\end{figure}\n\n\\begin{lemma}\\label{lemma relation A - A_1}\n\t\n\tThere is a strict $A_{\\infty}$-isomorphism $\\zeta_1 : \\mcA \\to \\mcA_1$ defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\zeta_1 \\left( \\Lambda^n (E) \\right) = \\Lambda_H^n (E)$,\n\t\t\n\t\t\\item on morphisms, $\\zeta_1$ sends a Reeb chord $c$ in $\\mcA \\left( \\Lambda^i (E), \\Lambda^j (E') \\right)$ to the Reeb chord \n\t\t\\[\\zeta_1 \\left( c \\right) = \\phi_H^{-1} \\circ c \\]\n\t\tin $\\mcA_1 \\left( \\Lambda_H^i (E), \\Lambda_H^j (E') \\right)$. \n\t\t\n\t\\end{enumerate} \n\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe have to show that $\\zeta_1$ is an $A_{\\infty}$-map.\n\tThis follows from the fact that the map \n\t\\[ u = \\left( \\sigma , v \\right) \\mapsto \\left( \\sigma , \\phi_H^{-1} \\circ v \\right) \\]\n\tinduces a bijection\n\t\\[\\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha_H \\right) \\xrightarrow{\\sim} \\mcM_{\\phi_H^{-1} (c_d) \\dots \\phi_H^{-1} (c_1), \\phi_H^{-1} (c_0)} \\left( \\mathbf{R} \\times \\Lambda_H, J_H, \\alpha \\right) . \\]\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma relation tau - tau_1}\n\t\n\tWe have \n\t\\[\\tau_1 = \\zeta_1 \\circ \\tau \\circ \\zeta_1^{-1} . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows from Theorem \\ref{thm invariance} using that $\\phi_H^* \\alpha_H = \\alpha$ and\n\t\\[\\left( \\left( \\Lambda_H^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J_H \\right) = \\left( \\left( \\phi_H^{-1} \\Lambda^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, \\phi_H^* J \\right). \\]\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma mapping torus of tau = mapping torus of tau_1}\n\t\n\tThe mapping torus of $\\tau : \\mcA \\to \\mcA$ is quasi-equivalent to the mapping torus of $\\tau_1 : \\mcA_1 \\to \\mcA_1$ (see Definition \\ref{definition mapping torus}).\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tAccording to Lemma \\ref{lemma relation tau - tau_1} the following diagram of Adams-graded $A_{\\infty}$-categories is commutative \n\t\\[\\begin{tikzcd}\n\t\\mcA \\ar[d, \"\\zeta_1\"] & \\mcA \\sqcup \\mcA \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\zeta_1 \\sqcup \\zeta_1\"] & \\mcA \\ar[d, \"\\zeta_1\"] \\\\\n\t\\mcA_1 & \\mcA_1 \\sqcup \\mcA_1 \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau_1\"] & \\mcA_1 .\n\t\\end{tikzcd} \\]\n\tMoreover, each vertical arrow is a quasi-equivalence according to Lemma \\ref{lemma relation A - A_1}. Thus the result follows from Proposition \\ref{prop invariance of homotopy colimits}.\n\t\n\\end{proof}\n\n\\subsection{Back to the original almost complex structure}\\label{subsection back to the original acs}\n\nIn this section, we introduce an $A_{\\infty}$-category $\\mcA_2$ which has same objects and morphisms as $\\mcA_1$, but whose operations count punctured discs in $\\mathbf{R} \\times V$ which are pseudo-holomorphic for the almost complex structure induced by $\\alpha$ and $J$ (instead of $J_H$). \n\n\\subsubsection{The $A_{\\infty}$-category $\\mcA_2$}\n\nRecall that we chose a contactomorphism $\\phi_H$ as in Lemma \\ref{lemma rectify contact form}, and recall that\n\\[\\Lambda_H^{\\theta} (E) := \\phi_H^{-1} \\left( \\Lambda^{\\theta} (E) \\right), \\, \\Lambda_H^n := \\bigcup\\limits_{E \\in \\mcE} \\Lambda_H^n (E), \\, \\Lambda_H := \\bigcup\\limits_{n \\in \\mathbf{Z}} \\Lambda_H^n \\text{ and } J_H := \\phi_H^* J. \\]\n\n\\begin{defin}\n\t\n\tWe consider the $A_{\\infty}$-category $\\mcA_2$ defined as follows\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\mcA_2$ are the Legendrians $\\Lambda_H^n (E)$, $(n, E) \\in \\mathbf{Z} \\times \\mcE$,\n\t\t\n\t\t\\item the vector space $\\mcA_2 \\left( \\Lambda_H^i (E), \\Lambda_H^j (E') \\right)$ is either generated by the $R_{\\alpha}$-chords from $\\Lambda_H^i (E)$ to $\\Lambda_H^j (E')$ if $(i,E)<(j,E')$, or $\\mathbf{F}$ if $(i,E)=(j,E')$, or $0$ otherwise, and\n\t\t\n\t\t\\item the operations are such that $1 \\in \\mcA_2 \\left( \\Lambda_H^n, \\Lambda_H^n \\right)$ is a strict unit, and for every sequence $(i_0, E_0) < \\dots < (i_d, E_d)$, for every sequence of Reeb chords \n\t\t\\[\\left( c_1, \\dots, c_d \\right) \\in \\mcR \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_1} (E_1) \\right) \\times \\dots \\times \\mcR \\left( \\Lambda_H^{i_{d-1}} (E_{d-1}), \\Lambda_H^{i_d} (E_d) \\right) \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcA_2} \\left( c_1, \\dots, c_d \\right) = \\sum \\limits_{c_0 \\in \\mcR \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_d} (E_d) \\right)} \\# \\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda_H, J, \\alpha \\right) c_0 . \\]\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\subsubsection{The quasi-autoequivalence $\\tau_2$}\n\n\\begin{defin}\\label{definition functor tau_2}\n\t\n\tWe denote by $\\tau_2 : \\mcA_1 \\to \\mcA_1$ the $A_{\\infty}$-functor defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\tau_2 \\left( \\Lambda_H^n (E) \\right) = \\Lambda_H^{n+1} (E)$,\n\t\t\n\t\t\\item on morphisms, the map \n\t\t\\[\\tau_2^d : \\mcA_1 \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcA_1 \\left( \\Lambda_H^{i_{d-1}} (E_{d-1}), \\Lambda_H^{i_d} (E_d) \\right) \\to \\mcA_1 \\left( \\Lambda_H^{i_0+1} (E_0), \\Lambda_H^{i_d+1} (E_d) \\right) \\]\n\t\tis obtained by dualizing the components of the DG-isomorphism\n\t\t\\[CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\]\n\t\tinduced by the path $ \\left( \\left( \\Lambda_H^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J \\right)_{0 \\leq t \\leq 1}$ (see Theorem \\ref{thm invariance} or \\cite[Proposition 2.6]{EES07}). \n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{lemma}\\label{lemma tau_2 is a quasi-equivalence}\n\t\n\tThe $A_{\\infty}$-functor $\\tau_2 : \\mcA_2 \\to \\mcA_2$ is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows from the fact that $\\tau_2$ is defined by dualizing the components of a DG-isomorphism (see the proof of Lemma \\ref{lemma tau is a quasi-equivalence}). \n\t\n\\end{proof}\n\nHere the $\\mathbf{Z}$-splitting \n\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA_2 \\right), \\quad (n, E) \\mapsto \\Lambda_H^n (E) \\]\nis compatible with the quasi-autoequivalence $\\tau_2$ in the sense of Definition \\ref{definition group-action}. As explained there, this turns $\\mcA_2$ into an Adams-graded $A_{\\infty}$-category. \n\n\\subsubsection{Relation between $\\left( \\mcA_1, \\tau_1 \\right)$ and $\\left( \\mcA_2, \\tau_2 \\right)$}\n\n\\begin{lemma}\\label{lemma relation A_1 - A_2}\n\t\n\tChoose a generic path $(J_t^{12})_{0 \\leq t \\leq 1}$ such that $J_0^{12} = J$ and $J_1^{12} = J_H$. \n\tThere is an $A_{\\infty}$-isomorphism $\\zeta_{12} : \\mcA_1 \\to \\mcA_2$ defined as follows\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\zeta_{12} \\left( \\Lambda_H^n (E) \\right) = \\Lambda_H^n (E)$, \n\t\t\n\t\t\\item on morphisms, the map\n\t\t\\[\\zeta_{12} : \\mcA_1 \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcA_1 \\left( \\Lambda_H^{i_{d-1}} (E_{d-1}), \\Lambda_H^{i_d} (E_d) \\right) \\to \\mcA_2 \\left( \\Lambda_H^{i_0} (E_0), \\Lambda_H^{i_d} (E_d) \\right) \\]\n\t\tis obtained by dualizing the components of the DG-isomorphism \n\t\t\\[CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\]\n\t\tinduced by the path $(\\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_t^{12})_{0 \\leq t \\leq 1}$ (see Theorem \\ref{thm invariance}).\n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe have to prove that $\\zeta_{12}$ is an isomorphism. This follows from the fact that it is defined by dualizing the components of a DG-isomorphism (see the proof of Lemma \\ref{lemma tau is a quasi-equivalence}). \n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma relation tau_1 - tau_2}\n\t\n\tThe $A_{\\infty}$-functor $\\tau_2$ is homotopic to $\\zeta_{12} \\circ \\tau_1 \\circ \\zeta_{12}^{-1}$ (see \\cite[paragraph (1h)]{Sei08}). \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tFirst recall that $\\tau_1$ is obtained by dualizing the components of the DG-map\n\t\\[CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\]\n\tinduced by the path $\\left( \\left( \\Lambda_H^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J_H \\right)_{0 \\leq t \\leq 1}$. Thus, $\\zeta_{12} \\circ \\tau_1$ is obtained by dualizing the components of the composition\n\t\\[CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) . \\]\n\tOn the other hand, $\\tau_2$ is obtained by dualizing the components of the DG-map \n\t\\[CE_{-*} \\left(\\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\]\n\tinduced by the path $\\left( \\left( \\Lambda_H^{n+1-t} \\right)_{i_0 \\leq n \\leq i_d}, J \\right)_{0 \\leq t \\leq 1}$. Thus, $\\tau_2 \\circ \\zeta_{12}$ is obtained by dualizing the components of the composition\n\t\\[CE_{-*} \\left( \\left( \\Lambda_H^{n+1} \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J, \\alpha \\right) \\to CE_{-*} \\left( \\left( \\Lambda_H^n \\right)_{i_0 \\leq n \\leq i_d}, J_H, \\alpha \\right) . \\]\n\tAccording to Theorem \\ref{thm invariance}, the DG-maps used to define $\\zeta_{12} \\circ \\tau_1$ and $\\tau_2 \\circ \\zeta_{12}$ are DG-homotopic. Therefore the $A_{\\infty}$-functors $\\zeta_{12} \\circ \\tau_1$ and $\\tau_2 \\circ \\zeta_{12}$ are homotopic. This concludes the proof.\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma mapping torus of tau_1 = mapping torus of tau_2}\n\t\n\tThe mapping torus of $\\tau_1 : \\mcA_1 \\to \\mcA_1$ is quasi-equivalent to the mapping torus of $\\tau_2 : \\mcA_2 \\to \\mcA_2$ (see Definition \\ref{definition mapping torus}).\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tLet $\\tau_{12} := \\zeta_{12} \\circ \\tau_1 \\circ \\zeta_{12}^{-1}$.\n\tConsider the following commutative diagram of Adams-graded $A_{\\infty}$-categories\n\t\\[\\begin{tikzcd}\n\t\\mcA_1 \\ar[d, \"\\zeta_{12}\"] & \\mcA_1 \\sqcup \\mcA_1 \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau_1\"] \\ar[d, \"\\zeta_{12} \\sqcup \\zeta_{12}\"] & \\mcA_1 \\ar[d, \"\\zeta_{12}\"] \\\\\n\t\\mcA_2 & \\mcA_2 \\sqcup \\mcA_2 \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau_{12}\"] & \\mcA_2 .\n\t\\end{tikzcd} \\]\n\tEach vertical arrow is a quasi-equivalence according to Lemma \\ref{lemma relation A_1 - A_2}, so it follows from Proposition \\ref{prop invariance of homotopy colimits} that the mapping torus of $\\tau_1$ is quasi-equivalent to the mapping torus of $\\tau_{12}$. \n\tNow according to Lemma \\ref{lemma relation tau_1 - tau_2}, $\\tau_{12}$ is homotopic to $\\tau_2$. Thus the result follows from Proposition \\ref{prop homotopic functors induce quasi-isomorphic homotopy colimits}.\n\t\n\\end{proof}\n\n\\subsection{Projection to $P$}\\label{subsection projection on P}\n\n\\subsubsection{The $A_{\\infty}$-category $\\mcO$}\n\nIn order to define the $A_{\\infty}$-category $\\mcO$, we need to introduce moduli spaces of pseudo-holomorphic discs in $P$. \n\n\\begin{defin}\\label{definition moduli spaces for Lagrangians}\n\t\n\tLet $\\mathbf{L} = (L^n (E))_{(n, E) \\in \\mathbf{Z} \\times \\mcE}$ be a family of mutually transverse connected compact exact Lagrangians in $(P, \\lambda)$. \n\tConsider a sequence of integers $i_0 < \\dots < i_d$, and a family of intersection points $\\left( x_0, x_1, \\dots, x_d \\right)$, where\n\t\\[x_0 \\in L^{i_0} (E_0) \\cap L^{i_d} (E_d) \\text{ and } x_k \\in L^{i_{k-1}} (E_{k-1}) \\cap L^{i_k} (E_k), \\, 1 \\leq k \\leq d. \\] \n\t\\begin{enumerate}\n\t\t\n\t\t\\item If $d=1$, we denote by $\\mcM_{x_1, x_0} \\left( \\mathbf{L}, j \\right)$ the set of equivalence classes of maps $u : \\mathbf{R} \\times \\left[ 0, 1 \\right] \\to P$ such that\n\t\t\\begin{itemize} \n\t\t\t\\item $u$ maps $\\mathbf{R} \\times \\{ 0 \\}$ to $L^{i_0} (E_0)$ and $\\mathbf{R} \\times \\{ 1 \\}$ to $L^{i_1 (E_1)}$,\n\t\t\t\\item $u$ satisfies the asymptotic conditions\n\t\t\t\\[u \\left( s, t \\right) \\underset{s \\to - \\infty}{\\longrightarrow} x_1 \\text{ and } u \\left( s, t \\right) \\underset{s \\to + \\infty}{\\longrightarrow} x_0, \\]\n\t\t\t\\item $u$ is $(i, j)$-holomorphic,\n\t\t\\end{itemize}\n\t\twhere two maps $u$ and $u'$ are identified if there exists $s_0 \\in \\mathbf{R}$ such that $u' (\\cdot, \\cdot) = u (\\cdot + s_0, \\cdot)$.\n\t\t\n\t\t\\item If $d \\geq 2$, we denote by $\\mcM_{x_d, \\dots, x_1, x_0} \\left( \\mathbf{L}, j \\right)$ the set of of pairs $\\left( r, u \\right)$ such that\n\t\t\\begin{itemize}\n\t\t\t\\item $r \\in \\mcR^{d+1}$ and $u : \\Delta_r \\to P$ maps the boundary arc $\\left( \\zeta_{k+1}, \\zeta_k \\right)$ of $\\Delta_r$ to $L^{i_k} (E_k)$,\n\t\t\t\\item $u$ satisfies the asymptotic conditions\n\t\t\t\\[\\left( u \\circ \\epsilon_k (r) \\right) \\left( s, t \\right) \\underset{s \\to - \\infty}{\\longrightarrow} x_k \\text{ and } \\left( u \\circ \\epsilon_0 (r) \\right) \\left( s, t \\right) \\underset{s \\to + \\infty}{\\longrightarrow} x_0, \\]\n\t\t\t\\item $u$ is $(i, j)$-holomorphic.\n\t\t\\end{itemize}\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\nRecall that we chose a contactomorphism $\\phi_H$ as in Lemma \\ref{lemma rectify contact form}.\nWe set \n\\[L_H^n := \\Pi_{P} \\left( \\Lambda_H^n (E) \\right) \\subset P \\text{ and } \\mathbf{L}_H := \\left( L_H^n (E) \\right)_{(n, E) \\in \\mathbf{Z} \\times \\mcE}. \\]\n\n\\begin{defin}\\label{definition category O}\n\t\n\tWe denote by $\\mcO$ the $A_{\\infty}$-category defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\mcO$ are the Lagrangians $L_H^n (E)$, $(n, E) \\in \\mathbf{Z} \\times \\mcE$, \n\t\t\n\t\t\\item the space of morphisms from $L_H^i (E)$ to $L_H^j (E')$ is either generated by $L_H^i (E) \\cap L_H^j (E')$ if $(i, E) < (j, E')$, or $\\mathbf{F}$ if $(i, E) = (j, E')$, or $0$ otherwise, and\n\t\t\n\t\t\\item the operations are such that $1 \\in \\mcO \\left( L_H^n (E), L_H^n (E) \\right)$ is a strict unit, and for every sequence $(i_0, E_0) < \\dots < (i_d, E_d)$, for every sequence of intersection points \n\t\t\\[\\left( x_1, \\dots, x_d \\right) \\in \\left( L_H^{i_0} (E_0) \\cap L_H^{i_1} (E_1) \\right) \\times \\dots \\times \\left( L_H^{i_{d-1}} (E_{d-1}) \\cap L_H^{i_d} (E_d) \\right), \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcO} \\left( x_1, \\dots, x_d \\right) = \\sum \\limits_{x_0 \\in L_H^{i_0} (E_0) \\cap L_H^{i_d} (E_d)} \\# \\mcM_{x_d, \\dots, x_1, x_0} \\left( \\mathbf{L}_H, j \\right) x_0 . \\]\n\t\t\n\t\\end{enumerate} \t\n\t\n\\end{defin}\n\n\\subsubsection{The quasi-autoequivalence $\\gamma$}\n\nBefore defining the $A_{\\infty}$-functor $\\gamma : \\mcO \\to \\mcO$, we recall Legendrian contact homology as defined in \\cite{EES07}. To each generic Legendrian $\\Lambda$ in $\\mathbf{R} \\times P$, the authors associate a semi-free DG-algebra $A = A \\left( \\Lambda, j \\right)$ generated by the self-intersection points of $\\Pi_P \\left( \\Lambda \\right)$, with a differential $\\partial : A \\to A$ defined using $j$-holomorphic discs in $P$. In our case, the differential of $A \\left( \\bigsqcup_k \\Lambda_H^k (E), j \\right)$ on a generator $x_0 \\in L_H^{i_0} (E_0) \\cap L_H^{i_d} (E_d)$ is given by \n\\[\\partial x_0 = \\sum_{\\left( x_1, \\dots, x_d \\right)} \\# \\mcM_{x_d, \\dots, x_1, x_0} \\left( \\mathbf{L}_H, j \\right) x_d \\cdots x_1 \\]\nwhere the sum is over the sequences \n\\[\\left( x_1, \\dots, x_d \\right) \\in \\left( L_H^{i_0} (E_0) \\cap L_H^{i_1} (E_1) \\right) \\times \\dots \\times \\left( L_H^{i_{d-1}} (E_{d-1}) \\cap L_H^{i_d} (E_d) \\right) . \\]\nAccording to \\cite[Theorem 2.1]{DR16}, Legendrian contact homology as defined in \\cite{EES07} coincides with the version exposed in section \\ref{section Legendrian invariants}: \n\\[A \\left( \\Lambda, j \\right) = CE_* \\left( \\Lambda, \\left( D \\Pi_P \\right)_{|\\xi}^* j, \\alpha \\right) . \\]\nWe introduced this version only because it makes clearer the fact that some operations are defined using pseudo-holomorphics polygons in the base $P$.\n\n\\begin{defin}\\label{definition functor gamma}\n\t\n\tWe denote by $\\gamma : \\mcO \\to \\mcO$ the $A_{\\infty}$-functor defined as follows\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\gamma \\left( L_H^n (E) \\right) = L_H^{n+1} (E)$,\n\t\t\n\t\t\\item on morphisms, the map\n\t\t\\[\\gamma : \\mcO \\left( L_H^{i_0} (E_0), L_H^{i_1} (E_1) \\right) \\otimes \\dots \\otimes \\mcO \\left( L_H^{i_{d-1}} (E_{d-1}), L_H^{i_d} (E_d) \\right) \\to \\mcO \\left( L_H^{i_0 + 1} (E_0), L_H^{i_d+1} (E_d) \\right) \\]\n\t\tis obtained by dualizing the components of the DG-isomorphism\n\t\t\\begin{align*}\n\t\tA \\left( \\bigsqcup \\limits_{k=i_0}^{i_d} \\Lambda_H^{k+1}, j \\right) & = CE_{-*} \\left( \\mathbf{R} \\times \\bigsqcup \\limits_{k=i_0}^{i_d} \\Lambda_H^{k+1}, \\left( D \\Pi_P \\right)_{|\\xi}^* j, \\alpha \\right) \\\\\n\t\t & \\to CE_{-*} \\left( \\mathbf{R} \\times \\bigsqcup \\limits_{k=i_0}^{i_d} \\Lambda_H^k, \\left( D \\Pi_P \\right)_{|\\xi}^* j, \\alpha \\right) = A \\left( \\bigsqcup \\limits_{k=i_0}^{i_d} \\Lambda_H^k, j \\right)\n\t\t\\end{align*}\n\t\tinduced by the Legendrian isotopy $\\left( \\bigsqcup \\limits_{k=i_0}^{i_d} \\Lambda_H^{k+1-t} \\right)_{0 \\leq t \\leq 1}$ (see Theorem \\ref{thm invariance}).\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{lemma}\\label{lemma gamma is a quasi-equivalence}\n\t\n\tThe $A_{\\infty}$-functor $\\gamma : \\mcO \\to \\mcO$ is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows from the fact that $\\gamma$ is defined by dualizing the components of a DG-isomorphism (see the proof of Lemma \\ref{lemma tau is a quasi-equivalence}). \n\t\n\\end{proof}\n\nHere the $\\mathbf{Z}$-splitting \n\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcO \\right), \\quad (n, E) \\mapsto L_H^n (E) \\]\nis compatible with the quasi-autoequivalence $\\gamma$ in the sense of Definition \\ref{definition group-action}. As explained there, this turns $\\mcO$ into an Adams-graded $A_{\\infty}$-category. \n\n\\subsubsection{Relation between $\\left( \\mcA_2, \\tau_2 \\right)$ and $\\left( \\mcO, \\gamma \\right)$}\n\nWe now explain how the pairs $\\left( \\mcA_2, \\tau_2 \\right)$ and $\\left( \\mcO, \\gamma \\right)$ are related.\nSee Figure \\ref{figure action of the projection to P}, where we illustrate the action of the projection $\\Pi_P$ in the case \n\\[\\left( P, \\lambda \\right) = \\left( T^* S^1, p dq \\right), \\, L = 0_{S^1}, \\text{ and } H \\left( q, p \\right) = h(q), \\]\nwhere $h : S^1 \\to \\mathbf{R}$ is a Morse function.\n\n\\begin{figure}\n\t\\def1\\textwidth{1\\textwidth}\n\t\\input{Invariants_lift\/Figures\/dessin14.pdf_tex}\t\t\n\t\\caption{Action of the projection $\\Pi_{T^*S^1}$}\t\n\t\\label{figure action of the projection to P}\n\\end{figure}\n\t\n\\begin{lemma}\\label{lemma relation A_2 - O}\n\t\n\tThere is a strict $A_{\\infty}$-isomorphism $\\zeta_2 : \\mcA_2 \\to \\mcO$ defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\zeta_2 \\left( \\Lambda_H^n (E) \\right) = L_H^n (E)$,\n\t\t\n\t\t\\item on morphisms, $\\zeta_2$ sends a Reeb chord $c$ in $\\mcA_2 \\left( \\Lambda_H^i (E), \\Lambda_H^j (E') \\right)$ to the intersection point \n\t\t\\[\\zeta_2 \\left( c \\right) = \\Pi_P \\left( c \\right) \\]\n\t\tin $\\mcO \\left( L_H^i (E), L_H^j (E') \\right)$.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe have to show that $\\zeta_2$ is an $A_{\\infty}$-map.\n\tSince $J = \\left( D \\Pi_P \\right)_{|\\xi}^* j$, it follows from \\cite[Theorem 2.1]{DR16} that the map \n\t\\[ u = \\left( \\sigma , v \\right) \\mapsto \\Pi_P \\circ v \\]\n\tinduces a bijection\n\t\\[\\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda_H, J, \\alpha \\right) \\xrightarrow{\\sim} \\mcM_{\\Pi_P (c_d) \\dots \\Pi_P (c_1), \\Pi_P (c_0)} \\left( \\mathbf{L}_H, j \\right) . \\]\n\tThis implies the result.\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma relation tau_2 - gamma}\n\t\n\tWe have\n\t\\[\\gamma = \\zeta_2 \\circ \\tau_2 \\circ \\zeta_2^{-1} . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis follows from the definitions of $\\tau_2$, $\\gamma$, $\\zeta_2$ and the fact that $J = \\left( D \\Pi_P \\right)_{|\\xi}^* j$.\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma mapping torus of tau_2 = mapping torus of gamma}\n\t\n\tThe mapping torus of $\\tau_2 : \\mcA_2 \\to \\mcA_2$ is quasi-equivalent to the mapping torus of $\\gamma : \\mcO \\to \\mcO$ (see Definition \\ref{definition mapping torus}).\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tAccording to Lemma \\ref{lemma relation tau_2 - gamma} the following diagram of Adams-graded $A_{\\infty}$-categories is commutative \n\t\\[\\begin{tikzcd}\n\t\\mcA_2 \\ar[d, \"\\zeta_2\"] & \\mcA_2 \\sqcup \\mcA_2 \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau_2\"] \\ar[d, \"\\zeta_2 \\sqcup \\zeta_2\"] & \\mcA \\ar[d, \"\\zeta_2\"] \\\\\n\t\\mcO & \\mcO \\sqcup \\mcO \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\gamma\"] & \\mcO .\n\t\\end{tikzcd} \\]\n\tMoreover, each vertical arrow is a quasi-equivalence according to Lemma \\ref{lemma relation A_2 - O}. Thus, the result follows from Proposition \\ref{prop invariance of homotopy colimits}.\n\t\n\\end{proof}\n\n\\subsection{Mapping torus of $\\gamma$}\\label{subsection proof of the main result}\n\nIn this section, we show that we can apply Theorem \\ref{thm mapping torus in weak situation} (Theorem \\ref{thm mapping torus in weak situation introduction} in the introduction) in order to compute the mapping torus of $\\gamma : \\mcO \\to \\mcO$. This allows us to finish the proof of Theorem \\ref{thm mainthm restatement}. \n\nRecall that we fixed a contactomorphism $\\phi_H$ of $V$ such that $\\phi_H^* \\alpha_H = \\alpha$. Also recall that if $\\theta$ is some real number, then \n\\[\\Lambda^{\\theta} (E) = \\left\\{ \\left( f_E(x) + \\theta, x \\right) \\mid x \\in L \\right\\}, \\, \\Lambda_H^{\\theta} (E) = \\phi_H^{-1} \\left( \\Lambda^{\\theta} (E) \\right), \\text{ and } L_H^{\\theta} (E) = \\Pi_P \\left( \\Lambda_H^{\\theta} (E) \\right) . \\]\n\n\\subsubsection{Continuation elements}\n\nWe denote by $\\mcO_{2r}$ the $A_{\\infty}$-category obtained from $\\mcO$ by applying the functor of Definition \\ref{definition forgetful functor}. \nBesides, we denote by\n\\[\\Gamma = \\left\\{ c_n (E) \\in \\mcO_{2r} (L^n (E), L^{n+1} (E)) \\mid (n,E) \\in \\mathbf{Z} \\times \\mcE \\right\\} \\]\nthe set of continuation elements in $\\mcO_{2r}$ induced by the exact Lagrangian isotopies $(L_H^{n+t})_{0 \\leq t leq 1}$ (see for example \\cite[section 3.3]{GPS20}).\n\nRecall that if $\\mcC$ is an $A_{\\infty}$-category equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcC \\right)$, we denote by $\\mcC^0$ the full $A_{\\infty}$-subcategory of $\\mcC$ whose set of objects corresponds to $\\{ 0 \\} \\times \\mcE$.\n\n\\begin{lemma}\\label{lemma localization of O equals CF}\n\t\n\tThere are quasi-equivalences of $A_{\\infty}$-categories\n\t\\[\\mcO_{2r}^0 \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}_H) \\text{ and } \\mcO_{2r} \\left[ \\Gamma^{-1} \\right]^0 \\simeq \\mcF uk (\\mathbf{L}_H) . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThis is explained in \\cite[Lecture 10]{Sei13}. See also \\cite{GPS20}, where they define Fukaya categories by localization.\n\t\n\\end{proof}\n\n\\subsubsection{The $\\mcO_{2r}$-bimodule map}\\label{subsection bimodule map}\n\nIn order to apply Theorem \\ref{thm mapping torus in weak situation}, we need a degree $0$ closed $\\mcO_{2r}$-module map $f : \\mcO_{2r} \\left( -, - \\right) \\to \\mcO_{2r} \\left( -, \\gamma \\left( - \\right) \\right)$ such that the elements in $f \\left( \\text{units} \\right)$ satisfy certain hypotheses. \nAs usual, we would like to find such an $f$ ``geometrically'', i.e. using some Lagrangian (or Legendrian) isotopy. However, here the unit $1 = e_{L_H^k (E)} \\in \\mcO \\left( L_H^k (E), L_H^k (E) \\right)$, which is not a ``geometric'' morphism, is supposed to be sent by $f$ to something in $\\mcO \\left( L_H^k (E), L_H^{k+1} (E) \\right)$, which is generated by the ``geometric'' elements of $L_H^k (E) \\cap L_H^{k+1} (E)$.\nTherefore, we need to somehow replace this unit by some intersection point between Lagrangians. The idea is that we will slightly perturb $L_H^k (E)$ to $L_H^{k+\\delta} (E)$, and then replace $e_{L_H^k (E)}$ by the continuation element in the vector space generated by $L_H^k (E) \\cap L_H^{k+\\delta} (E)$. \n\nObserve that if $\\delta$ is small enough, $L_H^{k+\\delta} (E)$ is a small perturbation of $L_H^k (E)$. Therefore, in a Weinstein neighborhood of $L_H^k (E)$, the Lagrangian $L_H^{k+\\delta} (E)$ is the graph of $d h_{\\delta, k, E}$, where $h_{\\delta, k, E}$ is some Morse function on $L (E)$.\nIn particular, the intersection points between $L_H^k (E)$ and $L_H^{k+\\delta} (E)$ correspond to the critical points of $h_{\\delta, k, E}$.\nMoreover, the continuation element in the vector space generated by $L_H^k (E) \\cap L_H^{k + \\delta} (E)$ corresponds to the sum of the minima of $h_{\\delta, k, E}$. \n\n\\begin{exa}\n\t\n\tAssume that we are in the case\n\t\\[\\left( P, \\lambda \\right) = \\left( T^* M, p dq \\right), \\, L = 0_M, \\text{ and } H \\left( q, p \\right) = h(q), \\]\n\twhere $h : M \\to \\mathbf{R}$ is a Morse function. \n\tAs explained in example \\ref{example rectify contact form in cotangent case}, in this case we have \n\t\\[L_H^{\\theta} = \\Pi_{T^*M} \\left( j^1 \\left( \\theta e^h \\right) \\right) = \\mathrm{graph} \\left( d \\left( \\theta e^h \\right) \\right) . \\]\n\tThus, $L_H^{k+\\delta}$ is the graph of $d \\left( \\delta e^h \\right)$ over $L_H^k$. \n\t\n\\end{exa}\n\nThe following result will allow us to ``replace'' the units by geometric morphisms as explained above. We denote by $g = - d\\lambda (-, j -)$ the metric on $P$ induced by $j$ and $\\left( -d \\lambda \\right)$.\n\n\\begin{lemma}\\label{lemma discs with boundary on small deformation}\n\t\n\tFor every positive integer $n$, there exists $\\delta_n > 0$ such that the following holds for every $\\delta \\in \\left] 0, \\delta_n \\right]$. For every sequence of integers\n\t\\[(-n, E_0) \\leq (j_0, E_0) < \\dots < (j_p, E_p) \\leq (\\ell_0, E_0') < \\dots < (\\ell_q, E_q') \\leq (n, E_q'), \\quad p,q \\geq 0, \\]\n\tthe rigid $j$-holomorphic discs in $P$ with boundary on \n\t\\[L_H^{j_0} (E_0) \\cup \\dots \\cup L_H^{j_p} (E_p) \\cup L_H^{\\ell_0 + \\delta} (E_0') \\cup \\dots \\cup L_H^{\\ell_q+\\delta} (E_q') \\]\n\tare \n\t\\begin{enumerate}\n\t\t\n\t\t\\item in bijection with the rigid $j$-holomorphic discs in $P$ with boundary on \n\t\t\\[L_H^{j_0} (E_0) \\cup \\dots \\cup L_H^{j_p} (E_p) \\cup L_H^{\\ell_0} (E_0') \\cup \\dots \\cup L_H^{\\ell_q} (E_q') \\]\n\t\tif $(j_p, E_p) < (\\ell_0, E_0')$, or\n\t\t\n\t\t\\item in bijection with the rigid $j$-holomorphic discs in $P$ with boundary on \n\t\t\\[L_H^{j_0} (E_0) \\cup \\dots \\cup L_H^{j_{p-1}} (E_{p-1}) \\cup L_H^{\\ell_0} (E_0') \\cup L_H^{\\ell_1} (E_1') \\cup \\dots \\cup L_H^{\\ell_q} (E_q') \\]\n\t\twith a flow line of $\\left( - \\nabla_g h_{\\delta, k, E_0'} \\right)$ attached on the component in $L_H^{\\ell_0} (E_0')$ if $(j_p, E_p) = (\\ell_0, E_0')$.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThe case $j_p < \\ell_0$ follows from transversality of the moduli spaces in consideration. The case $j_p = \\ell_0$ follows from the main analytic theorem of \\cite{EES09} (Theorem 3.6). \n\t\n\\end{proof}\n\nIn order to define the $\\mcO_{2r}$-bimodule map $f$ properly, we will use Lemma \\ref{lemma discs with boundary on small deformation} to modify the $A_{\\infty}$-category $\\mcO_{2r}$.\nIn the following, we fix a \\emph{decreasing} sequence of positive numbers $\\left( \\delta_n \\right)_{n \\geq 1}$ such that, for every $n$,\n\\begin{enumerate}\n\t\n\t\\item Lemma \\ref{lemma discs with boundary on small deformation} holds with $\\delta_n$, and\n\t\n\t\\item $\\delta_n$ is small enough so that there is no handle slide instant in the Legendrian isotopy\n\t\\[\\bigcup_{\\ell = -n}^n \\Lambda_H^{\\ell + \\delta_n t} = \\bigcup_{\\ell = -n}^n \\bigcup_{E \\in \\mcE} \\Lambda_H^{\\ell + \\delta_n t} (E), \\quad t \\in \\left[ 0, 1 \\right]. \\]\n\t\n\\end{enumerate}\n\nWe define two families of $A_{\\infty}$-categories $\\left( \\mcO_{n, k} \\right)_{n,k}$ and $\\left( \\widetilde{\\mcO}_{n,k} \\right)_{n,k}$ indexed by the couples $(n,k)$, where $n \\geq 1$ and $-n \\leq k \\leq n$. \nThe $A_{\\infty}$-category $\\mcO_{n, k}$ is basically obtained from $\\mcO_{2r}$ by restricting to objects $L_H^i (E)$, $-n \\leq i \\leq n$, and adding a copy of the object $L_H^k (E)$.\n\n\\begin{defin}\\label{definition category O n,k}\n\t\n\tFor every $(j,E) \\in \\mathbf{Z} \\times \\mcE$, let $\\overline{L_H^j} (E)$ be a copy of $L_H^j (E)$.\n\tWe denote by $\\mcO_{n,k}$ the $A_{\\infty}$-category defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the set of objects of $\\mcO_{n,k}$ is \n\t\t\\[\\mathrm{ob} \\left( \\mcO_{n,k} \\right) = \\left\\{ L_H^j (E) \\mid -n \\leq j \\leq k, E \\in \\mcE \\right\\} \\cup \\left\\{ \\overline{L_H^{\\ell}} (E) \\mid k \\leq \\ell \\leq n, E \\in \\mcE \\right\\}, \\]\n\t\t\n\t\t\\item the spaces of morphisms in $\\mcO_{n,k}$ are the corresponding spaces of morphisms in $\\mcO_{2r}$ when we replace $\\overline{L_H^{\\ell}} (E)$, $k \\leq \\ell \\leq n$, by $L_H^{\\ell} (E)$, except that \n\t\t\\[\\mcO_{n,k} \\left( \\overline{L_H^k} (E), L_H^k (E) \\right) = \\{ 0 \\}, \\]\n\t\t\n\t\t\\item the operations are the same as in $\\mcO_{2r}$.\n\t\t\n\t\\end{enumerate} \t\n\t\n\\end{defin}\n\nThe $A_{\\infty}$-category $\\widetilde{\\mcO}_{n,k}$ is obtained from $\\mcO_{n,k}$ by perturbing the objects $\\overline{L_H^{\\ell}} (E)$, $k \\leq \\ell \\leq n$, to $L_H^{\\ell + \\delta_n} (E)$.\n\n\\begin{defin}\\label{definition category O n,k tilde}\n\t\n\tLet \n\t\\[\\Theta_{n,k} := \\left\\{ -n, \\dots, k \\right\\} \\cup \\left\\{ \\ell + \\delta_n \\mid k \\leq \\ell \\leq n \\right\\} \\subset \\mathbf{R}, \\text{ and } \\widetilde{\\mathbf{L}_H} := \\left( L_H^{\\theta} (E) \\right)_{(\\theta, E) \\in \\Theta_{n,k} \\times \\mcE}. \\]\n\tWe denote by $\\widetilde{\\mcO}_{n,k}$ the $A_{\\infty}$-category defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects of $\\widetilde{\\mcO}_{n,k}$ are the Lagrangians $L_H^{\\theta} (E)$, $(\\theta, E) \\in \\Theta_{n,k} \\times \\mcE$,\n\t\t\n\t\t\\item the space of morphisms from $L_H^{\\theta} (E)$ to $L_H^{\\theta'} (E')$ is either generated by $L_H^{\\theta} (E) \\cap L_H^{\\theta'} (E')$ if $(\\theta, E) < (\\theta', E')$, or $\\mathbf{F}$ if $(\\theta, E) = (\\theta', E')$, or $0$ otherwise,\n\n\t\t\\item the operations are such that $e_{L_H^{\\theta} (E)} = 1 \\in \\widetilde{\\mcO}_{n,k} \\left( L_H^{\\theta} (E), L_H^{\\theta} (E) \\right)$ is a strict unit, and for every sequence $(\\theta_0, E_0) < \\dots < (\\theta_d, E_d)$, for every sequence of intersection points \n\t\t\\[\\left( x_1, \\dots, x_d \\right) \\in \\left( L_H^{\\theta_0} (E_0) \\cap L_H^{\\theta_1} (E_1) \\right) \\times \\dots \\times \\left( L_H^{\\theta_{d-1}} (E_{d-1}) \\cap L_H^{\\theta_d} (E_d) \\right), \\]\n\t\twe have \n\t\t\\[\\mu_{\\widetilde{\\mcO}_{n,k}} \\left( x_1, \\dots, x_d \\right) = \\sum \\limits_{x_0 \\in L_H^{\\theta_0} (E_0) \\cap L_H^{\\theta_d} (E_d)} \\# \\mcM_{x_d, \\dots, x_1, x_0} \\left( \\widetilde{\\mathbf{L}_H}, j \\right) x_0 . \\]\n\t\t\n\t\\end{enumerate} \t\n\t\n\\end{defin}\n\nThese $A_{\\infty}$-categories being defined, Lemma \\ref{lemma discs with boundary on small deformation} implies the following result.\n\n\\begin{lemma}\\label{lemma rho is a functor}\n\t\n\tThere is a strict $A_{\\infty}$-functor $\\rho_{n,k} : \\mcO_{n,k} \\to \\widetilde{\\mcO}_{n,k}$ defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\rho_{n,k} \\left( L_H^j (E) \\right) = L_H^j (E)$ if $-n \\leq j \\leq k$ and $\\rho_{n,k} \\left( \\overline{L_H^{\\ell}} (E) \\right) = L_H^{\\ell + \\delta_n} (E)$ if $k \\leq \\ell \\leq n$,\n\t\t\n\t\t\\item on morphisms, $\\rho_{n,k}$ sends the unit of $\\mcO_{n,k} \\left( L_H^k (E), \\overline{L_H^k} (E) \\right) = \\mathbf{F}$ to the continuation element in $\\widetilde{\\mcO}_{n,k} \\left( L_H^k (E), L_H^{k + \\delta_n} (E) \\right)$, and it sends any other morphism of $\\mcO_{n,k}$ to the corresponding one in $\\widetilde{\\mcO}_{n,k}$. \n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tConsider a sequence $\\left( x_0, \\dots, x_{d-1} \\right)$ of morphisms in $\\mcO_{n,k}$.\n\tIf in this sequence there is no morphism from $L_H^k (E)$ to $\\overline{L_H^k} (E)$, then the relation \n\t\\[\\mu_{\\widetilde{\\mcO}_{n,k}} \\left( \\rho_{n,k} x_0, \\dots, \\rho_{n,k} x_d \\right) = \\rho_{n,k} \\left( \\mu_{\\mcO_{n,k}} \\left( x_0, \\dots, x_d \\right) \\right) \\]\n\tfollows directly from the first item of Lemma \\ref{lemma discs with boundary on small deformation}.\n\tNow assume that there is $p \\in \\{ 0, \\dots, d-1 \\}$ such that $x_p = e_{L_H^k (E)} \\in \\mcO_{n,k} \\left( L_H^k (E), \\overline{L_H^k} (E) \\right)$. Recall that the continuation element in $\\widetilde{\\mcO}_{n,k} \\left( L_H^k (E), L_H^{k + \\delta_n} (E) \\right)$ corresponds to the sum of the minima of $h_{\\delta_n, k, E}$. \n\tThen the second item of Lemma \\ref{lemma discs with boundary on small deformation} implies that \n\t\\[\\mu_{\\widetilde{\\mcO}_{n,k}} \\left( \\rho_{n,k} x_0, \\dots, \\rho_{n,k} x_d \\right) = \\left\\{\n\t\\begin{array}{ll}\n\t\\rho_{n,k} x_1 & \\text{if } d=1 \\text{ and } p=0 \\\\\n\t\\rho_{n,k} x_0 & \\text{if } d=1 \\text{ and } p=1 \\\\\n\t0 & \\text{otherwise} .\n\t\\end{array}\n\t\\right. \\]\n\tThus, the $A_{\\infty}$-relation for $\\rho_{n,k}$ is still satisfied according to the behavior of the operations $\\mu_{\\mcO_{n,k}}$ with respect to the unit $e_{L_H^k (E)}$.\n\t\n\\end{proof}\n\nNow that we have in some sense ``replace'' the unit $e_{L_H^k (E)} \\in \\mcO_{n,k} \\left( L_H^k (E), \\overline{L_H^k} (E) \\right)$ by the continuation element in $\\widetilde{\\mcO}_{n,k} \\left( L_H^k (E), L_H^{k + \\delta_n} (E) \\right)$, we can define geometrically an $A_{\\infty}$-functor that will finally allow us to define the $\\mcO_{2r}$-bimodule map $f$.\n\n\\begin{defin}\\label{definition functor nu}\n\t\n\tWe denote by $\\nu_{n,k} : \\widetilde{\\mcO}_{n, k} \\to \\mcO_{2r}$ the $A_{\\infty}$-functor defined as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item on objects, $\\nu_{n,k} \\left( L_H^j (E) \\right) = L_H^j (E)$ if $-n \\leq j \\leq k$, and $\\nu_{n,k} \\left( L_H^{\\ell + \\delta_n} (E) \\right) = L_H^{\\ell + 1} (E)$ if $k \\leq \\ell \\leq n$,\n\t\t\n\t\t\\item on morphisms, $\\nu_{n,k}$ is obtained by dualizing the components of the DG-isomorphism\n\t\t\\[A \\left( \\bigsqcup\\limits_{i=-n}^{n+1} \\Lambda_H^i \\right) \\xrightarrow{\\sim} A \\left( \\bigsqcup\\limits_{j=-n}^{k} \\Lambda_H^j \\sqcup \\bigsqcup\\limits_{\\ell=k}^{n} \\Lambda_H^{\\ell+\\delta_n} \\right) . \\]\n\t\tinduced by the Legendrian isotopy \n\t\t\\[\\left( \\bigsqcup\\limits_{j=-n}^{k} \\Lambda_H^j \\right) \\sqcup \\left( \\bigsqcup\\limits_{\\ell=k}^{n} \\Lambda_H^{\\ell+1-t(1-\\delta_n)} \\right), \\quad t \\in \\left[ 0,1 \\right] \\] \n\t\t(see Theorem \\ref{thm invariance} or \\cite[Proposition 2.6]{EES07}).\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{rmks}\\label{rmk properties of sigma}\n\t\n\tWe point out some properties of the $A_{\\infty}$-functors\n\t\\[\\sigma_{n,k} := \\nu_{n,k} \\circ \\rho_{n,k} : \\mcO_{n,k} \\to \\mcO_{2r} . \\]\n\t\\begin{enumerate}\n\t\t\n\t\t\\item Let $n \\leq p$ be two positive integers, and let $k \\in \\{ -n, \\dots, n \\}$. Recall that we chose $\\delta_n$ small enough so that there is no handle slide instant in the Legendrian isotopy \n\t\t\\[\\bigsqcup_{\\ell=-n}^n \\Lambda_H^{\\ell + \\delta_n t}, 0 \\leq t \\leq 1. \\]\n\t\tSince $\\delta_p \\leq \\delta_n$, neither is there any handle slide instant in the Legendrian isotopy \n\t\t\\[\\bigsqcup_{\\ell=-n}^n \\Lambda_H^{\\ell + \\delta_p t}, 0 \\leq t \\leq 1. \\]\n\t\tTherefore, $\\sigma_{p,k}$ agrees with $\\sigma_{n,k}$ on $\\mcO_{n,k} \\subset \\mcO_{p,k}$.\n\t\t\n\t\t\\item \tConsider a sequence of integers \n\t\t\\[-n \\leq j_0 < \\dots < j_p \\leq k_1 < k_2 \\leq \\ell_0 < \\dots < \\ell_q \\leq n , \\]\n\t\tand a sequence of morphisms \n\t\t\\begin{align*}\n\t\t\t& \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) \\in \\mcO_{n,k_i} \\left( L_H^{j_0} (E_0), L_H^{j_1} (E_1) \\right) \\times \\dots \\times \\mcO_{n,k_i} \\left( L_H^{j_{p-1}} (E_{p-1}), L_H^{j_p} (E_p) \\right) \\\\\n\t\t\t& \\times \\mcO_{n,k_i} \\left( L_H^{j_p} (E_p), \\overline{L_H^{\\ell_0}} (E_0') \\right) \\times \\mcO_{n,k_i} \\left( \\overline{L_H^{\\ell_0}} (E_0'), \\overline{L_H^{\\ell_1}} (E_1') \\right) \\times \\dots \\times \\mcO_{n,k_i} \\left( \\overline{L_H^{\\ell_{q-1}}} (E_{q-1}'), \\overline{L_H^{\\ell_q}} (E_q') \\right) .\n\t\t\\end{align*}\n\t\tSince the Legendrian isotopy defining $\\nu_{n,k_i}$ is \n\t\t\\[\\left( \\bigsqcup\\limits_{j=-n}^{k_i} \\Lambda_H^j \\right) \\sqcup \\left( \\bigsqcup\\limits_{\\ell=k_i}^{n} \\Lambda_H^{\\ell+1-t(1-\\delta_n)} \\right), \\quad t \\in \\left[ 0,1 \\right], \\] \n\t\twe have\n\t\t\\[\\left\\{\n\t\t\\begin{array}{l}\n\t\t\\sigma_{n, k_1} \\left( x_0, \\dots, x_{p-1} \\right) = \\delta_{1 p} x_0 \\\\\n\t\t\\sigma_{n, k_2} \\left( y_0, \\dots, y_{q-1} \\right) = \\gamma \\left( y_0, \\dots, y_{q-1} \\right) \\\\\n\t\t\\sigma_{n, k_2} \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) = \\sigma_{n, k_1} \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) .\n\t\t\\end{array}\n\t\t\\right. \\]\n\t\t\n\t\t\\item By construction, the $A_{\\infty}$-functor $\\nu_{n,k}$ sends the continuation element in $\\widetilde{\\mcO}_{n,k} \\left( L_H^k (E), L_H^{k + \\delta_n} (E) \\right)$ (corresponding to the sum of the minima of $h_{\\delta_n, k, E}$) to the continuation element $c_k (E)$ in $\\mcO_{2r} \\left( L_H^k (E), L_H^{k+1} (E) \\right)$.\n\t\tIn other words, $\\sigma_{n,k}$ sends the unit $e_{L_H^k (E)} \\in \\mcO_{n,k} \\left( L_H^k (E), \\overline{L_H^k} (E) \\right)$ to $c_k (E)$.\n\t\t\n\t\t\\item The map $\\sigma_{n,k} : \\mcO_{n,k} \\left( L_H^j(E) , \\overline{L_H^k}(E') \\right) \\to \\mcO_{2r} \\left( L_H^j(E), L_H^{k+1} (E') \\right)$ is a quasi-somorphism for every $j < k$ and $E, E' \\in \\mcE$\n\t\t\n\t\\end{enumerate}\n\t\n\\end{rmks}\n\nWe can now state and prove the desired result. \n\n\\begin{lemma}\\label{lemma bimodule map}\n\t\n\tThere exists a degree $0$ closed $\\mcO_{2r}$-bimodule map $f : \\mcO_{2r} \\left( -, - \\right) \\to \\mcO_{2r} \\left( -, \\gamma (-) \\right)$ which sends the unit $e_{L_H^k (E)} \\in \\mcO_{2r} \\left( L_H^k (E), L_H^k (E) \\right)$ to the continuation element $c_k (E) \\in \\mcO_{2r} \\left( L_H^k (E), L_H^{k+1} (E) \\right) \\cap \\Gamma$, and such that $f : \\mcO_{2r} \\left( L_H^j(E) , L_H^k(E') \\right) \\to \\mcO_{2r} \\left( L_H^j(E), L_H^{k+1} (E') \\right)$ is a quasi-somorphism for every $j< k$ and $E, E' \\in \\mcE$.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tConsider a sequence \n\t\\[(j_0, E_0) < \\dots < (j_p, E_p) \\leq (k ,E) = (\\ell_0, E_0') < \\dots < (\\ell_q, E_q') , \\]\n\tand a sequence of morphisms \n\t\\begin{align*}\n\t& \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) \\in \\mcO_{2r} \\left( L_H^{j_0} (E_0), L_H^{j_1} (E_1) \\right) \\times \\dots \\times \\mcO_{2r} \\left( L_H^{j_{p-1}} (E_{p-1}), L_H^{j_p} (E_p) \\right) \\\\\n\t& \\times \\mcO_{2r} \\left( L_H^{j_p} (E_p), L_H^{k} (E_0') \\right) \\times \\mcO_{2r} \\left( L_H^{k} (E_0'), L_H^{\\ell_1} (E_1') \\right) \\times \\dots \\times \\mcO_{2r} \\left( L_H^{\\ell_{q-1}} (E_{q-1}'), L_H^{\\ell_q} (E_q') \\right) .\n\t\\end{align*}\n\tWe choose $n \\geq 1$ such that $-n \\leq j_0 \\leq \\ell_q \\leq n$, and we set \n\t\\[f \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) := \\sigma_{n, k} \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) \\in \\mcO_{2r} \\left( L_H^{j_0} (E_0), \\gamma L_H^{\\ell_q} (E_q') \\right), \\]\n\twhere on the right hand side we consider that \n\t\\begin{align*}\n\t& \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) \\in \\mcO_{n,k} \\left( L_H^{j_0} (E_0), L_H^{j_1} (E_1) \\right) \\times \\dots \\times \\mcO_{n,k} \\left( L_H^{j_{p-1}} (E_{p-1}), L_H^{j_p} (E_p) \\right) \\\\\n\t& \\times \\mcO_{n,k} \\left( L_H^{j_p} (E_p), \\overline{L_H^k} (E_0') \\right) \\times \\mcO_{n,k} \\left( \\overline{L_H^k} (E_0'), \\overline{L_H^{\\ell_1}} (E_1') \\right) \\times \\dots \\times \\mcO_{n,k} \\left( \\overline{L_H^{\\ell_{q-1}}} (E_{q-1}'), \\overline{L_H^{\\ell_q}} (E_q') \\right) .\n\t\\end{align*}\n\tObserve that $f$ is well defined (it does not depend on the choice of $n$) according to the first item of Remark \\ref{rmk properties of sigma}.\n\n\tWe now verify that $f$ is closed. \n\tAccording to Definition \\ref{definition bimodule}, we have \n\t\\begin{align*}\n\t\\mu_{\\mathrm{Mod}_{\\mcC, \\mcC}}^1 & \\left( f \\right) \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) \\\\\n\t& = \\sum \\sigma_{n, k} \\left( \\dots, \\mu_{\\mcO_{2r}} \\left( \\dots \\right), \\dots, u, \\dots \\right) \\\\ \n\t& + \\sum \\sigma_{n, \\ell_s} \\left( \\dots, \\mu_{\\mcO_{2r}} \\left( x_r, \\dots, x_{p-1}, u, y_0, \\dots, y_{s-1} \\right), \\dots \\right) \\\\\n\t& + \\sum \\sigma_{n, k} \\left( \\dots, u, \\dots, \\mu_{\\mcO_{2r}} \\left( \\dots \\right), \\dots \\right) \\\\ \n\t& + \\sum \\mu_{\\mcO_{2r}} \\left( \\dots, \\sigma_{n, k} \\left( \\dots, u, \\dots \\right), \\gamma \\left( \\dots \\right), \\dots, \\gamma \\left( \\dots \\right) \\right) .\n\t\\end{align*}\n\tNow according to the second item of Remark \\ref{rmk properties of sigma}, we have \n\t\\begin{align*}\n\t\\sum \\sigma_{n, \\ell_s} & \\left( \\dots, \\mu_{\\mcO_{2r}} \\left( x_r, \\dots, x_{p-1}, u, y_0, \\dots, y_{s-1} \\right), \\dots \\right) \\\\\n\t& = \\sum \\sigma_{n, k} \\left( \\dots, \\mu_{\\mcO_{2r}} \\left( x_r, \\dots, x_{p-1}, u, y_0, \\dots, y_{s-1} \\right), \\dots \\right)\n\t\\end{align*}\n\tand \n\t\\begin{align*}\n\t\\sum & \\mu_{\\mcO_{2r}} \\left( \\dots, \\sigma_{n, k} \\left( \\dots, u, \\dots \\right), \\gamma \\left( \\dots \\right), \\dots, \\gamma \\left( \\dots \\right) \\right) \\\\\n\t& = \\sum \\mu_{\\mcO_{2r}} \\left( \\sigma_{n, k} \\left( \\dots \\right), \\dots, \\sigma_{n, k} \\left( \\dots \\right), \\sigma_{n, k} \\left( \\dots, u, \\dots \\right), \\sigma_{n, k} \\left( \\dots \\right), \\dots, \\sigma_{n, k} \\left( \\dots \\right) \\right) .\n\t\\end{align*}\n\tTherefore, we get \n\t\\[\\mu_{\\mathrm{Mod}_{\\mcC, \\mcC}}^1 \\left( f \\right) \\left( x_0, \\dots, x_{p-1}, u, y_0, \\dots, y_{q-1} \\right) = 0 \\]\n\tfrom the fact that $\\sigma_{n, k}$ is an $A_{\\infty}$-functor. \n\t\n\tNow $f$ sends the unit $e_{L_H^k (E)} \\in \\mcO_{2r} \\left( L_H^k (E), L_H^k (E) \\right)$ to the continuation element $c_k (E) \\in \\mcO_{2r} \\left( L_H^k (E), L_H^{k+1} (E) \\right) \\cap \\Gamma$ according to the third item of Remark \\ref{rmk properties of sigma}. \n\tFinally, the map $f : \\mcO_{2r} \\left( L_H^j(E) , L_H^k(E') \\right) \\to \\mcO_{2r} \\left( L_H^j(E), L_H^{k+1} (E') \\right)$ is a quasi-somorphism for every $j< k$ and $E, E' \\in \\mcE$ according to the last item of Remark \\ref{rmk properties of sigma}.\n\t\n\\end{proof}\n\n\\subsubsection{Proof of the main result}\n\nWe end the section with the proof of Theorem \\ref{thm mainthm restatement} (Theorem \\ref{thm mainthm} in the Introduction). \n\nRecall that we denote by $\\mathbf{F} \\left[ t_m \\right]$ the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $(m, 1)$, and by $t_m \\mathbf{F} \\left[ t_m \\right]$ its augmentation ideal (or equivalently, the ideal generated by $t_m$).\nThe key result is the following.\n\n\\begin{lemma}\\label{lemma mapping torus of gamma}\n\t\n\tThe mapping torus of $\\gamma$ is quasi-equivalent to the Adams-graded $A_{\\infty}$-category $\\overrightarrow{\\mcF uk} (\\mathbf{L}) \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcF uk (\\mathbf{L}) \\right)$.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tLet $f : \\mcO_{2r} \\left( -, - \\right) \\to \\mcO_{2r} \\left( -, \\gamma (-) \\right)$ be the degree $0$ closed bimodule map of Lemma \\ref{lemma bimodule map}. According to the latter, the hypotheses of Theorem \\ref{thm mapping torus in weak situation} are satisfied, and $f(\\mathrm{units}) = \\Gamma$. \n\tThus the mapping torus of $\\gamma$ is quasi-equivalent to the Adams-graded $A_{\\infty}$-algebra $\\mcO_{2r}^0 \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcO_{2r} \\left[ \\Gamma^{-1} \\right]^0 \\right)$ (recall that if $\\mcC$ is an $A_{\\infty}$-category equipped with a $\\mathbf{Z}$-splitting $\\mathbf{Z} \\times \\mcE \\simeq \\mathrm{ob} \\left( \\mcC \\right)$, we denote by $\\mcC^0$ the full $A_{\\infty}$-subcategory of $\\mcC$ whose set of objects corresponds to $\\{ 0 \\} \\times \\mcE$).\n\tAccording to Lemma \\ref{lemma localization of O equals CF} we have\n\t\\[\\mcO_{2r}^0 \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}_H) \\text{ and } \\mcO_{2r} \\left[ \\Gamma^{-1} \\right]^0 \\simeq \\mcF uk (\\mathbf{L}_H) . \\]\n\tThe result follows from invariance of the Fukaya category\n\t\\[\\overrightarrow{\\mcF uk} (\\mathbf{L}_H) \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}) \\text{ and } \\mcF uk (\\mathbf{L}_H) \\simeq \\mcF uk (\\mathbf{L}) . \\]\n\t\n\\end{proof}\n\nWe now give the proof of Theorem \\ref{thm mainthm restatement} (Theorem \\ref{thm mainthm} in the introduction).\nAccording to \\cite[Theorem 2.4]{LPWZ08}, Koszul duality holds for the augmented Adams-graded DG-algebra $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ because it is \\emph{Adams connected} (see \\cite[Definition 2.1]{LPWZ08}). Indeed, recall from section \\ref{subsection invariants and statement of the result} that the Adams-degree in $CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ of a Reeb chord $c$ is the number of times $c$ winds around the fiber. \nBesides, recall from section \\ref{subsection invariants and statement of the result} that there is a coaugmented Adams-graded $A_{\\infty}$-cocategory $LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)$ such that \n\\[CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) = \\Omega \\left( LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right) \\text{ and } LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) = LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)^{\\#}. \\]\nSince there is a quasi-isomorphism $B \\left( \\Omega C \\right) \\simeq C$ for every $A_{\\infty}$-cocategory $C$ (see \\cite[section 2.2.2]{EL21}), it follows that \n\\[E \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right) = B \\left( CE_{-*}^r \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\right)^{\\#} \\simeq LC_* \\left( \\mathbf{\\Lambda^{\\circ}} \\right)^{\\#} = LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\]\n(graded dual preserves quasi-isomorphisms).\nNow the quasi-equivalence \n\\[LA^* \\left( \\mathbf{\\Lambda^{\\circ}} \\right) \\simeq \\overrightarrow{\\mcF uk} (\\mathbf{L}) \\oplus \\left( t_{2r} \\mathbf{F} \\left[ t_{2r} \\right] \\otimes \\mcF uk (\\mathbf{L}) \\right) \\]\nfollows from Lemmas \\ref{lemma A^rond = mapping torus of tau}, \\ref{lemma mapping torus of tau = mapping torus of tau_1}, \\ref{lemma mapping torus of tau_1 = mapping torus of tau_2}, \\ref{lemma mapping torus of tau_2 = mapping torus of gamma} and \\ref{lemma mapping torus of gamma}.\nThis concludes the proof.\n\\section{Chekanov-Eliashberg DG-category}\\label{section Legendrian invariants}\n\nIn this section, we recall the definition of the Chekanov-Eliashberg DG-category associated to a family of Legendrians in a contact manifold equipped with a hypertight contact form $\\alpha$, which means that the Reeb vector field of $\\alpha$ has no contractible periodic orbits. . We also describe the behavior of the Chekanov-Eliashberg DG-category under change of data. \n\nIn the following, $(V, \\xi)$ is a contact manifold of dimension $(2n+1)$.\nIn order to have well-defined gradings in $\\mathbf{Z}$, we assume that $H_1 \\left( V \\right)$ is free and that the first Chern class of $\\xi$ (equipped with any compatible almost complex structure) is $2$-torsion.\nWe will need the following definition.\n\n\\begin{defin}\\label{definition chord generic}\n\t\n\tWe say that a Legendrian submanifold $\\Lambda$ in $\\left( V, \\xi \\right)$ is \\emph{chord generic} with respect to a contact form $\\alpha$ if the following holds:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item for every Reeb chord $c : \\left[ 0,T \\right] \\to V$ of $\\Lambda$, the space $D \\varphi_{R_{\\alpha}}^T \\left( T_{c(0)} \\Lambda \\right)$ is transverse to $T_{c(T)} \\Lambda$ in $\\xi$,\n\t\t\n\t\t\\item different Reeb chords belong to different Reeb trajectories. \n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\subsection{Conley-Zehnder index}\\label{subsection Conley-Zehnder index}\n\nLet $\\alpha$ be a hypertight contact form on $(V, \\xi)$ and let $\\Lambda$ be a chord generic Legendrian submanifold of $(V, \\alpha)$.\nIn the following, we define the Conley-Zehnder index of a Reeb chord of $\\Lambda$ starting and ending on the same connected component (such chords are called \\emph{pure}).\n\nWe briefly recall what is the Maslov index of a loop in the Grassmanian of Lagrangian subspaces in $\\mathbf{C}^n$. We refer to \\cite{RS93} for a precise exposition. Fix a Lagrangian subspace $K$, and denote by $\\Sigma_k \\left( K \\right)$ the set of Lagrangian subspaces in $\\mathbf{C}^n$ whose intersection with $K$ is $k$ dimensional. Consider the \\emph{Maslov cycle} \n\\[\\Sigma = \\Sigma_1 \\left( K \\right) \\cup \\dots \\cup \\Sigma_n \\left( K \\right) . \\]\nThis is an algebraic variety of codimension one in the Lagrangian Grassmanian. Now if $\\Gamma$ is a loop in the Lagrangian Grassmanian, its Maslov index $\\mu \\left( \\Gamma \\right) \\in \\mathbf{Z}$ is the intersection number of $\\Gamma$ with $\\Sigma$. The contribution of an intersection instant $t_0$ is computed as follows. Choose a Lagrangian complement $W$ of $K$ in $\\mathbf{C}^n$. Then for each $v$ in $\\Gamma (t_0) \\cap K$, there exists a vector $w(t)$ in $W$ such that $v + w(t)$ is in $\\Gamma (t)$ for every $t$ near $t_0$. Consider the quadratic form\n\\[Q(v) = \\frac{d}{dt} \\omega \\left( v, w(t) \\right)_{|t=t_0} \\]\non $\\Gamma (t_0) \\cap K$. Without loss of generality, $Q$ can be assumed non-singular and the contribution of $t_0$ to $\\mu (\\Gamma)$ is the signature of $Q$.\n\nRecall that $H_1 \\left( V \\right)$ is assumed to be free. We choose a family $(h_1, \\dots, h_r)$ of embedded circles in $V$ which represent a basis of $H_1 ( V )$, and a symplectic trivialization of $\\xi$ over each $h_i$. If $\\gamma$ is some loop in $\\Lambda$, there is a unique family $(a_1, \\dots, a_r)$ of integers such that $\\left[ \\gamma_c - \\sum_i a_i h_i \\right]$ is zero in $H_1 (V)$. Choose a surface $\\Sigma_{\\gamma}$ in $V$ such that \n\\[ \\partial \\Sigma_{\\gamma} = \\gamma - \\sum_i a_i h_i . \\]\nThere is a unique trivialization of $\\xi$ over $\\Sigma_{\\gamma}$ which extends the chosen trivializations over $h_i$.\nThus we get a trivialization $\\gamma^{-1} \\xi \\simeq S^1 \\times \\mathbf{C}^n$ (where $n$ is the dimension of $\\Lambda$). We denote by $\\Gamma$ the loop of Lagrangian planes in $\\mathbf{C}^n$ corresponding, via the latter trivialization, to the loop $t \\mapsto T_{\\gamma (t)} \\Lambda$. The Maslov index of $\\Gamma$ does not depend on the choice of the surface $\\Sigma_{\\gamma}$ because we assumed $2 c_1 (\\xi) = 0$. This construction defines a morphism $H_1 \\left( \\Lambda, \\mathbf{Z} \\right) \\to \\mathbf{Z}$, and the \\emph{Maslov number} $m(\\Lambda)$ of $\\Lambda$ is the generator of its image. \nIn the following, we assume that the Maslov number of $\\Lambda$ is zero. \n\nNow, let $c$ be a pure Reeb chord of $\\Lambda$. We choose a path $\\gamma_c : \\left[ 0,1 \\right] \\to \\Lambda$ which starts at the endpoint of $c$, and ends at its starting point ($\\gamma_c$ is called a \\emph{capping path} of $c$). We denote by $\\overline{\\gamma}_c$ the loop obtained by concatenating $\\gamma$ and $c$. Let $(a_1, \\dots, a_r)$ be the unique family of integers such that $\\left[ \\overline{\\gamma}_c - \\sum_i a_i h_i \\right]$ is zero in $H_1 (V)$, and choose a surface $\\Sigma_c$ in $V$ such that \n\\[ \\partial \\Sigma_c = \\overline{\\gamma}_c - \\sum_i a_i h_i . \\]\nThere is a unique trivialization of $\\xi$ over $\\Sigma_c$ which extends the chosen trivializations over $h_i$.\nThus we get a trivialization $\\overline{\\gamma}_c^{-1} \\xi \\simeq S^1 \\times \\mathbf{C}^n$ (where $n$ is the dimension of $\\Lambda$). We denote by $\\Gamma_c$ the path of Lagrangian planes in $\\mathbf{C}^n$ corresponding, via the latter trivialization, to the concatenation of $t \\mapsto T_{\\gamma (t)} \\Lambda$ and $t \\mapsto D \\varphi_{R_{\\alpha}}^t \\left( T_{c(0)} \\Lambda \\right)$. Since $\\Lambda$ is chord generic, $\\Gamma_c$ is not a loop: we close it in the following way. Let $I$ be a complex structure on $\\mathbf{C}^n$ which is compatible with the standard symplectic form on $\\mathbf{C}^n$ and such that $I \\left( \\Gamma_c (1) \\right) = \\Gamma_c (0)$. Then we let $\\overline{\\Gamma}_c$ be the loop of Lagrangian subspaces obtained by concatenating $\\Gamma_c$ and the path $t \\mapsto e^{tI} \\Gamma_c (1)$. The Conley-Zehnder index of $c$ is the Maslov index of $\\overline{\\Gamma}_c$: \n\\[CZ (c) := \\mu \\left( \\overline{\\Gamma}_c \\right). \\]\nThe Conley-Zehnder index of a Reeb chord does not depend on the choice of $\\Sigma_c$ because the first Chern class of $\\xi$ is $2$-torsion, and it does not depend on the choice of $\\gamma_c$ because the Maslov number of $\\Lambda$ vanishes. \n\n\\subsection{Moduli spaces}\\label{subsection moduli spaces}\n\nRecall that $(V, \\xi)$ is a contact manifold such that $H_1 (V)$ is free and the first Chern class of $\\xi$ (equipped with any compatible almost complex structure) is 2-torsion. \nLet $\\alpha$ be a hypertight contact form on $(V, \\xi)$ and let $\\Lambda$ be a chord generic Legendrian submanifold of $(V, \\alpha)$ with vanishing Maslov number.\nIn the following, we introduce the moduli spaces needed to define the Chekanov-Eliashberg category of $\\Lambda$. \n\n\\begin{defin}\\label{definition Riemann disk}\n\t\n\tA Riemann $(d+1)$-pointed disk is a triple $\\left( D, \\boldsymbol{\\zeta}, j \\right)$ such that \n\t\\begin{enumerate}\n\t\t\n\t\t\\item $D$ is a smooth oriented manifold-with-boundary diffeomorphic to the closed unit disk in $\\mathbf{C}$,\n\t\t\n\t\t\\item a family $\\boldsymbol{\\zeta} = \\left( \\zeta_d, \\dots, \\zeta_1, \\zeta_0 \\right)$ is a cyclically ordered family of distinct points on $\\partial D$,\n\t\t\n\t\t\\item $j$ is an integrable almost complex structure on $D$ which induces the given orientation on $D$.\n\t\t\n\t\\end{enumerate}\n\tIf $\\left( D, \\boldsymbol{\\zeta}, j \\right)$ is a Riemann pointed disk, we denote by $\\Delta := D \\setminus \\{ \\zeta_d, \\dots, \\zeta_1, \\zeta_0 \\}$ the corresponding punctured disk.\n\t\n\\end{defin}\n\n\\begin{defin}\n\t\n\tA family of Riemann $(d+1)$-pointed discs is a bundle $\\mcS \\to \\mcR$ with \n\t\\begin{enumerate}\n\t\t\n\t\t\\item a family $\\boldsymbol{\\zeta} = \\left( \\zeta_d, \\dots, \\zeta_1, \\zeta_0 \\right)$ of non-intersecting sections $\\zeta_k : \\mcR \\to \\mcS$ and\n\t\t\n\t\t\\item a section $j : \\mcR \\to \\mathrm{End} (T \\mcS )$\n\t\t\n\t\\end{enumerate}\n\tsuch that $(\\mcS_r, \\boldsymbol{\\zeta} (r), j(r))$ is a Riemann $(d+1)$-pointed disk for every $r \\in \\mcR$.\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition choice of strip-like ends}\n\t\n\tLet $\\mcS \\to \\mcR$ be a family of Riemann $(d+1)$-pointed discs.\n\tA choice of strip-like ends for $\\mcS \\to \\mcR$ is a family of sections\n\t\\[\\epsilon_d, \\dots, \\epsilon_1 : \\mcR \\times \\mathbf{R}_{\\leq 0} \\times \\left[ 0,1 \\right] \\to \\Delta_r, \\quad \\epsilon_0 : \\mcR \\times \\mathbf{R}_{\\geq 0} \\times \\left[ 0,1 \\right] \\to \\Delta_r \\]\n\tsuch that\n\t\\begin{enumerate}\n\t\t\n\t\t\\item $\\epsilon_d (r), \\dots, \\epsilon_1 (r), \\epsilon_0 (r)$ are proper embeddings with\n\t\t\\[\\epsilon_k (r) \\left( \\mathbf{R}_{\\leq 0} \\times \\{ 0, 1 \\} \\right) \\subset \\partial \\Delta_r \\text{ and } \\epsilon_0 (r) \\left( \\mathbf{R}_{\\geq 0} \\times \\{ 0, 1 \\} \\right) \\subset \\partial \\Delta_r, \\]\n\t\t\n\t\t\\item $\\epsilon_d (r), \\dots, \\epsilon_1 (r), \\epsilon_0 (r)$ satisfy the asymptotic conditions\n\t\t\\[\\epsilon_k (r) \\left( s, t \\right) \\underset{s \\to - \\infty}{\\longrightarrow} \\zeta_k (r) \\text{ and } \\epsilon_0 (r) \\left( s, t \\right) \\underset{s \\to + \\infty}{\\longrightarrow} \\zeta_0 (r) \\]\n\t\t\n\t\t\\item $\\epsilon_d (r), \\dots, \\epsilon_1 (r), \\epsilon_0 (r)$ are $\\left( i, j(r) \\right)$-holomorphic, where $i$ is the standard complex structure on $\\mathbf{C}$.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\nAs explained in \\cite[section (9c)]{Sei08}, there is a universal family $\\mcS^{d+1} \\to \\mcR^{d+1}$ of Riemann $(d+1)$-pointed discs when $d \\geq 2$, which means that any other family $\\mcS \\to \\mcR$ is isomorphic to the pullback of $\\mcS^{d+1} \\to \\mcR^{d+1}$ by a map $\\mcR \\to \\mcR^{d+1}$.\nIn the following, we fix a choice of strip-like ends for the unversal family $\\mcS^{d+1} \\to \\mcR^{d+1}$.\n\n\\begin{defin}\\label{definition moduli spaces}\n\t\n\tLet $J$ be an almost complex structure on $\\xi$ compatible with $\\left( d \\alpha \\right)_{| \\xi}$. We denote by $J^{\\alpha}$ the unique almost complex structure on $\\mathbf{R}_{\\sigma} \\times V$ which sends $\\partial_{\\sigma}$ to $R_{\\alpha}$ and which restricts to $J$ on $\\xi$.\n\tLet $c_d, \\dots, c_1, c_0$ be a Reeb chords of $\\Lambda$, where $c_k : [0, T_k] \\to V$. \n\t\\begin{enumerate}\n\t\t\n\t\t\\item If $d=1$, we denote by $\\widetilde{\\mcM}_{c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right)$ the set of equivalence classes of maps $u : \\mathbf{R} \\times \\left[ 0, 1 \\right] \\to \\mathbf{R} \\times V$ such that\n\t\t\\begin{itemize} \n\t\t\t\\item $u$ maps the boundary of $\\mathbf{R} \\times \\left[ 0, 1 \\right]$ to $\\mathbf{R} \\times \\Lambda$,\n\t\t\t\\item $u$ satisfies the asymptotic conditions\n\t\t\t\\[u \\left( s, t \\right) \\underset{s \\to - \\infty}{\\longrightarrow} \\left( - \\infty, c_1 (T_1 t) \\right) \\text{ and } u \\left( s, t \\right) \\underset{s \\to + \\infty}{\\longrightarrow} \\left( + \\infty, c_0 (T_0 t) \\right), \\]\n\t\t\t\\item $u$ is $(i, J^{\\alpha})$-holomorphic,\n\t\t\\end{itemize}\n\t\twhere two maps $u$ and $u'$ are identified if there exists $s_0 \\in \\mathbf{R}$ such that $u' (\\cdot, \\cdot) = u (\\cdot + s_0, \\cdot)$.\n\t\t\n\t\t\\item If $d \\geq 2$, we denote by $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right)$ the set of of pairs $\\left( r, u \\right)$ such that\n\t\t\\begin{itemize}\n\t\t\t\\item $r \\in \\mcR^{d+1}$ and $u : \\Delta_r \\to \\mathbf{R} \\times V$ maps the boundary of $\\Delta_r$ to $\\mathbf{R} \\times \\Lambda$,\n\t\t\t\\item $u$ satisfies the asymptotic conditions\n\t\t\t\\[\\left( u \\circ \\epsilon_k (r) \\right) \\left( s, t \\right) \\underset{s \\to - \\infty}{\\longrightarrow} \\left( -\\infty, c_k (T_k t) \\right) \\text{ and } \\left( u \\circ \\epsilon_0 (r) \\right) \\left( s, t \\right) \\underset{s \\to + \\infty}{\\longrightarrow} \\left( + \\infty, c_0 (T_0 t) \\right), \\]\n\t\t\t\\item $u$ is $(i, J^{\\alpha})$-holomorphic.\n\t\t\\end{itemize}\n\t\t\n\t\\end{enumerate}\n\tObserve that $\\mathbf{R}$ acts on $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right)$ by translation in the $\\mathbf{R}_{\\sigma}$-coordinate. \n\tWe set\n\t\\[\\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right) := \\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right) \/ \\mathbf{R} . \\] \n\t\n\\end{defin}\n\nThe moduli space $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right)$ can be realized as the zero-set of a section $\\overline{\\partial} : \\mcB \\to \\mcE$ of a Banach bundle $\\mcE \\to \\mcB$ (see for example \\cite{EES07}). We say that $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J \\right)$ is transversely cut out if $\\overline{\\partial}$ is transverse to the 0-section. \n\n\\begin{defin}\\label{definition regular almost complex structure}\n\t\n\tWe say that $J$ is \\emph{regular} (with repect to $\\alpha$ and $\\Lambda$) if the moduli spaces $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right)$ are all transversely cut out.\n\t\n\\end{defin}\n\n\\begin{prop}[\\cite{DR16bis} Proposition 3.13]\\label{prop transverslity moduli spaces}\n\n\tThe set of regular almost complex structures on $\\xi$ is Baire. \n\tMoreover, the dimension of a transversely cut out moduli space is \n\t\\[\\mathrm{dim} \\, \\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right) = CZ \\left( a \\right) - \\left( \\sum_{k=1}^{d} CZ \\left( b_k \\right) \\right) + d-1 . \\]\n\n\\end{prop}\n\n\\subsection{Chekanov-Eliashberg DG-category}\\label{subsection Chekanov-Eliashberg algebra}\n\nRecall that $(V, \\xi)$ is a contact manifold such that $H_1 (V)$ is free and the first Chern class of $\\xi$ (equipped with any compatible almost complex structure) is 2-torsion. \nLet $\\alpha$ be a hypertight contact form on $(V, \\xi)$ and let $\\mathbf{\\Lambda} = (\\Lambda (E))_{E \\in \\mcE}$ be a family of Legendrian submanifolds of $(V, \\xi)$. We set $\\Lambda := \\bigcup_{E \\in \\mcE} \\Lambda(E)$ and we assume that $\\Lambda$ is chord generic with vanishing Maslov number. \nMoreover, we denote by $\\mcC (\\Lambda(E), \\Lambda(E'))$ the graded vector space generated by the words of Reeb chords $c_1 \\cdots c_d$ where $c_1$ starts on $\\Lambda(E)$, $c_d$ ends on $\\Lambda(E')$, and the ending component of $c_i$ is the starting component of $c_{i+1}$ for every $1 \\leq i \\leq d-1$, with grading \n\\[\\vert c_1 \\cdots c_d \\vert := \\sum_{i=1}^{d} \\left( CZ \\left( c_i \\right) - 1 \\right). \\]\nFinally, let $J$ be a regular almost complex structure on $\\xi$.\n\n\\begin{defin}\\label{definition Chekanov-Eliashberg}\n\t\n\tWe denote by $CE_* \\left( \\mathbf{\\Lambda} \\right) = CE_* \\left( \\mathbf{\\Lambda}, J, \\alpha \\right)$ the graded category defined as follows \n\t\\begin{enumerate}\n\t\t\n\t\t\\item the objects are the Legendrians $\\Lambda (E)$, $E \\in \\mcE$,\n\t\t\n\t\t\\item the space of morphisms from $\\Lambda (E)$ to $\\Lambda (E')$ is either $\\mcC (\\Lambda(E), \\Lambda(E'))$ if $E \\ne E'$, or $\\mathbf{F} \\oplus \\mcC (\\Lambda(E), \\Lambda(E'))$ if $E = E'$,\n\t\t\n\t\t\\item the composition is given by concatenation of words.\n\t\t\n\t\\end{enumerate}\n\tIf $c_0$ is a Reeb chord in $CE_* \\left( \\mathbf{\\Lambda} \\right)$, we set \n\t\\[\\partial (c_0) := \\sum \\limits_{c_d, \\dots,c_1} \\# \\mcM_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda, J, \\alpha \\right) c_d \\cdots c_1, \\]\n\twhere $\\# \\mcM \\in \\mathbf{F}$ denotes the number of elements modulo $2$ in $\\mcM$ if $\\mcM$ is finite, and $0$ otherwise. Finally, we extend $\\partial$ to $CE_* \\left( \\mathbf{\\Lambda} \\right)$ so that it is linear and satisfies the Leibniz rule with respect to the concatenation product.\n\t\n\\end{defin}\n\n\\begin{thm}\\label{thm d rond d egal 0}\n\t\n\t$\\partial : CE_* \\left( \\mathbf{\\Lambda} \\right) \\to CE_* \\left( \\mathbf{\\Lambda} \\right)$ decreases the grading by $1$ and satisfies $\\partial \\circ \\partial = 0$.\n\tAs a result, $(CE_{-*} \\left( \\mathbf{\\Lambda} \\right), \\partial)$ is a DG-category. \n\t\n\\end{thm}\n\n\\begin{proof}\n\t\n\tThis follows from Proposition \\ref{prop transverslity moduli spaces}, SFT compactness (see \\cite{BEHW03} and \\cite{Abb14}, in particular \\cite[Theorem 3.20]{Abb14}) and pseudo-holomorphic gluing. See \\cite{Ekh08}, \\cite{EES05} and \\cite{EES07} for details.\n\t\n\\end{proof}\n\n\\paragraph{Augmentations and Legendrian $A_{\\infty}$-(co)category.} \n\nLet $\\mathbf{F}_{\\mcE}$ be the category with $\\mcE$ as set of objects, and morphism space from $E$ to $E'$ equal to $\\mathbf{F}$ if $E = E'$, or $0$ if $E \\ne E'$.\nAssume that we have an augmentation of $CE_{-*} \\left( \\mathbf{\\Lambda} \\right)$, i.e. a DG-functor $\\varepsilon : CE_{-*} \\left( \\mathbf{\\Lambda} \\right) \\to \\mathbf{F}_{\\mcE}$. Denote by $\\phi_{\\varepsilon}$ the automorphism of $CE_{-*} \\left( \\mathbf{\\Lambda} \\right)$ defined by \n\\[\\phi_{\\varepsilon} (c) = c + \\varepsilon (c) \\]\nfor every Reeb chord $c$ of $\\Lambda$. We denote by $CE_{-*}^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)$ the DG-category whose underlying graded category is the same as for $CE_{-*} \\left( \\mathbf{\\Lambda} \\right)$, but the differential is $\\partial_{\\varepsilon} = \\phi_{\\varepsilon} \\circ \\partial \\circ \\phi_{\\varepsilon}^{-1}$. Now let $\\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)}$ be the graded pre-category (no composition) with\n\\begin{enumerate}\n\t\n\t\\item objects the Legendrians $\\Lambda (E)$, $E \\in \\mcE$,\n\t\n\t\\item space of morphisms from $\\Lambda (E)$ to $\\Lambda (E')$ generated by Reeb chords $c$ which starts on $\\Lambda(E)$ and ends on $\\Lambda(E')$, with grading \n\t\\[\\vert c \\vert := -CZ(c). \\]\n\\end{enumerate} \nObserve that, as a graded pre-category, we have\n\\[CE_{-*}^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right) = \\mathbf{F}_{\\mcE} \\oplus \\left( \\bigoplus \\limits_{d \\geq 1} \\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)} \\left[ -1 \\right]^{\\otimes d} \\right) . \\] \nIf we write\n\\[\\left( \\partial_{\\varepsilon} \\right)_{| \\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)} } = \\sum_{d \\geq 0} \\partial_{\\varepsilon}^d \\text{ with } \\partial_{\\varepsilon}^d : \\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)} \\to \\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)}^{\\otimes d}, \\]\nthen $\\partial_{\\varepsilon}^0 = \\varepsilon \\circ \\partial = 0$. Moreover, the operations $\\left( \\partial_{\\varepsilon}^d \\right)_{d \\geq 1}$ make $\\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)}$ a (non-counital) $A_{\\infty}$-cocategory (see Definition \\ref{definition coalgebra}). We define the coaugmented $A_{\\infty}$-cocategory of $\\left( \\mathbf{\\Lambda}, \\varepsilon \\right)$ to be\n\\[LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right) := \\mathbf{F}_{\\mcE} \\oplus \\overline{LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)} \\]\n(the $A_{\\infty}$-cooperations are naturally extended so that $1 \\in \\mathbf{F}_{\\mcE} (E, E)$, $E \\in \\mcE$ are counits).\nNow observe that, as a DG-category, \n\\[CE_{-*}^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right) = \\Omega \\left( LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right) \\right) \\]\n(see \\cite[section 2.2]{EL21} for the cobar construction).\nFinally, we define the augmented $A_{\\infty}$-category of $\\left( \\mathbf{\\Lambda}, \\varepsilon \\right)$ to be the graded dual (see \\cite[section 2.1.3]{EL21}) of $LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)$:\n\\[LA^*_{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right) = LC_*^{\\varepsilon} \\left( \\mathbf{\\Lambda} \\right)^{\\#} . \\]\n\n\\subsection{Functoriality}\\label{subsection invariance}\n\nRecall that $(V, \\xi)$ is a contact manifold such that $H_1 (V)$ is free and the first Chern class of $\\xi$ (equipped with any compatible almost complex structure) is 2-torsion. \nLet $\\mathbf{M}=(M(E))_{E \\in \\mcE}$ be a family of $n$-dimensional manifolds.\nWhen we write a map $\\mathbf{\\Lambda} : \\mathbf{M} \\to V$, we mean that $\\mathbf{\\Lambda}$ is a family of maps $\\Lambda (E) : M(E) \\to V$ indexed by $\\mcE$, and we set\n\\[\\Lambda = \\bigsqcup_{E \\in \\mcE} \\Lambda (E) : \\bigsqcup_{E \\in \\mcE} M (E) \\to V . \\]\n\n\\begin{defin}\\label{definition geometric category}\n\t\n\tLet $\\alpha$ be a hypertight contact form on $(V, \\xi)$.\n\tWe denote by $\\mcL_\\mathbf{M} (\\alpha)$ the bicategory where\n\t\\begin{enumerate}\n\t\t\n\t\t\\item objects are the pairs $(\\mathbf{\\Lambda}, J)$, where $\\mathbf{\\Lambda} : \\mathbf{M} \\to V$ is a family of Legendrian embedding such that $\\Lambda$ is chord generic with vanishing Maslov number, and $J$ is a regular almost complex structure on $\\xi$,\n\t\t\\item morphisms from $(\\mathbf{\\Lambda}_0, J_0)$ to $(\\mathbf{\\Lambda}_1, J_1)$ are the smooth paths $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1}$ going from $(\\mathbf{\\Lambda}_0, J_0)$ to $(\\mathbf{\\Lambda}_1, J_1)$, where $\\mathbf{\\Lambda}_t : \\mathbf{M} \\to V$ is a family of Legendrian embeddings and $J_t$ is an almost complex structure on $\\xi$, \n\t\t\\item homotopies from a morphism $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1} : (\\mathbf{\\Lambda}_0, J_0) \\to (\\mathbf{\\Lambda}_1, J_1)$ to another morphism $\\Phi' = (\\mathbf{\\Lambda}_t', J_t')_{0 \\leq t \\leq 1} : (\\mathbf{\\Lambda}_0, J_0) \\to (\\mathbf{\\Lambda}_1, J_1)$ are the smooth families $(\\mathbf{\\Lambda}_{s,t}, J_{s,t})_{0 \\leq s \\leq S, 0 \\leq t \\leq 1}$, where $\\mathbf{\\Lambda}_{s,t} : \\mathbf{M} \\to V$ is a family of Legendrian embeddings, $J_{s,t}$ is an almost complex structure on $\\xi$, and\n\t\t\\[(\\mathbf{\\Lambda}_{s,0}, J_{s,0}) = (\\mathbf{\\Lambda}_0, J_0), \\; (\\mathbf{\\Lambda}_{s,1}, J_{s,1}) = (\\mathbf{\\Lambda}_1, J_1), \\quad (\\mathbf{\\Lambda}_{0,t}, J_{0,t}) = (\\mathbf{\\Lambda}_t, J_t), \\; (\\mathbf{\\Lambda}_{S,t}, J_{S,t}) = (\\mathbf{\\Lambda}_t', J_t'). \\]\n\t\t\n\t\\end{enumerate}\n\t\t\n\\end{defin}\n\n\\begin{defin}\\label{definition action of contact isotopy}\n\t\n\tLet $\\alpha$, $\\alpha'$ be hypertight contact forms on $(V, \\xi)$, and let $\\varphi$ be a contactomorphism of $(V, \\xi)$ such that $\\varphi^* \\alpha = \\alpha'$.\n\tIf $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1}$ is a morphism in $\\mcL_\\mathbf{M} (\\alpha)$, we denote by \n\t\\[\\varphi^* \\Phi = (\\varphi^{-1} (\\mathbf{\\Lambda}_t), \\varphi^* J_t)_{0 \\leq t \\leq 1} \\]\n\tthe corresponding morphism in $\\mcL_\\mathbf{M} (\\alpha')$, and by \n\t\\[f_{(\\mathbf{\\Lambda}_t, J_t)}^{\\varphi} : CE_{-*} (\\mathbf{\\Lambda}_t, J_t, \\alpha) \\to CE_{-*} (\\varphi^{-1} (\\mathbf{\\Lambda}_t), \\varphi^* J_t, \\alpha') \\]\n\tthe DG-functor which sends a Reeb chord $c$ to $\\varphi^{-1} (c)$.\n\t\n\\end{defin}\n\n\\begin{defin}\\label{definition handle slide instant}\n\t\n\tLet $\\alpha$ be a hypertight contact form on $(V, \\xi)$, and let $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1}$ be a morphism in $\\mcL_\\mathbf{M} (\\alpha)$.\n\tA \\emph{handle slide instant} in $\\Phi$ is a time $t_0$ where $\\mathbf{\\Lambda}_{t_0}$ is chord generic and has Reeb chords $c_d, \\dots, c_1, c_0$ such that the moduli space $\\widetilde{\\mcM}_{c_d, \\dots, c_1, c_0} \\left( \\mathbf{R} \\times \\Lambda_{t_0}, J_{t_0}, \\alpha \\right)$ is not transversely cut out.\n\t\n\\end{defin}\n\n\\begin{conj}\\label{conjecture functoriality}\n\t\n\tThere exist functors $\\mcF_{\\alpha}$ from $\\mcL_\\mathbf{M} (\\alpha)$ to the bicategory\\footnote{Homotopies between DG-maps are DG-homotopies, see for example \\cite[section 2.1]{PR21}.} of DG-categories such that \n\t\\begin{enumerate}\n\t\t\n\t\t\\item $\\mcF_{\\alpha}$ sends an object $( \\mathbf{\\Lambda}, J )$ to $CE_{-*} ( \\mathbf{\\Lambda}, J, \\alpha)$,\n\t\t\n\t\t\\item $\\mcF_{\\alpha}$ sends a morphism to a homotopy equivalence,\n\t\t\n\t\t\\item if $\\varphi$ is a contactomorphism of $(V, \\xi)$ such that $\\varphi^* \\alpha = \\alpha'$ and if $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1}$ is a morphism in $\\mcL_\\mathbf{M} (\\alpha)$, then \n\t\t\\[\\mcF_{\\alpha'} (\\varphi^* \\Phi) = f_{(\\mathbf{\\Lambda}_1, J_1)}^{\\varphi} \\circ \\mcF_{\\alpha} (\\Phi) \\circ (f_{(\\mathbf{\\Lambda}_0, J_0)}^{\\varphi})^{-1} \\]\n\t\t\n\t\t\\item if $(\\varphi_t)_{0 \\leq t \\leq 1}$ is a contact isotopy of $(V, \\xi)$ satisfying $\\varphi_t^* \\alpha = \\alpha'$ for every $t$, and if $(\\mathbf{\\Lambda}, J)$ is an object of $\\mcL_\\mathbf{M} (\\alpha)$ such that there is neither birth\/death of Reeb chords nor handle slide instants in the path $\\Phi' = (\\varphi_t^{-1} (\\mathbf{\\Lambda}), \\varphi_t^* J)$, then \n\t\t\\[\\mcF_{\\alpha'} (\\Phi') = f_{(\\mathbf{\\Lambda}, J)}^{\\varphi_1} \\circ (f_{(\\mathbf{\\Lambda}, J)}^{\\varphi_0})^{-1} . \\]\n\t\t\t\t\n\t\\end{enumerate}\n\t\n\\end{conj}\n\nWe chose to state the latter result as a conjecture out of caution, even if some parts of it have already been proved, or at least detailed strategies of proofs have been given. The existence of such functors at the category level (without homotopies) has been completely established in the case $(V, \\alpha) = (\\mathbf{R} \\times P, dz - \\lambda)$, see \\cite[section 2.4]{EES07}. Detailed strategies of proofs for the general case can be found in \\cite[section 4]{Ekh08} and \\cite[section 5]{EO17}.\n \nNote that I proved a weaker version of this result in my thesis by generalizing methods of \\cite{EES05} and \\cite{PR21}.\nThe following is the only version of Conjecture \\ref{conjecture functoriality} that we will use in this paper. \n\n\\begin{thm}[\\cite{Pet22} Theorem 3.8]\\label{thm invariance}\n\t\n\tConjecture \\ref{conjecture functoriality} is true if we replace the categories $\\mcL_\\mathbf{M} (\\alpha)$ by the subcategories $\\mcL_\\mathbf{M}^0 (\\alpha)$ where\n\t\\begin{enumerate}\n\t\t\n\t\t\\item objects are the pairs $(\\mathbf{\\Lambda}, J)$ such that $\\Lambda$ has finitely many Reeb chords,\n\t\t\n\t\t\\item morphisms from $(\\mathbf{\\Lambda}_0, J_0)$ to $(\\mathbf{\\Lambda}_1, J_1)$ are the families $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1}$ such that $\\Lambda_t$ is chord generic and has finitely many Reeb chords for every $t$,\n\t\t\n\t\t\\item homotopies from a morphism $\\Phi = (\\mathbf{\\Lambda}_t, J_t)_{0 \\leq t \\leq 1} : (\\mathbf{\\Lambda}_0, J_0) \\to (\\mathbf{\\Lambda}_1, J_1)$ to another morphism $\\Phi' = (\\mathbf{\\Lambda}_t', J_t')_{0 \\leq t \\leq 1} : (\\mathbf{\\Lambda}_0, J_0) \\to (\\mathbf{\\Lambda}_1, J_1)$ are the families $(\\mathbf{\\Lambda}_{s,t}, J_{s,t})_{0 \\leq s \\leq S, 0 \\leq t \\leq 1}$ such that $\\Lambda_{s,t}$ is chord generic and has finitely many Reeb chords for every $s, t$.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{thm}\n\n\\begin{rmk}\n\n\tWe expect that the finiteness of Reeb chords condition in Theorem \\ref{thm invariance} (which is very restrictive) can be easily dropped using (homotopy) colimits of DG-categories diagrams. \n\tOn the other hand, studying birth\/death of Reeb chords phenomena is a more serious issue that we will address in a future work.\n\n\\end{rmk}\n\\section{Mapping torus of an $A_{\\infty}$-autoequivalence}\\label{subsection mapping torus}\n\nIn this section, we introduce the notion of mapping torus for a quasi-autoequivalence of an $A_{\\infty}$-category, by analogy with the mapping torus associated to an automorphism of a topological space. This terminology was also used in \\cite{Kar21}, but we do not know if the two notions coincide. The two main theorems of this section allow us to compute this mapping torus under different hypotheses. \n\n\\begin{rmk}\n\t\n\tIn this section, $A_{\\infty}$-categories are always assumed to be \\emph{strictly unital} (see \\cite[paragraph (2a)]{Sei08}).\n\t\n\\end{rmk}\n\n\\subsection{Definitions and main results}\n\n\\subsubsection{Definitions}\n\n\\begin{defin}\\label{definition mapping torus}\n\t\n\tLet $\\tau$ be a quasi-autoequivalence of an Adams-graded $A_{\\infty}$-category $\\mcA$. \n\tThe mapping torus of $\\tau$ is the $A_{\\infty}$-category\n\t\\[\\mathrm{MT} (\\tau) := \\mathrm{hocolim} \\left( \n\t\\begin{tikzcd}\n\t\\mcA \\sqcup \\mcA \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\mathrm{id} \\sqcup \\mathrm{id}\" left] & \\mcA \\\\\n\t\\mcA\n\t\\end{tikzcd} \\right) \\]\n\t(see Definition \\ref{definition Grothendieck construction}).\n\t\n\\end{defin}\n\n\\begin{rmks}\n\t\n\t\\begin{enumerate}\n\t\t\n\t\t\\item We use the terminology ``mapping torus'' by analogy with the analogous situation in the category of topological spaces. Indeed, if $f$ is an automorphism of some topological space $X$, then the mapping torus of $f$\n\t\t\\[M_f = \\left( X \\times \\left[ 0,1 \\right] \\right) \/ \\left( \\left( x, 0 \\right) \\sim \\left( f(x), 1 \\right) \\right) \\]\n\t\tis the homotopy colimit of the following diagram\n\t\t\\[\\begin{tikzcd}\n\t\t\tX \\sqcup X \\ar[r, \"\\mathrm{id} \\sqcup f\"] \\ar[d, \"\\mathrm{id} \\sqcup \\mathrm{id}\" left] & X \\\\\n\t\t\tX .\n\t\t\\end{tikzcd} \\]\n\t\n\t\t\\item The terminology ``mapping torus of an autoequivalence of $A_{\\infty}$-categories'' also appears in \\cite{Kar21}, and it is used in \\cite{Kar21bis} in order to distinguish open symplectic mapping tori. We do not know if the two notions (the one of \\cite{Kar21} and the one of Definition \\ref{definition mapping torus}) coincide.\n\t\t\n\t\t\\item The mapping torus of a quasi-autoequivalence is also Adams-graded, because it is the localization of an Adams-graded $A_{\\infty}$-category at morphisms of Adams-degree $0$. \n\t\t\n\t\\end{enumerate}\n\t\n\\end{rmks}\n\n\\begin{defin}\\label{definition group-action}\n\t\n\tLet $\\mcA$ be an $A_{\\infty}$-category. A $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ is a bijection \n\t\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA \\right), \\quad \\left( n, E \\right) \\mapsto X^n \\left( E \\right) . \\]\n\tIf such a splitting has been chosen, we define the Adams-grading of an homogeneous element $x \\in \\mcA \\left( X^i (E), X^j (E) \\right)$ to be $(j-i)$. This turns $\\mcA$ into an Adams-graded $A_{\\infty}$-category.\n\t\n\tLet $\\tau$ be a quasi-autoequivalence of $\\mcA$.\n\tWe say that a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ is compatible with $\\tau$ if \n\t\\[\\tau \\left( X^n \\left( E \\right) \\right) = X^{n+1} \\left( E \\right) \\]\n\tfor every $n \\in \\mathbf{Z}$ and $E \\in \\mcE$. \n\t\n\tWe say that $\\mcA$ is weakly directed with respect to a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ if \n\t\\[\\mcA \\left( X^i (E), X^j (E') \\right) = 0 \\]\n\tfor every $i > j$ and $E, E' \\in \\mcE$ (we use the term ``weakly directed'' $A_{\\infty}$-category because the notion is slightly more general than that of directed $A_{\\infty}$-category defined by Seidel in \\cite[paragraph (5m)]{Sei08}).\n\t\n\\end{defin}\n\n\\begin{rmk}\n\t\n\tCompatible $\\mathbf{Z}$-splittings naturally arise in the context of $\\mathbf{Z}$-actions. \n\tA strict $\\mathbf{Z}$-action on an $A_{\\infty}$-category $\\mcA$ is a family of $A_{\\infty}$-endofunctors $\\left( \\tau_n \\right)_{n \\in \\mathbf{Z}}$ such that $\\tau_0 = \\mathrm{id}_{\\mcA}$ and $\\tau_{i+j} = \\tau_i \\circ \\tau_j$ (see \\cite[paragraph (10b)]{Sei08}). \n\tIf the induced $\\mathbf{Z}$-action on $\\mathrm{ob} \\left( \\mcA \\right)$ is free, then any section $\\sigma$ of the projection $\\mathrm{ob} ( \\mcA ) \\to \\mcE$, where $\\mcE$ is the set of equivalence classes of objects in $\\mcA$ under the $\\mathbf{Z}$-action, gives a $\\mathbf{Z}$-splitting \n\t\\[\\mathbf{Z} \\times \\mcE \\xrightarrow{\\sim} \\mathrm{ob} \\left( \\mcA \\right), \\quad \\left( n, E \\right) \\mapsto \\tau_n \\left( \\sigma \\left( E \\right) \\right) \\]\n\twhich is compatible with the automorphism $\\tau_1$. \n\t\n\\end{rmk}\n\n\\subsubsection{Main results}\n\n\\paragraph{First result.}\n\n\\begin{defin}\\label{definition category of coinvariants}\n\t\n\tLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$ equipped with a compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\n\tAssume that $\\tau$ is strict, i.e $\\tau^d=0$ for $d \\geq 2$, and acts bijectively on hom-sets. \n\tIn this case, we define an Adams-graded $A_{\\infty}$-category $\\mcA_{\\tau}$ as follows:\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the set of objects of $\\mcA_{\\tau}$ is $\\mcE$,\n\t\t\n\t\t\\item the space of morphisms $\\mcA_{\\tau} \\left( E, E' \\right)$ is the Adams-graded vector space given by\n\t\t\\[\\mcA_{\\tau} \\left( E, E' \\right) = \\left( \\bigoplus \\limits_{i,j \\in \\mathbf{Z}} \\mcA \\left( X^i \\left( E \\right), X^j \\left( E' \\right) \\right) \\right) \/ \\left( \\tau (x) \\sim x \\right) \\]\n\t\t\n\t\t\\item the operations are the unique linear maps such that for every sequence \n\t\t\\[ \\left( x_0, \\dots, x_{d-1} \\right) \\in \\mcA \\left( X^{i_0} \\left( E_0 \\right), X^{i_1} \\left( E_1 \\right) \\right) \\times \\dots \\times \\mcA \\left( X^{i_{d-1}} \\left( E_{d-1} \\right), X^{i_d} \\left( E_d \\right) \\right), \\]\n\t\twe have \n\t\t\\[\\mu_{\\mcA_{\\tau}} \\left( [ x_0 ], \\dots, [ x_{d-1} ] \\right) = \\left[ \\mu_{\\mcA} \\left( x_0, \\dots, x_{d-1} \\right) \\right], \\]\n\t\twhere $[\\cdot] : \\mcA \\left( X^i \\left( E \\right), X^j \\left( E' \\right) \\right) \\to \\mcA_{\\tau} \\left( E, E' \\right)$ denotes the projection. (It is not hard to see that such operations exist and satisfy the $A_{\\infty}$-relations.) \n\t\t\n\t\\end{enumerate}\n\t\n\\end{defin}\n\n\\begin{thm}\\label{thm mapping torus in strict situation}\n\n\tLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$ equipped with a compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\n\tAssume that $\\tau$ is strict and acts bijectively on hom-sets.\n\tThen there is a quasi-equivalence of Adams-graded $A_{\\infty}$-categories \n\t\\[\\mathrm{MT} (\\tau) \\simeq \\mcA_{\\tau}. \\] \n\t\n\\end{thm}\n\n\\begin{rmk}\n\t\n\tThe $A_{\\infty}$-category $\\mcA_{\\tau}$ is the (ordinary) colimit of the diagram used to define $\\mathrm{MT} (\\tau)$. Thus, Theorem \\ref{thm mapping torus in strict situation} can be thought of as a ``homotopy colimit equals colimit\" result.\n\t\n\\end{rmk}\n\n\\paragraph{Second result.}\n\nWe denote by $\\mathbf{F} \\left[ t_m \\right]$ the augmented Adams-graded associative algebra generated by a variable $t_m$ of bidegree $(m, 1)$.\nObserve that if $\\mcC$ is a subcategory of an $A_{\\infty}$-category $\\mcD$ with $\\mathrm{ob} (\\mcC) = \\mathrm{ob} (\\mcD)$, then $\\mcC \\oplus (t \\mathbf{F} [t] \\otimes \\mcD)$ is naturally an Adams-graded $A_{\\infty}$-category, where the Adams degree of $t^k \\otimes x$ equals $k$.\nBesides, if $\\mcC$ is an $A_{\\infty}$-category equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcC \\right)$, we denote by $\\mcC^0$ the full $A_{\\infty}$-subcategory of $\\mcC$ whose set of objects corresponds to $\\{ 0 \\} \\times \\mcE$.\nFinally, we use the functor $\\mcC \\mapsto \\mcC_m$ of Definition \\ref{definition forgetful functor}. \n\n\\begin{thm}[Theorem \\ref{thm mapping torus in weak situation introduction} in the Introduction]\\label{thm mapping torus in weak situation}\n\t \n\tLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$, weakly directed with respect to some compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\n\tAssume that there exists a closed degree $0$ bimodule map $f : \\mcA_m \\left( -, - \\right) \\to \\mcA_m \\left( -, \\tau(-) \\right)$ such that $f : \\mcA_m \\left( X^i(E) , X^j(E') \\right) \\to \\mcA_m \\left( X^i(E), X^{j+1} (E') \\right)$ is a quasi-somorphism for every $i < j$ and $E, E' \\in \\mcE$.\n\tThen there is a quasi-equivalence of Adams-graded $A_{\\infty}$-categories \n\t\\[\\mathrm{MT} (\\tau) \\simeq \\mcA_m^0 \\oplus \\left( t_m \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ f \\left( \\mathrm{units} \\right)^{-1} \\right]^0 \\right) . \\]\n\t\n\\end{thm}\n\n\\paragraph{Outline of the section} \n\nIn section \\ref{subsection resuts about a specific module}, we consider an $A_{\\infty}$-category $\\mcA$ equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ and a choice of a closed degree $0$ morphism $c_n (E) \\in \\mcA \\left( X^n (E), X^{n+1} (E) \\right)$ for every $n \\in \\mathbf{Z}$ and every $E \\in \\mcE$. We give technical results about specific modules associated to this data. This will be used in the proof of Theorem \\ref{thm mapping torus in weak situation} with $c_n \\left( E \\right) = f \\left( e_{X^n(E)} \\right)$.\n\nIn section \\ref{subsection category and modules for the mapping torus}, we consider the Grothendieck construction $\\mcG$ of a slightly different diagram than the one in Definition \\ref{definition mapping torus}, together with a set $W_{\\mcG}$ of closed degree 0 morphisms. The idea is that the localization $\\mcH = \\mcG \\left[ W_{\\mcG}^{-1} \\right]$ is the homotopy colimit of a diagram obtained from the one in Definition \\ref{definition mapping torus} by a cofibrant replacement of the diagonal functor $\\mcA \\sqcup \\mcA \\to \\mcA$. Thus it is not surprising that $\\mcH$ is quasi-equivalent to the mapping torus of $\\tau$.\nMoreover, we prove technical results about specific modules over $\\mcG$ that will be used in the proofs of Theorems \\ref{thm mapping torus in strict situation} and \\ref{thm mapping torus in weak situation}. \n \nIn section \\ref{subsection proof of the first result}, we prove Theorem \\ref{thm mapping torus in strict situation}. We first define an $A_{\\infty}$-functor $\\Phi : \\mcG \\to \\mcA_{\\tau}$ which sends $W_{\\mcG}$ to the set of units in $\\mcA_{\\tau}$. Then we prove that the induced $A_{\\infty}$-functor $\\widetilde{\\Phi} : \\mcH \\to \\mcA_{\\tau} \\left[ \\{ \\text{units} \\}^{-1} \\right]$ is a quasi-equivalence. To do that, our strategy is to apply Proposition \\ref{prop quasi-isomorphism between localizations} using the results of section \\ref{subsection category and modules for the mapping torus} about the specific $\\mcG$-modules. \n\nIn section \\ref{subsection proof of the second result}, we prove Theorem \\ref{thm mapping torus in weak situation}. We use the fact that $\\mcG$ is ``big enough'' (there are more objects and morphisms than in the Grothendieck construction of the diagram in Definition \\ref{definition mapping torus}) in order to define an $A_{\\infty}$-functor $\\Psi_m : \\mcG_m \\to \\mcA_m$ (see Definition \\ref{definition forgetful functor}). This induces an $A_{\\infty}$-functor $\\widetilde{\\Psi} : \\mcH \\to \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ f \\left( \\{ \\text{units} \\} \\right)^{-1} \\right]$. Then we prove that for every Adams degree $j \\geq 1$, and for every objects $X,Y$ in $\\mcH$, the map\n\\[\\widetilde{\\Psi} : \\mcH \\left( X,Y \\right)^{*,j} \\to \\left( \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ f \\left( \\{ \\text{units} \\} \\right)^{-1} \\right] \\right) \\left( \\Psi X , \\Psi Y \\right)^{*,j} \\]\nis a quasi-isomorphism (if $V$ is an Adams-graded vector space, $V^{*,j}$ denotes the subspace of Adams degree $j$ elements).\nTo do that, we apply once again Proposition \\ref{prop quasi-isomorphism between localizations} using the results of sections \\ref{subsection resuts about a specific module} and \\ref{subsection category and modules for the mapping torus} about the specific modules over $\\mcA_m$ and $\\mcG$ respectively.\nThis allows us to finish the proof of Theorem \\ref{thm mapping torus in weak situation}.\n\n\\subsection{Results about specific modules}\\label{subsection resuts about a specific module}\n\nIn this section, we give technical results that will allow us to apply Proposition \\ref{prop quasi-isomorphism between localizations} in the proof of Theorem \\ref{thm mapping torus in weak situation}.\n\nLet $\\mcA$ be an $A_{\\infty}$-category equipped with a $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\nAssume that we chose, for every $n \\in \\mathbf{Z}$ and every $E \\in \\mcE$, a closed degree $0$ morphism $c_n (E) \\in \\mcA \\left( X^n (E), X^{n+1} (E) \\right)$. Moreover, assume that we chose a set $W_{\\mcA}$ of closed degree $0$ morphisms which contains the morphisms $c_n(E)$.\n\n\\begin{rmk}\n\t\n\tAccording to Definition \\ref{definition group-action}, the $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ naturally induces an Adams-grading on $\\mcA$. However in this section, we do not consider $\\mcA$ as being Adams-graded. \n\t\n\\end{rmk}\n\nIn the following, we fix some element $E_{\\diamond} \\in \\mcE$. When we write an object $X^n$ or a morphism $c_n$ without specifying the element of $\\mcE$, we mean $X^n (E_{\\diamond})$ or $c_n (E_{\\diamond})$ respectively. \nRecall that $t_{c_n} : \\mcA \\left( -, X^n \\right) \\to \\mcA \\left( -, X^{n+1} \\right)$ denotes the $\\mcA$-module map induced by $c_n \\in \\mcA \\left( X^n, X^{n+1} \\right)$ (see Definition \\ref{definition Yoneda functor}). \n\n\\begin{defin}\\label{definition specific module}\n\t\n\tWe set $\\mcM_{\\mcA}$ to be the $\\mcA$-module\n\t\\[ \\mcM_{\\mcA} := \\left[\n\t\\begin{tikzcd}\n\t\\dots \\ar[rd, \"t_{c_{-1}}\"] & \\mcA \\left( - , X^0 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_0}\"] & \\mcA \\left( - , X^1 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_1}\"] & \\dots \\\\\n\t\\dots & \\mcA \\left( - , X^0 \\right) & \\mcA \\left( - , X^1 \\right) & \\dots\n\t\\end{tikzcd} \n\t\\right] \\]\n\t(see Definition \\ref{definition cone of a morphism between modules}).\n\tBesides, we set $t_{\\mcA}^n : \\mcA \\left( - , X^n \\right) \\to \\mcM_{\\mcA}$ to be the $\\mcA$-module inclusion for every $n \\in \\mathbf{Z}$.\n\t\n\\end{defin}\n\nThe first result highlights a key property of the module $\\mcM_{\\mcA}$.\n\n\\begin{lemma}\\label{lemma multiplication by continuation element becomes homotopic to inclusion}\n\t\n\tFor every $n \\in \\mathbf{Z}$, the closed $\\mcA$-module map $t_{\\mcA}^{n+1} \\circ t_{c_n} : \\mcA \\left( -, X^n \\right) \\to \\mcM_{\\mcA}$ is homotopic to $t_{\\mcA}^n : \\mcA \\left( -, X^n \\right) \\to \\mcM_{\\mcA}$. \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tConsider the degree $(-1)$ strict $\\mcA$-module map $s : \\mcA \\left( -, X^n \\right) \\to \\mcM_{\\mcA}$ which sends a morphism in $\\mcA \\left( X, X^n \\right)$ to the corresponding shifted element in $\\mcA \\left( X, X^n \\right) \\left[ 1 \\right]$.\n\tThen an easy computation gives\n\t\\[\\mu_{\\mathrm{Mod}_{\\mcA}}^1 \\left( s \\right) = t_{\\mcA}^{n+1} \\circ t_{c_n} + t_{\\mcA}^n . \\]\n\tThis concludes the proof. \n\t\n\\end{proof}\n\nIn the proof of the two results below, we will use specific $\\mcA$-modules. If $p$ is a fixed non negative integer, we set \n\\[K_p = \\left[\n\\begin{tikzcd}\n\\dots \\ar[rd, \"t_{c_{p-2}}\"] & \\mcA \\left( - , X^{p-1} \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_{p-1}}\"] & \\mcA \\left( - , X^p \\right) \\ar[d, \"\\mathrm{id}\"] \\\\\n\\dots & \\mcA \\left( - , X^{p-1} \\right) & \\mcA \\left( - , X^p \\right)\n\\end{tikzcd} \n\\right] \\]\nand\n\\[\\widetilde{K}_p = \\left[\n\\begin{tikzcd}\n\\mcA \\left( - , X^p \\right) \\ar[rd, \"t_{c_p}\"] & \\mcA \\left( - , X^{p+1} \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_{p+1}}\"] & \\dots \\\\\n& \\mcA \\left( - , X^{p+1} \\right) & \\dots \n\\end{tikzcd} \n\\right] . \\]\nMoreover, we will consider the sequences of $\\mcA$-modules $\\left( F_p^q \\right)_{q \\geq 0}$, $\\left( \\widetilde{F}_p^q \\right)_{q \\geq 0}$ starting at $F_p^0 = \\widetilde{F}_p^0 = 0$ and with\n\\[ F_p^q = \\left[\n\\begin{tikzcd}\n\\mcA \\left( - , X^{p-q+1} \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"t_{c_{p-q+1}}\"] & \\dots \\ar[rd, \"t_{c_{p-1}}\"] & \\mcA \\left( - , X^p \\right) \\ar[d, \"\\mathrm{id}\"] \\\\\n\\mcA \\left( - , X^{p-q+1} \\right) & \\dots & \\mcA \\left( - , X^p \\right)\n\\end{tikzcd}\n\\right] \\]\nand \n\\[\\widetilde{F}_p^q = \\left[\n\\begin{tikzcd}\n\\mcA \\left( - , X^p \\right) \\ar[rd, \"t_{c_p}\"] & \\dots \\ar[rd, \"t_{c_{p+q-1}}\"] & \\\\\n& \\dots & \\mcA \\left( - , X^{p+q} \\right)\n\\end{tikzcd} \n\\right] \\]\nfor $q \\geq 1$.\n\nThe following Lemma is mostly technical. It will be used in the proofs of Lemmas \\ref{lemma second result for specific module} and \\ref{lemma G-module map for relation G-A}.\n\n\\begin{lemma}\\label{lemma first result for specific module}\n\t\n\tAssume that for every $i < j$, for every $E \\in \\mcE$, the chain map\n\t\\[\\mu_{\\mcA}^2 \\left( - , c_j \\right) : \\mcA \\left( X^i (E) , X^j \\right) \\to \\mcA \\left( X^i (E) , X^{j+1} \\right) \\]\n\tis a quasi-isomorphism. \n\tThen for every $k < n$, for every $E \\in \\mcE$, the inclusion $\\mcA \\left( X^k (E), X^n \\right) \\hookrightarrow \\mcM_{\\mcA} \\left( X^k (E) \\right)$ is a quasi-isomorphism.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tThe cone of the inclusion $\\mcA \\left( X^k (E) , X^n \\right) \\hookrightarrow \\mcM_{\\mcA} \\left( X^k (E) \\right)$ is quasi-isomorphic to its cokernel, which is $K_{n-1} \\left( X^k (E) \\right) \\oplus \\widetilde{K}_n \\left( X^k (E) \\right)$.\n\t\n\tWe have to show that these complexes are acyclic. \n\tObserve that $\\left( F_{n-1}^q \\left( X^k (E) \\right) \\right)_{q \\geq 0}$ and $\\left( \\widetilde{F}_n^q \\left( X^k (E) \\right) \\right)_{q \\geq 0}$ are increasing, exhaustive, and bounded from below filtrations of $K_{n-1} \\left( X^k (E) \\right)$ and $\\widetilde{K}_n \\left( X^k (E) \\right)$ respectively.\n\tFor every $q \\geq 1$, we have \n\t\\[F_{n-1}^q \\left( X^k (E) \\right) \/ F_{n-1}^{q-1} \\left( X^k (E) \\right) = \\left[\n\t\\begin{tikzcd}\n\t\\mcA \\left( X^k (E) , X^{n-q} \\right) \\ar[d, \"\\mathrm{id}\"] \\\\\n\t\\mcA \\left( X^k (E) , X^{n-q} \\right)\n\t\\end{tikzcd} \\right] \\]\n\tand\n\t\\[\\widetilde{F}_n^q \\left( X^k (E) \\right) \/ \\widetilde{F}_n^{q-1} \\left( X^k (E) \\right) = \\left[\n\t\\begin{tikzcd}\n\t\\mcA \\left( X^k (E) , X^{n+q-1} \\right) \\ar[rd, \"t_{c_{n+q-1}}\"] & \\\\\n\t& \\mcA \\left( X^k (E) , X^{n+q} \\right) \n\t\\end{tikzcd} \\right] . \\]\n\tThe first of the two latter complexes is clearly acyclic, and the second one is acyclic by assumption on the morphisms $c_j$. Thus the entire complex $K_{n-1} \\left( X^k (E) \\right) \\oplus \\widetilde{K}_n \\left( X^k (E) \\right)$ is acyclic, which is what we needed to prove. \n\t\n\\end{proof}\n\nThe following two lemmas will be used later in order to apply Proposition \\ref{prop quasi-isomorphism between localizations}. \n\n\\begin{lemma}\\label{lemma second result for specific module}\n\t\n\tAssume that for every $i < j < k$, for every $E \\in \\mcE$, the chain maps\n\t\\[\\left\\{\n\t\\begin{array}{ccccl}\n\t\\mu_{\\mcA}^2 \\left( - , c_j \\right) & : & \\mcA \\left( X^i (E) , X^j \\right) & \\to & \\mcA \\left( X^i (E) , X^{j+1} \\right) \\\\\n\t\\mu_{\\mcA}^2 \\left( c_j (E) , - \\right) & : & \\mcA \\left( X^{j+1} (E), X^{k+1} \\right) & \\to & \\mcA \\left( X^j (E), X^{k+1} \\right)\n\t\\end{array}\n\t\\right. \\]\n\tare quasi-isomorphisms. \n\tThen for every $(n, E) \\in \\mathbf{Z} \\times \\mcE$, the complex $\\mcM_{\\mcA} \\left( \\mathrm{Cone} \\, c_n (E) \\right)$ is acyclic.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe have\n\t\\[\\mcM_{\\mcA} \\left( \\mathrm{Cone} \\, c_n (E) \\right) = \\mathrm{Cone} \\left( \\mcM_{\\mcA} \\left( X^{n+1} (E) \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcA}}^2 \\left( c_n (E) , - \\right)} \\mcM_{\\mcA} \\left( X^n (E) \\right) \\right), \\]\n\tso we have to prove that $\\mu_{\\mcM_{\\mcA}}^2 \\left( c_n (E) , - \\right) : \\mcM_{\\mcA} \\left( X^{n+1} (E) \\right) \\to \\mcM_{\\mcA} \\left( X^n (E) \\right)$ is a quasi-isomorphism. \n\tObserve that we have the following commutative diagram\n\t\\[\\begin{tikzcd}[column sep = 2.5cm]\n\t\\mcM_{\\mcA} \\left( X^{n+1} (E) \\right) \\ar[r, \"\\mu_{\\mcM_{\\mcA}}^2 {(c_n (E),-)}\"] & \\mcM_{\\mcA} \\left( X^n (E) \\right) \\\\\n\t\\mcA \\left( X^{n+1} (E) , X^{n+2} \\right) \\ar[u, hook] \\ar[r, \"\\mu_{\\mcA}^2 {\\left( c_n (E) , - \\right)}\"] & \\mcA \\left( X^n (E) , X^{n+2} \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tThe bottom horizontal map is a quasi-isomorphism by assumption on the morphisms $c_j (E)$. Moreover, the vertical maps are quasi-isomorphisms according to Lemma \\ref{lemma first result for specific module}. This implies that $\\mu_{\\mcM_{\\mcA}}^2 \\left( c_n (E) , - \\right)$ is indeed a quasi-isomorphism. \n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma third result for specific module}\n\t\n\tThe $\\mcA$-module map $_{W_{\\mcA}^{-1}} t_{\\mcA}^n : \\, _{W_{\\mcA}^{-1}} \\mcA \\left( - , X^n \\right) \\to \\, _{W_{\\mcA}^{-1}} \\mcM_{\\mcA}$ is a quasi-isomorphism for every $n \\in \\mathbf{Z}$.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tLet $X$ be some object of $\\mcA$. We want to prove that the chain map $_{W_{\\mcA}^{-1}} t_{\\mcA}^n : \\, _{W_{\\mcA}^{-1}} \\mcA \\left( X , X^n \\right) \\to \\, _{W_{\\mcA}^{-1}} \\mcM_{\\mcA} \\left( X \\right)$ is a quasi-isomorphism. Observe that \n\t\\[_{W_{\\mcA}^{-1}} \\mcM_{\\mcA} \\left( X \\right) = \\left[\n\t\\begin{tikzcd}\n\t\t\\dots \\ar[rd] & \\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^0 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"_{W_{\\mcA}^{-1}} t_{c_0}\" below = 2.5] & \\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^1 \\right) \\ar[d, \"\\mathrm{id}\"] \\ar[rd, \"_{W_{\\mcA}^{-1}} t_{c_1}\"] & \\dots \\\\\n\t\t\\dots & \\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^0 \\right) & \\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^1 \\right) & \\dots\n\t\\end{tikzcd} \n\t\\right] \\]\n\tand the chain map $_{W_{\\mcA}^{-1}} t_{\\mcA}^n : \\, _{W_{\\mcA}^{-1}} \\mcA \\left( X , X^n \\right) \\to \\, _{W_{\\mcA}^{-1}} \\mcM_{\\mcA} \\left( X \\right)$ is the inclusion. The cone of the latter is then quasi-isomorphic to its cokernel, which is $_{W_{\\mcA}^{-1}} K_{n-1} \\left( X \\right) \\oplus \\, _{W_{\\mcA}^{-1}} \\widetilde{K}_n \\left( X \\right)$.\n\tObserve that $\\left( _{W_{\\mcA}^{-1}} F_{n-1}^q \\left( X \\right) \\right)_{q \\geq 0}$, $\\left( _{W_{\\mcA}^{-1}} \\widetilde{F}_n^q \\left( X \\right) \\right)_{q \\geq 0}$ are increasing, exhaustive, and bounded from below filtrations of $_{W_{\\mcA}^{-1}} K_{n-1} \\left( X \\right)$, $_{W_{\\mcA}^{-1}} \\widetilde{K}_n \\left( X \\right)$ respectively. For every $q \\geq 1$, we have \n\t\\[_{W_{\\mcA}^{-1}} F_{n-1}^q \\left( X \\right) \/ \\, _{W_{\\mcA}^{-1}} F_{n-1}^{q-1} \\left( X \\right) = \\left[\n\t\\begin{tikzcd}\n\t\t\\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^{n-q} \\right) \\ar[d, \"\\mathrm{id}\"] \\\\\n\t\t\\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^{n-q} \\right)\n\t\\end{tikzcd} \\right] \\]\n\tand\n\t\\[_{W_{\\mcA}^{-1}} \\widetilde{F}_n^q \\left( X \\right) \/ \\, _{W_{\\mcA}^{-1}} \\widetilde{F}_n^{q-1} \\left( X \\right) =\n\t\\left[\n\t\\begin{tikzcd}\n\t\t\\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^{n-1+q} \\right) \\ar[rd, \"_{W_{\\mcA}^{-1}} t_{c_{n-1+q}}\"] & \\\\\n\t\t& \\mcA \\left[ W_{\\mcA}^{-1} \\right] \\left( X , X^{n+q} \\right) \n\t\\end{tikzcd} \\right] . \\]\n\tThe first of the two latter complexes is clearly acyclic, and the second one is acyclic because $c_{n-1+q}$ belongs to the set $W_{\\mcA}$ by which we localized (see \\cite[Lemma 3.12]{GPS20}). Thus the entire complex $_{W_{\\mcA}^{-1}} K_{n-1} \\left( X \\right) \\oplus \\, _{W_{\\mcA}^{-1}} \\widetilde{K}_n \\left( X \\right)$ is acyclic, which is what we needed to prove.\n\t\n\\end{proof}\n\n\\subsection{The $A_{\\infty}$-category and modules for the mapping torus}\\label{subsection category and modules for the mapping torus}\n\nIn this section, we consider an $A_{\\infty}$-category $\\mcG$, together with a set $W_{\\mcG}$ of closed degree 0 morphisms. We prove that $\\mcH = \\mcG \\left[ W_{\\mcG}^{-1} \\right]$ is quasi-equivalent to the mapping torus of $\\tau$, and we prove technical results about specific $\\mcG$-modules that will allow us to apply Proposition \\ref{prop quasi-isomorphism between localizations} in the proofs of Theorems \\ref{thm mapping torus in strict situation} and \\ref{thm mapping torus in weak situation}. \n\nLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$ equipped with a compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\nIf $\\mcA_{\\triangle}$ is a copy of $\\mcA$, we denote by $X_{\\triangle}^n \\left( E \\right)$ the object of $\\mcA_{\\triangle}$ corresponding to $\\left( n, E \\right) \\in \\mathbf{Z} \\times \\mcE$.\n\n\\subsubsection{The Grothendieck construction $\\mcG$}\n\nThe $A_{\\infty}$-category $\\mcG$ will be the Grothendieck construction of a slightly different diagram than the one in Definition \\ref{definition mapping torus}. The idea is to introduce an $A_{\\infty}$-category $\\mcC$ together with a set of closed degree $0$ morphisms $W_{\\mcC}$ such that the localization $\\mcC \\left[ W_{\\mcC}^{-1} \\right]$ is a cylinder object for $\\mcA$. \nObserve that this kind of cofibrant replacement is common in homotopy colimits computation, and indeed we need it to prove Theorem \\ref{thm mapping torus in weak situation}.\n\n\\begin{defin}\\label{definition cylinder object}\n\t\n\tLet $\\mcA_{\\bot}$, $\\mcA_I$ and $\\mcA_{\\top}$ be three copies of $\\mcA$. \n\tWe denote by $\\mcC$ the Grothendieck construction (see Definition \\ref{definition Grothendieck construction}) of the following diagram\n\t\\[\\begin{tikzcd}\n\t\\mcA_I \\ar[r, \"\\mathrm{id}\"] \\ar[d, \"\\mathrm{id}\" left] & \\mcA_{\\top} \\\\\n\t\\mcA_{\\bot} &\n\t\\end{tikzcd} \\]\n\tand we let $\\iota_{\\bot}, \\iota_I, \\iota_{\\top} : \\mcA \\to \\mcC$ be the strict inclusions with images $\\mcA_{\\bot}$, $\\mcA_I$, $\\mcA_{\\top}$ respectively. Finally, we denote by $W_{\\mcC}$ the set of adjacent units in $\\mcC$, and we let $\\mcC yl_{\\mcA} = \\mcC \\left[ W_{\\mcC}^{-1} \\right]$ be the homotopy colimit of the diagram above.\n\t\n\\end{defin}\n \n\\begin{defin}\\label{definition category G}\n\t\n\tLet $\\mcA_-$, $\\mcA_{+}$, $\\mcA_{\\bullet}$ be three copies of $\\mcA$.\n\tWe denote by $\\mcG$ the Grothendieck construction of the following diagram\n\t\\[\\begin{tikzcd}\n\t\\mcA_- \\sqcup \\mcA_+ \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\" left] & \\mcA_{\\bullet} \\\\\n\t\\mcC .\n\t\\end{tikzcd} \\]\n\tBesides, we denote by $W_{\\mcG}$ the union of $W_{\\mcC}$ and the set of adjacent units in $\\mcG$, and we set\n\t\\[\\mcH := \\mcG \\left[ W_{\\mcG}^{-1} \\right]. \\]\n\t\n\\end{defin}\n\nAccording to Proposition \\ref{prop cylinder object}, $\\mcC yl_{\\mcA}$ can be thought as a cylinder object for $\\mcA$.\nTherefore, the following result should not be surprising. \n\n\\begin{lemma}\\label{lemma relation G - mapping torus}\n\t\n\tThe mapping torus of $\\tau$ is quasi-equivalent to $\\mcH$. \n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tLet $\\pi : \\mcC \\to \\mcA$ be the $A_{\\infty}$-functor induced by the following commutative diagram \n\t\\[\\begin{tikzcd}\n\t\\mcA_I \\ar[r, \"\\mathrm{id}\"] \\ar[d, \"\\mathrm{id}\"] & \\mcA_{\\top} \\ar[d, \"\\mathrm{id}\"] \\\\\n\t\\mcA_{\\bot} \\ar[r, \"\\mathrm{id}\"] & \\mcA\n\t\\end{tikzcd} \\]\n\t(see Proposition \\ref{prop induced functor from Grothendieck construction}).\n\tWe get a commutative diagram \n\t\\[\\begin{tikzcd}\n\t\\mcC \\ar[d, \"\\pi\"] & \\mcA_- \\sqcup \\mcA_+ \\ar[l, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\" above] \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\mathrm{id}\"] & \\mcA_{\\bullet} \\ar[d, \"\\mathrm{id}\"] \\\\\n\t\\mcA & \\mcA_- \\sqcup \\mcA_+ \\ar[l, \"\\mathrm{id} \\sqcup \\mathrm{id}\"] \\ar[r, \"\\mathrm{id} \\sqcup \\tau\" below] & \\mcA_{\\bullet} \n\t\\end{tikzcd} \\]\n\twhich induces an $A_{\\infty}$-functor $\\chi$ from $\\mcG$ to the Grothendieck construction of the bottom line (see Proposition \\ref{prop invariance of homotopy colimits}).\n\tObserve that $\\chi$ sends $W_{\\mcC}$ to the set $U$ of units in $\\mcA$. \n\tNow, according to Proposition \\ref{prop cylinder object}, the $A_{\\infty}$-functor $\\widetilde{\\pi} : \\mcC yl_{\\mcA} = \\mcC \\left[ W_{\\mcC}^{-1} \\right] \\to \\mcA \\left[ U^{-1} \\right]$ is a quasi-equivalence. According to Lemma A.6 in \\cite{GPS19} (called ``localization and homotopy colimits commute''), this implies that the $A_{\\infty}$-functor induced by $\\chi$\n\t\\[\\mathrm{hocolim} \\left( \n\t\\begin{tikzcd}[column sep = 0.5cm]\n\t\\mcA_- \\sqcup \\mcA_+ \\ar[d, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\"] \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] & \\mcA_{\\bullet} \\\\\n\t\\mcC\n\t\\end{tikzcd}\n\t\\right) \\left[ W_{\\mcC}^{-1} \\right] \\xrightarrow{\\widetilde{\\chi}}\n\t\\mathrm{hocolim} \\left( \n\t\\begin{tikzcd}[column sep = 0.5cm]\n\t\\mcA_- \\sqcup \\mcA_+ \\ar[d, \"\\mathrm{id} \\sqcup \\mathrm{id}\"] \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] & \\mcA_{\\bullet} \\\\\n\t\\mcA\n\t\\end{tikzcd}\n\t\\right) \\left[ U^{-1} \\right] \\]\n\tis a quasi-equivalence.\n\tThis completes the proof because the source of $\\widetilde{\\chi}$ is exactly $\\mcH$, and its target is quasi-equivalent to the mapping torus of $\\tau$. \n\t\n\\end{proof}\n\n\\subsubsection{Modules over $\\mcG$}\n\nIn the following, we fix some element $E_{\\diamond} \\in \\mcE$. When we write an object $X_{\\triangle}^n$ without specifying the element of $\\mcE$, we mean $X_{\\triangle}^n (E_{\\diamond})$.\nMoreover, we denote by \n\\[t_{\\triangle \\square}^n : \\mcG \\left( -, X_{\\triangle}^n \\right) \\to \\mcG \\left( -, X_{\\square}^{n+\\delta_{\\triangle \\square}} \\right) \\]\nthe $\\mcG$-module map induced by the adjacent unit in $\\mcG \\left( X_{\\triangle}^n, X_{\\square}^{n+\\delta_{\\triangle \\square}} \\right)$ (see Definition \\ref{definition Yoneda functor}), where \n\\[\\delta_{\\triangle \\square} = \\left\\{\n\\begin{array}{ll}\n1 & \\text{if } \\left( \\triangle, \\square \\right) = \\left( +, \\bullet \\right) \\\\\n0 & \\text{otherwise} .\n\\end{array}\n\\right. \\]\n\n\\begin{defin}\\label{definition module for G}\n\t\n\tWe denote by $\\mcM_{\\mcG}$ the $\\mcG$-module defined by\n\t\\[\\mcM_{\\mcG} = \n\t\\left[ \n\t\\begin{tikzcd}[column sep = scriptsize]\n\t\\dots \\ar[rd, \"t_{+ \\bullet}^{-1}\"] & \\mcG \\left( - , X_-^{0} \\right) \\ar[d, \"t_{- \\bullet}^0\"] \\ar[rd, \"t_{- \\bot}^0\"] & \\mcG \\left( - , X_I^{0} \\right) \\ar[d, \"t_{I \\bot}^0\"] \\ar[rd, \"t_{I \\top}^0\"] & \\mcG \\left( - , X_+^{0} \\right) \\ar[d, \"t_{+ \\top}^0\"] \\ar[rd, \"t_{+ \\bullet}^0\"] & \\mcG \\left( - , X_{-}^{1} \\right) \\ar[d, \"t_{- \\bullet}^1\"] \\ar[rd, \"t_{- \\bot}^1\"] & \\dots \\\\\n\t\\dots & \\mcG \\left( - , X_{\\bullet}^{0} \\right) & \\mcG \\left( - , X_{\\bot}^0 \\right) & \\mcG \\left( - , X_{\\top}^0 \\right) & \\mcG \\left( - , X_{\\bullet}^{1} \\right) & \\dots \n\t\\end{tikzcd}\n\t\\right] \\]\n\t(see Definition \\ref{definition cone of a morphism between modules}). \n\tFor practical reasons, we also consider the $\\mcG$-modules\n\t\\[ \\mcM_{\\star}^n := \n\t\\left[\n\t\\begin{tikzcd}\n\t& \\mcG \\left( - , X_I^n \\right) \\ar[ld, \"t_{I \\bot}^n\"] \\ar[rd, \"t_{I \\top}^n\"] & \\\\\n\t\\mcG \\left( - , X_{\\bot}^n \\right) & & \\mcG \\left( - , X_{\\top}^n \\right)\n\t\\end{tikzcd}\n\t\\right], \\quad n \\in \\mathbf{Z}. \\]\n\tBesides, we denote by $t_{\\mcG} : \\mcG \\left( - , X_{\\bullet}^0 \\right) \\to \\mcM_{\\mcG}$ the $\\mcG$-module inclusion.\n\t\n\\end{defin}\n\n\\begin{rmk}\n\t\n\tWe can write \n\t\\[ \\mcM_{\\mcG} = \n\t\\left[\n\t\\begin{tikzcd}\n\t& \\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( \\mcG \\left( - , X_-^n \\right) \\oplus \\mcG \\left( - , X_+^n \\right) \\right) \\ar[ld, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bot}^n \\oplus t_{+ \\top}^n \\right)\"] \\ar[rd, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bullet}^n \\oplus t_{+ \\bullet}^n \\right)\"] & \\\\\n\t\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\mcM_{\\star}^n & & \\bigoplus \\limits_{n \\in \\mathbf{Z}} \\mcG \\left( - , X_{\\bullet}^n \\right)\n\t\\end{tikzcd}\n\t\\right] . \\]\n\t\n\\end{rmk}\n\nThe following two lemmas are analogs of Lemmas \\ref{lemma second result for specific module} and \\ref{lemma third result for specific module} respectively. They will be used later in order to apply Proposition \\ref{prop quasi-isomorphism between localizations}.\n\n\\begin{lemma}\\label{lemma properties of modules}\n\t\n\tFor every $w$ in $W_{\\mcG}$, the complex $\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right)$ is acyclic.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tLet $w$ be the morphism in $W_{\\mcG} \\cap \\mcG \\left( X_I^k (E) , X_{\\top}^k (E) \\right)$ (the proof is analogous for the morphism in $W_{\\mcG} \\cap \\mcG \\left( X_I^k (E) , X_{\\bot}^k (E) \\right)$). Then \n\t\\begin{align*}\n\t\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right) & = \\mathrm{Cone} \\left( \\mcM_{\\mcG} \\left( X_{\\top}^k (E) \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} \\mcM_{\\mcG} \\left( X_I^k (E) \\right) \\right) \\\\\n\t& = \\bigoplus\\limits_{n} \\mathrm{Cone} \\left( \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} \\mcM_{\\star}^n \\left( X_I^k (E) \\right) \\right) .\n\t\\end{align*}\n\tWe want to prove that $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to \\mcM_{\\star}^n \\left( X_I^k (E) \\right)$ is a quasi-isomorphism for every $n$. Observe that the following diagram of chain complexes is commutative \n\t\\[\\begin{tikzcd}\n\t\\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\ar[rr, \"{ \\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) }\"] & & \\mcM_{\\star}^n \\left( X_I^k (E) \\right) \\\\\n\t\\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\ar[u, equal] \\ar[rr, \"{\\mu_{\\mcG}^2 \\left( w, - \\right)}\"] & & \\mcG \\left( X_I^k (E) , X_{\\top}^n \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tThe rightmost vertical arrow is injective, so its cone is quasi-isomorphic to its cokernel, which is the cone of $t_{I \\bot}^n : \\mcG \\left( X_I^k (E), X_I^n \\right) \\to \\mcG \\left( X_I^k (E), X_{\\bot}^n \\right)$. Since the latter map is a quasi-isomorphism, the cone of $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to \\mcM_{\\star}^n \\left( X_I^k (E) \\right)$ is quasi-isomorphic to the cone of $\\mu_{\\mcG}^2 \\left( w, - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to \\mcG \\left( X_I^k (E) , X_{\\top}^n \\right)$. The latter map is a quasi-isomorphism, so we conclude that $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to \\mcM_{\\star}^n \\left( X_I^k (E) \\right)$ is a quasi-isomorphism for every $n$, and thus $\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right)$ is acyclic. \n\t\n\tNow let $w$ be the morphism in $W_{\\mcG} \\cap \\mcG \\left( X_+^k (E) , X_{\\top}^k (E) \\right)$ (the proof is analogous for the morphism in $W_{\\mcG} \\cap \\mcG \\left( X_-^k (E) , X_{\\bot}^k (E) \\right)$). Then \n\t\\begin{align*}\n\t\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right) & = \\mathrm{Cone} \\left( \\mcM_{\\mcG} \\left( X_{\\top}^k (E) \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} \\mcM_{\\mcG} \\left( X_+^k (E) \\right) \\right) \\\\\n\t& = \\bigoplus\\limits_{n} \\mathrm{Cone} \\left( \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} K^n \\right).\n\t\\end{align*}\n\twhere \n\t\\[K^n = \\left[ \n\t\\begin{tikzcd}\n\t& \\mcG \\left( X_+^k (E) , X_+^n \\right) \\ar[ld, \"t_{+ \\top}^n\"] \\ar[rd, \"t_{+ \\bullet}^n\"] \\\\\n\t\\mcG \\left( X_+^k (E) , X_{\\top}^n \\right) & & \\mcG \\left( X_+^k (E) , X_{\\bullet}^{n+1} \\right)\n\t\\end{tikzcd}\n\t\\right] . \\]\n\tObserve that $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to K^n$ is injective, so its cone is quasi-isomorphic to its cokernel, which is the cone of $t_{+ \\bullet}^n : \\mcG \\left( X_+^k (E) , X_+^n \\right) \\to \\mcG \\left( X_+^k (E) , X_{\\bullet}^{n+1} \\right)$. The latter map is a quasi-isomorphism because $\\tau$ is a quasi-equivalence. This implies that the cone of $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\top}^k (E) , X_{\\top}^n \\right) \\to K^n$ is acyclic for every $n$, and thus $\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right)$ is acyclic.\n\t\n\tIt remains to consider a morphism $w$ in $W_{\\mcG} \\cap \\mcG \\left( X_+^k (E) , X_{\\bullet}^{k+1} (E) \\right)$ (the proof is analogous for the morphism in $W_{\\mcG} \\cap \\mcG \\left( X_-^k (E) , X_{\\bullet}^k (E) \\right)$). Then \n\t\\begin{align*}\n\t\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right) & = \\mathrm{Cone} \\left( \\mcM_{\\mcG} \\left( X_{\\bullet}^{k+1} (E) \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} \\mcM_{\\mcG} \\left( X_+^k (E) \\right) \\right) \\\\\n\t& = \\bigoplus\\limits_{n} \\mathrm{Cone} \\left( \\mcG \\left( X_{\\bullet}^{k+1} (E) , X_{\\bullet}^{n} \\right) \\xrightarrow{\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right)} K^n \\right).\n\t\\end{align*}\n\twhere \n\t\\[K^n = \\left[ \n\t\\begin{tikzcd}\n\t& \\mcG \\left( X_+^k (E) , X_+^{n-1} \\right) \\ar[ld, \"t_{+ \\top}^{n-1}\"] \\ar[rd, \"t_{+ \\bullet}^{n-1}\"] \\\\\n\t\\mcG \\left( X_+^k (E) , X_{\\top}^{n-1} \\right) & & \\mcG \\left( X_+^k (E) , X_{\\bullet}^{n} \\right)\n\t\\end{tikzcd}\n\t\\right] . \\]\n\tObserve that $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\bullet}^{k+1} (E) , X_{\\bullet}^{n} \\right) \\to K^n$ is injective, so its cone is quasi-isomorphic to its cokernel, which is the cone of $t_{+ \\top}^{n-1} : \\mcG \\left( X_+^k (E) , X_+^{n-1} \\right) \\to \\mcG \\left( X_+^k (E) , X_{\\top}^{n-1} \\right)$. The latter map is a quasi-isomorphism, so we conclude that the cone of $\\mu_{\\mcM_{\\mcG}}^2 \\left( w , - \\right) : \\mcG \\left( X_{\\bullet}^{k+1} (E) , X_{\\bullet}^{n} \\right) \\to K^n$ is acyclic for every $n$, and thus $\\mcM_{\\mcG} \\left( \\mathrm{Cone} \\, w \\right)$ is acyclic.\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma properties of modules bis}\n\t\n\tThe $\\mcH$-module map $_{W_{\\mcG}^{-1}} t_{\\mcG} : \\, _{W_{\\mcG}^{-1}} \\mcG \\left( - , X_{\\bullet}^0 \\right) \\to \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG}$ is a quasi-isomorphism.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe fix an object $X$ in $\\mcG$, and we want to prove that $_{W_{\\mcG}^{-1}} t_{\\mcG} : \\, _{W_{\\mcG}^{-1}} \\mcG \\left( X , X_{\\bullet}^0 \\right) \\to \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X \\right)$ is a quasi-isomorphism. \n\tObserve that\n\t\\[_{W_{\\mcG}^{-1}} \\mcM_{\\mcG} := \\left[ \n\t\\begin{tikzcd}\n\t\t\\dots \\ar[rd] & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( - , X_+^{-1} \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{+ \\top}^{-1}\"] \\ar[rd, \"_{W_{\\mcG}^{-1}} t_{+ \\bullet}^{-1}\" below = 2] & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( - , X_{-}^{0} \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{- \\bullet}^0\"] \\ar[rd, \"_{W_{\\mcG}^{-1}} t_{- \\bot}^0\"] & \\dots \\\\\n\t\t\\dots & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( - , X_{\\top}^{-1} \\right) & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( - , X_{\\bullet}^0 \\right) & \\dots \n\t\\end{tikzcd}\n\t\\right] \\]\n\tand that the chain map $_{W_{\\mcG}^{-1}} t_{\\mcG} : \\, _{W_{\\mcG}^{-1}} \\mcG \\left( X , X_{\\bullet}^0 \\right) \\to \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X \\right)$ is the inclusion. The cone of the latter is then quasi-isomorphic to its cokernel, which can be written $K' \\oplus K''$ with\n\t\\[K' = \\left[ \n\t\\begin{tikzcd}\n\t\t\\dots \\ar[rd] & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_I^{-1} \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{I \\bot}^{-1}\"] \\ar[rd, \"_{W_{\\mcG}^{-1}} t_{I \\top}^{-1}\" below = 2] & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_+^{-1} \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{+ \\top}^{-1}\"] \\\\\n\t\t\\dots & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_{\\bot}^{-1} \\right) & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_{\\top}^{-1} \\right) & \n\t\\end{tikzcd}\n\t\\right] \\]\n\tand \n\t\\[K'' = \\left[ \n\t\\begin{tikzcd}\t\n\t\t\\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_-^0 \\right) \\ar[rd, \"_{W_{\\mcG}^{-1}} t_{- \\bot}^0\" below=2] & \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_I^0 \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{I \\bot}^0\" left=1] \\ar[rd, \"_{W_{\\mcG}^{-1}} t_{I \\top}^0\"] & \\dots \\\\\n\t\t& \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\left( X , X_{\\bot}^0 \\right) & \\dots\n\t\\end{tikzcd}\n\t\\right] . \\]\n\tObserve that the maps defining the chain complexes structures in $K'$ and $K''$ are all quasi-isomorphisms (see \\cite[Lemma 3.12]{GPS20}). Thus it is not difficult to show using an increasing exhaustive and bounded from below filtration of $K'$ and $K''$ that these complexes are acyclic (compare the proof of Lemma \\ref{lemma third result for specific module}). This implies that the map $_{W_{\\mcG}^{-1}} t_{\\mcG} : \\, _{W_{\\mcG}^{-1}} \\mcG \\left( X , X_{\\bullet}^0 \\right) \\to \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X \\right)$ is a quasi-isomorphism. \n\t\n\\end{proof}\n\n\\subsection{Proof of the first result}\\label{subsection proof of the first result}\n\nLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$ equipped with a compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\nAssume that $\\tau$ is strict and acts bijectively on hom-sets.\n\nObserve that there is a strict $A_{\\infty}$-functor $\\sigma : \\mcA \\to \\mcA_{\\tau}$ which sends $X^n (E)$ to $E$, and which sends $x \\in \\mcA \\left( X^i (E_1) , X^j (E_2) \\right)$ to $[x] \\in \\mcA_{\\tau} \\left( E_1, E_2 \\right)$. \nBesides, let $\\pi : \\mcC \\to \\mcA$ be the $A_{\\infty}$-functor induced by the following commutative diagram \n\\[\\begin{tikzcd}\n\\mcA \\ar[r, \"\\mathrm{id}\"] \\ar[d, \"\\mathrm{id}\"] & \\mcA \\ar[d, \"\\mathrm{id}\"] \\\\\n\\mcA \\ar[r, \"\\mathrm{id}\"] & \\mcA\n\\end{tikzcd} \\]\n(see Proposition \\ref{prop induced functor from Grothendieck construction}).\nThen the diagram of Adams-graded $A_{\\infty}$-categories\n\\[\\begin{tikzcd}\n\\mcA_- \\sqcup \\mcA_+ \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\" left] & \\mcA_{\\bullet} \\ar[d, \"\\sigma\"] \\\\\n\\mcC \\ar[r, \"\\sigma \\circ \\pi\"] & \\mcA_{\\tau}\n\\end{tikzcd} \\]\nis commutative because $\\sigma \\circ \\tau = \\sigma$. Moreover, the induced $A_{\\infty}$-functor $\\Phi : \\mcG \\to \\mcA_{\\tau}$ is strict, and it sends $W_{\\mcG}$ to the set of units in $\\mcA_{\\tau}$.\nLet \n\\[ \\widetilde{\\Phi} : \\mcH \\to \\mcA_{\\tau} \\left[ \\{ \\mathrm{units} \\}^{-1} \\right] \\]\nbe the $A_{\\infty}$-functor induced by $\\Phi$.\n\nAccording to Lemma \\ref{lemma relation G - mapping torus}, $\\mcH$ is quasi-equivalent to the mapping torus of $\\tau$. Moreover, $\\mcA_{\\tau} \\left[ \\{ \\mathrm{units} \\}^{-1} \\right]$ is quasi-equivalent to $\\mcA_{\\tau}$. Thus, Theorem \\ref{thm mapping torus in strict situation} will follow if we prove that $\\widetilde{\\Phi}$ is a quasi-equivalence. \nOur strategy is to apply Proposition \\ref{prop quasi-isomorphism between localizations}. \nObserve that it suffices to prove that \n\\[ \\widetilde{\\Phi} : \\mcH (X, Y) \\to \\mcA_{\\tau} \\left[ \\{ \\mathrm{units} \\}^{-1} \\right] (\\Phi X, \\Phi Y) \\]\nis a quasi-isomorphism for every object $X$, $Y$ in $\\mcA_{\\bullet}^0$ (recall that $\\mcA_{\\bullet}^0$ denotes the subcategory of $\\mcA_{\\bullet}$ generated by the objects $X_{\\bullet}^0 (E)$, $E \\in \\mcE$) because every object of $\\mcG$ can be related to one of $\\mcA_{\\bullet}^0$ by a zigzag of morphisms in $W_{\\mcG}$, which are quasi-isomorphisms in $\\mcH$ (see \\cite[Lemma 3.12]{GPS20}). \n\nIn the following, we fix some element $E_{\\diamond} \\in \\mcE$. When we write an object $X_{\\triangle}^n$ without specifying the element of $\\mcE$, we mean $X_{\\triangle}^n (E_{\\diamond})$.\nWe consider the corresponding $\\mcG$-module $\\mcM_{\\mcG}$ and the $\\mcG$-module map $t_{\\mcG} : \\mcG \\left( -, X_{\\bullet}^0 \\right) \\to \\mcG$ of Definition \\ref{definition module for G}. \nMoreover, we set \n\\[\\mcM_{\\mcA_{\\tau}} := \\mcA_{\\tau} \\left( -, E_{\\diamond} \\right) \\text{ and } t_{\\mcA_{\\tau}} := \\mathrm{id} : \\mcA_{\\tau} \\left( -, E_{\\diamond} \\right) \\to \\mcM_{\\mcA_{\\tau}}. \\]\n\n\\begin{lemma}\\label{lemma G-module map for relation G - coinvariants}\n\t\n\tThere exists a $\\mcG$-module map $t_0 : \\mcM_{\\mcG} \\to \\Phi^* \\mcM_{\\mcA_{\\tau}}$ (see Definition \\ref{definition pullback functor} for the pullback functor) such that\n\t\\begin{enumerate}\n\t\t\n\t\t\\item the following diagram of $\\mcG$-modules commutes\n\t\t\\[\\begin{tikzcd}\n\t\t\t\\mcG \\left( - , X_{\\bullet}^0 \\right) \\ar[d, \"t_{\\mcG}\"] \\ar[r, \"t_{\\Phi}\"] & \\Phi^* \\mcA_{\\tau} \\left( - , E_{\\diamond} \\right) \\ar[d, \"\\Phi^* t_{\\mcA_{\\tau}} = \\mathrm{id}\"] \\\\\n\t\t\t\\mcM_{\\mcG} \\ar[r, \"t_0\"] & \\Phi^* \\mcM_{\\mcA_{\\tau}}\n\t\t\\end{tikzcd} \\]\n\t\t(see Definition \\ref{definition modules morphism induced by functor} for the map $t_{\\Phi}$),\n\t\t\n\t\t\\item for every $E \\in \\mcE$, the map $t_0 : \\mcM_{\\mcG} (X^0_{\\bullet} (E)) \\to \\Phi^* \\mcM_{\\mcA_{\\tau}} (X^0_{\\bullet} (E))$ is a quasi-isomorphism.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tObserve that the diagram of $\\mcG$-modules \n\t\\[\\begin{tikzcd}\n\t\\mcG \\left( - , X_I^n \\right) \\ar[r, \"t_{I \\top}^n\"] \\ar[d, \"t_{I \\bot}^n\"] & \\mcG \\left( - , X_{\\top}^n \\right) \\ar[d, \"t_{\\Phi}\"] \\\\\n\t\\mcG \\left( - , X_{\\bot}^n \\right) \\ar[r, \"t_{\\Phi}\"] & \\Phi^* \\mcM_{\\mcA_{\\tau}}\n\t\\end{tikzcd} \\]\n\tis commutative, so that it induces a $\\mcG$-module map $\\mcM_{\\star}^n \\to \\Phi^* \\mcM_{\\mcA_{\\tau}}$ (see Proposition \\ref{prop morphism induced by homotopy}). \n\tNow observe that the following diagram of $\\mcG$-modules commutes\n\t\\[\\begin{tikzcd}[column sep = huge]\n\t\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( \\mcG \\left( - , X_-^n \\right) \\oplus \\mcG \\left( - , X_+^n \\right) \\right) \\ar[r, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bullet}^n \\oplus t_{+ \\bullet}^n \\right)\"] \\ar[d, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bot}^n \\oplus t_{+ \\top}^n \\right)\"] & \\bigoplus \\limits_{n \\in \\mathbf{Z}} \\mcG \\left( - , X_{\\bullet}^n \\right) \\ar[d, \"t_{\\Phi}\"] \\\\\n\t\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\mcM_{\\star}^n \\ar[r] & \\Phi^* \\mcM_{\\mcA_{\\tau}}\n\t\\end{tikzcd} . \\]\n\tWe let $t_0 : \\mcM_{\\mcG} \\to \\Phi^* \\mcM_{\\mcA_{\\tau}}$ be the induced $\\mcG$-module map. It is then easy to verify that the following diagram of $\\mcG$-modules is commutative\n\t\\[\\begin{tikzcd}\n\t\\mcG \\left( - , X_{\\bullet}^0 \\right) \\ar[d, \"t_{\\mcG}\"] \\ar[r, \"t_{\\Phi}\"] & \\Phi^* \\mcA_{\\tau} \\left( - , E_{\\diamond} \\right) \\ar[d, \"\\Phi^* t_{\\mcA_{\\tau}} = \\mathrm{id}\"] \\\\\n\t\\mcM_{\\mcG} \\ar[r, \"t_0\"] & \\Phi^* \\mcM_{\\mcA_{\\tau}} .\n\t\\end{tikzcd} \\]\n\t\n\tWe now prove the second part of the lemma.\n\tWe have \n\t\\[\\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right) = \\bigoplus\\limits_n \\mcG \\left( X_{\\bullet}^0 (E), X_{\\bullet}^n \\right) = \\bigoplus\\limits_n \\mcA \\left( X^k (E) , X^n \\right) \\]\t\n\tand \n\t\\[t_0 : \\bigoplus\\limits_n \\mcA \\left( X^k (E) , X^n \\right) = \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right) \\to \\Phi^* \\mcM_{\\mcA_{\\tau}} \\left( X_{\\bullet}^0 (E) \\right) = \\mcA_{\\tau} \\left( E, E_{\\diamond} \\right) \\]\n\tis the sum of the projections, which is an isomorphism. This concludes the proof\n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma relation G - coinvariants}\n\t\n\tFor every $E \\in \\mcE$, the chain map \n\t\\[ \\widetilde{\\Phi} : \\mcH \\left( X_{\\bullet}^0 (E), X_{\\bullet}^0 \\right) \\to \\mcA_{\\tau} \\left[ \\{ \\mathrm{units} \\}^{-1} \\right] \\left( E, E_{\\diamond} \\right) \\]\n\tis a quasi-isomorphism.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tAccording to Lemmas \\ref{lemma properties of modules}, \\ref{lemma properties of modules bis} and \\ref{lemma G-module map for relation G - coinvariants}, the assumptions of Proposition \\ref{prop quasi-isomorphism between localizations} are satisfied. This concludes the proof. \n\t\n\\end{proof}\n\nAs explained above, Theorem \\ref{thm mapping torus in strict situation} follows from Lemma \\ref{lemma relation G - coinvariants} since $\\mcH$ is quasi-equivalent to the mapping torus of $\\tau$ (see Lemma \\ref{lemma relation G - mapping torus}) and $\\mcA_{\\tau} \\left[ \\{ \\mathrm{units} \\}^{-1} \\right]$ is quasi-equivalent to $\\mcA_{\\tau}$.\n\n\\subsection{Proof of the second result}\\label{subsection proof of the second result}\n\nLet $\\tau$ be a quasi-autoequivalence of an $A_{\\infty}$-category $\\mcA$ equipped with a compatible $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$.\nAssume that the following holds: \n\\begin{enumerate}\n\t\n\t\\item $\\mcA$ is weakly directed with respect to the $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ (see Definition \\ref{definition group-action}),\n\t\n\t\\item there exists a closed degree $0$ bimodule map $f : \\mcA_m \\left( -, - \\right) \\to \\mcA_m \\left( -, \\tau(-) \\right)$ (see Definitions \\ref{definition bimodule} and \\ref{definition pullback bimodule}) such that $f : \\mcA_m \\left( X^i(E) , X^j(E') \\right) \\to \\mcA_m \\left( X^i(E), X^{j+1} (E') \\right)$ is a quasi-somorphism for every $i < j$ and $E, E' \\in \\mcE$.\n\t\n\\end{enumerate}\n\n\\begin{rmk}\\label{rmk multiplication by f(unit) is quasi-iso}\n\t\n\tIt follows from Lemma \\ref{coro closed module map homotopic to Yoneda module map} and $\\tau$ being a quasi-equivalence that the chain maps\n\t\\[\\left\\{\n\t\\begin{array}{ccccl}\n\t\t\\mu_{\\mcA_m}^2 \\left( - , f(e_{X^j(E)}) \\right) & : & \\mcA_m \\left( X^i(E') , X^j(E) \\right) & \\to & \\mcA_m \\left( X^i(E') , X^{j+1} (E) \\right) \\\\\n\t\t\\mu_{\\mcA_m}^2 \\left( f(e_{X^j (E)}) , - \\right) & : & \\mcA_m \\left( X^{j+1} (E) , X^{k+1} (E') \\right) & \\to & \\mcA_m \\left( X^j (E) , X^{k+1} (E') \\right)\n\t\\end{array}\n\t\\right. \\]\n\tare quasi-isomorphisms for every $i < j < k$ and $E, E' \\in \\mcE$. \n\t\n\\end{rmk}\n\nIn the following, we set \n\\[c_n (E) := f \\left( e_{X^n (E)} \\right) \\]\nfor every $n \\in \\mathbf{Z}$, $E \\in \\mcE$, and\n\\[W_{\\mcA_m} := \\left\\{ c_n (E) \\mid n \\in \\mathbf{Z}, E \\in \\mcE \\right\\} \\cup \\left\\{ \\text{units of } \\mcA_m \\right\\} . \\]\n\n\\subsubsection{Generalized homotopy}\\label{subsection generalized homotopy}\n\nRecall that we introduced a functor $\\mcB \\mapsto \\mcB_m$ from the category of Adams-graded $A_{\\infty}$-categories to the category of (non Adams-graded) $A_{\\infty}$-categories. \nAlso, recall that we introduced Adams-graded $A_{\\infty}$-categories $\\mcC$ and $\\mcG$ in Definitions \\ref{definition cylinder object} and \\ref{definition category G} respectively.\nObserve that $\\mcC_m$ and $\\mcG_m$ are the Grothendieck constructions of the diagrams\n\\[\\begin{tikzcd}\n(\\mcA_I)_m \\ar[r, \"\\mathrm{id}\"] \\ar[d, \"\\mathrm{id}\" left] & (\\mcA_{\\top})_m \\\\\n(\\mcA_{\\bot})_m &\n\\end{tikzcd} \\text{ and }\n\\begin{tikzcd}\n(\\mcA_-)_m \\sqcup (\\mcA_+)_m \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\" left] & (\\mcA_{\\bullet})_m \\\\\n\\mcC_m\n\\end{tikzcd}\n\\]\nrespectively. \nWe denote by $W_{\\mcC_m}$ the set of adjacent units in $\\mcC_m$, and by $W_{\\mcG_m}$ the union of $W_{\\mcC_m}$ and the set of adjacent units in $\\mcG_m$. \n\nWe would like to define an $A_{\\infty}$-functor $\\Psi_m : \\mcG_m \\to \\mcA_m$ which sends $W_{\\mcG_m}$ to $W_{\\mcA_m}$. According to Proposition \\ref{prop induced functor from Grothendieck construction}, it is enough to prove the following result.\n\n\\begin{lemma}\\label{lemma relation tau continuation}\n\t\n\tThere exists an $A_{\\infty}$-functor $\\eta : \\mcC_m \\to \\mcA_m$ which sends $W_{\\mcC_m}$ to $W_{\\mcA_m}$, and such that\n\t\\[\\eta \\circ \\iota_I = \\eta \\circ \\iota_{\\bot} = \\mathrm{id}, \\quad \\eta \\circ \\iota_{\\top} = \\tau . \\]\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tWe first define $\\eta$ to be $\\mathrm{id}$ on $(\\mcA_{\\bot})_m$, $(\\mcA_I)_m$. and to be $\\tau$ on $(\\mcA_{\\top})_m$. Observe that this completely defines $\\eta$ on the objects.\n\tBesides, we ask for $\\eta$ to act as the identity on the sequences involving an adjacent morphism from $(\\mcA_I)_m$ to $(\\mcA_{\\bot})_m$.\n\t\n\tIt remains to define $\\eta$ on the sequences involving an adjacent morphism from $(\\mcA_I)_m$ to $(\\mcA_{\\top})_m$.\n\tConsider a sequence of morphisms\n\t\\begin{align*}\n\t\\left( x_0, \\dots, x_{p+q} \\right) & \\in \\mcC_m \\left( X_I^{i_0} (E_0), X_I^{i_1} (E_1) \\right) \\times \\dots \\times \\mcC_m \\left( X_I^{i_{p-1}} (E_{p-1}), X_I^{i_p} (E_p) \\right) \\\\ \n\t& \\times \\mcC_m \\left( X_I^{i_p} (E_p), X_{\\top}^{i_{p+1}} (E_{p+1}) \\right) \\\\\n\t& \\times \\mcC_m \\left( X_{\\top}^{i_{p+1}} (E_{p+1}), X_{\\top}^{i_{p+2}} (E_{p+2}) \\right) \\times \\dots \\times \\mcC_m \\left( X_{\\top}^{i_{p+q}} (E_{p+q}), X_{\\top}^{i_{p+q+1}} (E_{p+q+1}) \\right) .\n\t\\end{align*} \n\tObserve that \n\t\\[\\mcC_m \\left( X_I^i (E), X_I^j (E') \\right) = \\mcC_m \\left( X_I^i (E), X_ {\\top}^j (E') \\right) = \\mcC_m \\left( X_{\\top}^i (E), X_{\\top}^j (E') \\right) = \\mcA_m \\left( X^i (E), X^j (E) \\right) . \\]\n\tThen we set \n\t\\[\\eta \\left( x_0, \\dots, x_{p+q} \\right) := f \\left( x_0, \\dots, x_{p-1}, x_p, x_{p+1}, \\dots, x_{p+q+1} \\right) \\in \\mcA_m \\left( X^{i_0} (E_0), \\tau X^{i_{p+q+1}} (E_{p+q+1}) \\right) . \\]\n\tThe functor $\\eta$ we defined satisfies the $A_{\\infty}$-relations because $f : \\mcA_m \\left( -, - \\right) \\to \\mcA_m \\left( -, \\tau (-) \\right)$ is a closed degree $0$ bimodule map. Moreover, $\\eta$ sends $W_{\\mcC_m}$ to $W_{\\mcA_m}$ by construction. \n\t\n\\end{proof}\n\n\\begin{rmk}\n\t\n\tFirst observe that\n\t\\[\\mcC yl_{\\mcA_m} = \\mcC_m \\left[ W_{\\mcC_m}^{-1} \\right] = \\left( \\mcC \\left[ W_{\\mcC}^{-1} \\right] \\right)_m = \\left( \\mcC yl_{\\mcA} \\right)_m . \\]\n\tAccording to Lemma \\ref{lemma relation tau continuation}, the functor $\\eta$ induces an $A_{\\infty}$-functor $\\widetilde{\\eta} : \\mcC yl_{\\mcA_m} \\to \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right]$. Moreover, Lemma \\ref{lemma relation tau continuation} implies that the following diagram commutes\n\t\\[\\begin{tikzcd}\n\t(\\mcA_+)_m \\ar[d, \"\\lambda_{\\mcC_m} \\circ \\iota_{\\top}\" left] \\ar[rd, bend left, \"\\lambda_{\\mcA_m} \\circ \\tau\"] & \\\\\n\t\\mcC yl_{\\mcA_m} \\ar[r, \"\\widetilde{\\eta}\"] & \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\\\\n\t(\\mcA_-)_m \\ar[u, \"\\lambda_{\\mcC_m} \\circ \\iota_{\\bot}\"] \\ar[ru, bend right, \"\\lambda_{\\mcA_m}\"] & \n\t\\end{tikzcd}\n\t\\] \n\t($\\lambda_{\\mcA_m} : \\mcA_m \\to \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right]$ and $\\lambda_{\\mcC_m} : \\mcC_m \\to \\mcC_m \\left[ W_{\\mcC_m}^{-1} \\right]$ denote the localization functors). Since $\\mcC yl_{\\mcA_m}$ should be thought as a cylinder object for $\\mcA_m$ (see Proposition \\ref{prop cylinder object}), we should think that the functors $\\lambda_{\\mcA_m}$ and $\\lambda_{\\mcA_m} \\circ \\tau$ are homotopic (even if they do not act the same way on objects) and that $\\widetilde{\\eta}$ is a generalized homotopy between them (see Proposition \\ref{prop homotopy is generalized homotopy} for a justification of this terminology).\n\t\n\\end{rmk}\n\n\\subsubsection{Relation between $\\mcG$ and $\\mcA_m$}\n\nUsing the $A_{\\infty}$-functor $\\eta : \\mcC_m \\to \\mcA_m$ of Lemma \\ref{lemma relation tau continuation}, we get a commutative diagram of (non Adams-graded) $A_{\\infty}$-categories \n\\[\\begin{tikzcd}\n(\\mcA_-)_m \\sqcup (\\mcA_+)_m \\ar[r, \"\\mathrm{id} \\sqcup \\tau\"] \\ar[d, \"\\iota_{\\bot} \\sqcup \\iota_{\\top}\" left] & (\\mcA_{\\bullet})_m \\ar[d, \"\\mathrm{id}\"] \\\\\n\\mcC_m \\ar[r, \"\\eta\"] & \\mcA_m\n\\end{tikzcd} \\]\nand the induced $A_{\\infty}$-functor $\\Psi_m : \\mcG_m \\to \\mcA_m$ (see Proposition \\ref{prop induced functor from Grothendieck construction}) sends $W_{\\mcG_m}$ to $W_{\\mcA_m}$ (see Lemma \\ref{lemma relation tau continuation}). Let \n\\[ \\widetilde{\\Psi_m} : \\mcH_m = \\mcG_m \\left[ W_{\\mcG_m}^{-1} \\right] \\to \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\]\nbe the $A_{\\infty}$-functor induced by $\\Psi_m$. \nObserve that, since $\\mcA$ is assumed to be weakly directed and since the Adams-degree of $\\mcA$ comes from the $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$ (see Definition \\ref{definition group-action}), $\\mcH$ is concentrated in non-negative Adams-degree.\nIn particular, we can apply the adjunction of Definition \\ref{definition adjunction} to $\\widetilde{\\Psi_m}$, which gives an $A_{\\infty}$-functor\n\\[ \\widetilde{\\Psi} : \\mcH = \\mcG \\left[ W_{\\mcG}^{-1} \\right] \\to \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] . \\]\nWe would like to prove that for every Adams degree $j \\geq 1$, and for every objects $X,Y$ in $\\mcA_{\\bullet}^0$ (recall that $\\mcA_{\\bullet}^0$ denotes the subcategory of $\\mcA_{\\bullet}$ generated by the objects $X_{\\bullet}^0 (E)$, $E \\in \\mcE$), the map\n\\[\\widetilde{\\Psi} : \\mcH \\left( X,Y \\right)^{*,j} \\to \\left( \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\right) \\left( \\Psi X , \\Psi Y \\right)^{*,j} = \\mathbf{F}_m^j \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\left( \\Psi X , \\Psi Y \\right) \\]\nis a quasi-isomorphism, ($\\mathbf{F}_m^j$ is the vector space generated by $t_m^j$ ; also recall that if $V$ is an Adams-graded vector space, $V^{*,j}$ denotes the subspace of Adams degree $j$ elements). Our strategy is once again to apply Proposition \\ref{prop quasi-isomorphism between localizations}.\n\nIn the following we fix some element $E_{\\diamond} \\in \\mcE$. When we write $X_{\\triangle}^n$ or $c_n$ without specifying the element of $\\mcE$, we mean $X_{\\triangle}^n (E_{\\diamond})$ or $c_n (E_{\\diamond})$ respectively.\nWe consider the corresponding $\\mcG$-module $\\mcM_{\\mcG}$, and the $\\mcG$-module map $t_{\\mcG} : \\mcG \\left( -, X_{\\triangle}^n \\right) \\to \\mcG$ of Definition \\ref{definition module for G}. \nMoreover, we consider the $\\mcA_m$-module $\\mcM_{\\mcA_m}$ and the $\\mcA_m$-module maps\n\\[t_{\\mcA_m}^n : \\mcA_m \\left( - , X^n \\right) \\to \\mcM_{\\mcA_m}, \\quad n \\in \\mathbf{Z}, \\]\nassociated to the morphisms $\\left( c_n \\right)_{n \\in \\mathbf{Z}}$ as in Definition \\ref{definition specific module}.\n\nThe following result is a first step in order to define a $\\mcG_m$-module map $(t_0)_m : (\\mcM_{\\mcG})_m \\to \\Psi_m^* \\mcM_{\\mcA_m}$ as in Proposition \\ref{prop quasi-isomorphism between localizations}.\n\n\\begin{lemma}\\label{lemma first step to define the module map}\n\t\n\tFor every $n \\in \\mathbf{Z}$, the diagram of $\\mcG_m$-modules \n\t\\[\\begin{tikzcd}[column sep = huge]\n\t\\mcG_m \\left( - , X_I^n \\right) \\ar[r, \"t_{I \\top}^n\"] \\ar[d, \"t_{I \\bot}^n\"] & \\mcG_m \\left( - , X_{\\top}^n \\right) \\ar[d, \"\\Psi_m^* t_{\\mcA_m}^{n+1} \\circ t_{\\Psi_m}\"] \\\\\n\t\\mcG_m \\left( - , X_{\\bot}^n \\right) \\ar[r, \"\\Psi_m^* t_{\\mcA_m}^n \\circ t_{\\Psi_m}\"] & \\Psi_m^* \\mcM_{\\mcA_m}\n\t\\end{tikzcd} \\]\n\tcommutes up to homotopy.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tFirst observe that $\\iota_I^* \\mcG_m \\left( - , X_I^n \\right) = \\mcA_m \\left( -, X^n \\right)$, and $\\iota_I^* \\Psi_m^* \\mcM_{\\mcA_m} = \\mcM_{\\mcA_m}$\tbecause $\\Psi_m \\circ \\iota_I = \\mathrm{id}$ (see Remark \\ref{rmk composition of pullback functors}).\n\tMoreover, it suffices to show that the $\\mcA_m$-module maps \n\t\\[\\iota_I^* \\left( \\Psi_m^* t_{\\mcA_m}^n \\circ t_{\\Psi_m} \\circ t_{I \\bot}^n \\right) = t_{\\mcA_m}^n \\circ \\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\bot}^n \\right) : \\mcA_m \\left( -, X^n \\right) \\to \\mcM_{\\mcA_m} \\]\n\tand \n\t\\[\\iota_I^* \\left( \\Psi_m^* t_{\\mcA_m}^{n+1} \\circ t_{\\Psi_m} \\circ t_{I \\top}^n \\right) = t_{\\mcA_m}^{n+1} \\circ \\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\top}^n \\right) : \\mcA_m \\left( -, X^n \\right) \\to \\mcM_{\\mcA_m} \\]\n\tare homotopic because \n\t\\[\\mcG_m \\left( X_{\\triangle}^k, X_I^n \\right) = 0 \\text{ if } \\triangle \\ne I . \\]\n\tOn the one hand, \n\t\\[t_{\\mcA_m}^n \\circ \\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\bot}^n \\right) = t_{\\mcA_m}^n .\\]\n\tOn the other hand, $\\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\top}^n \\right) : \\mcA_m \\left( -, X^n \\right) \\to \\mcA_m \\left( -, X^{n+1} \\right)$ is closed (as composition and pullback of closed module maps), and \n\t\\[\\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\top}^n \\right) \\left( e_{X^n} \\right) = \\eta \\left( e_{X^n} \\right) = c_n \\]\n\taccording to Lemma \\ref{lemma relation tau continuation}. \n\tTherefore, $\\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\top}^n \\right)$ is homotopic to $t_{c_n}$ according to Corollary \\ref{coro closed module map homotopic to Yoneda module map}, and thus $t_{\\mcA_m}^{n+1} \\circ \\iota_I^* \\left( t_{\\Psi_m} \\circ t_{I \\top}^n \\right)$ is homotopic to $t_{\\mcA_m}^{n+1} \\circ t_{c_n}$. Now according to Lemma \\ref{lemma multiplication by continuation element becomes homotopic to inclusion}, $t_{\\mcA_m}^{n+1} \\circ t_{c_n}$ is homotopic to $t_{\\mcA_m}^n$. This concludes the proof.\n\t\n\\end{proof}\n\nWe can now state the result establishing the existence of a $\\mcG_m$-module map $(t_0)_m : (\\mcM_{\\mcG})_m \\to \\Psi_m^* \\mcM_{\\mcA_m}$ as in Proposition \\ref{prop quasi-isomorphism between localizations}.\n\n\\begin{lemma}\\label{lemma G-module map for relation G-A}\n\t\n\tThere exists a $\\mcG_m$-module map $(t_0)_m : (\\mcM_{\\mcG})_m \\to \\Psi_m^* \\mcM_{\\mcA_m}$ such that the following holds. \n\t\\begin{enumerate}\n\t\t\n\t\t\\item The following diagram of $\\mcG_m$-modules commutes\n\t\t\\[\\begin{tikzcd}\n\t\t\t\\mcG_m \\left( - , X_{\\bullet}^0 \\right) \\ar[d, \"t_{\\mcG_m}\"] \\ar[r, \"t_{\\Psi_m}\"] & \\Psi_m^* \\mcA_m \\left( - , X^0 \\right) \\ar[d, \"\\Psi_m^* t_{\\mcA_m}^0\"] \\\\\n\t\t\t(\\mcM_{\\mcG})_m \\ar[r, \"(t_0)_m\"] & \\Psi_m^* \\mcM_{\\mcA_m}\n\t\t\\end{tikzcd} \\]\n\t\n\t\t\\item for every $E \\in \\mcE$ and $j \\geq 1$, the induced map $t_0 : \\mcM_{\\mcG} (X_{\\bullet}^0 (E)) \\to \\mathbf{F} \\left[ t_m \\right] \\otimes \\Psi_m^* \\mcM_{\\mcA_m} (X_{\\bullet}^0 (E))$ (see Definition \\ref{definition adjunction}) is a quasi-isomorphism in each positive Adams-degree.\n\t\t\n\t\\end{enumerate}\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tUsing Lemma \\ref{lemma first step to define the module map} and Proposition \\ref{prop morphism induced by homotopy}, we get a $\\mcG_m$-module map $t_{\\star}^n : \\left( \\mcM_{\\star}^n \\right)_m \\to \\Psi_m^* \\mcM_{\\mcA_m}$ for every $n \\in \\mathbf{Z}$ (see Definition \\ref{definition module for G} for the $\\mcG$-modules $\\mcM_{\\star}^n$). \n\tObserve that the diagram of $\\mcG_m$-modules \n\t\\[\\begin{tikzcd}[column sep = huge]\n\t\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( \\mcG_m \\left( - , X_-^n \\right) \\oplus \\mcG_m \\left( - , X_+^n \\right) \\right) \\ar[r, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bullet}^n \\oplus t_{+ \\bullet}^n \\right)\"] \\ar[d, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( t_{- \\bot}^n \\oplus t_{+ \\top}^n \\right)\"] & \\bigoplus \\limits_{n \\in \\mathbf{Z}} \\mcG_m \\left( - , X_{\\bullet}^n \\right) \\ar[d, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\Psi_m^* t_{\\mcA_m}^n \\circ t_{\\Psi_m}\"] \\\\\n\t\\bigoplus \\limits_{n \\in \\mathbf{Z}} \\left( \\mcM_{\\star}^n \\right)_m \\ar[r, \"\\bigoplus \\limits_{n \\in \\mathbf{Z}} t_{\\star}^n\"] & \\Psi_m^* \\mcM_{\\mcA_m}\n\t\\end{tikzcd} \\]\n\tis commutative (the composition is $\\mathrm{id}$ for $-$-terms and $\\tau$ for $+$-terms), so that it induces a $\\mcG_m$-module map $(t_0)_m : (\\mcM_{\\mcG})_m \\to \\Psi_m^* \\mcM_{\\mcA_m}$. It is then easy to verify that the following diagram of $\\mcG_m$-modules is commutative\n\t\\[\\begin{tikzcd}\n\t\\mcG_m \\left( - , X_{\\bullet}^0 \\right) \\ar[d, \"t_{\\mcG_m}\"] \\ar[r, \"t_{\\Psi_m}\"] & \\Psi_m^* \\mcA_m \\left( - , X^0 \\right) \\ar[d, \"\\Psi_m^* t_{\\mcA_m}^0\"] \\\\\n\t(\\mcM_{\\mcG})_m \\ar[r, \"(t_0)_m\"] & \\Psi_m^* \\mcM_{\\mcA_m} .\n\t\\end{tikzcd} \\]\n\t\n\tIt remains to show that the map $t_0 : \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)^{*,j} \\to \\mathbf{F}_m^j \\otimes \\mcM_{\\mcA_m} \\left( X^0 (E) \\right)$ is a quasi-isomorphism for every $E \\in \\mcE$ and $j \\geq 1$.\n\tNote that \n\t\\[\\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)^{*,j} = \\mcG \\left( X_{\\bullet}^0 (E), X_{\\bullet}^j \\right) = \\mcA \\left( X^0 , X^j \\right) \\\\ \\]\n\tand the map\n\t\\[\\mcA \\left( X^0 (E) , X^j \\right) = \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)^{*,j} \\xrightarrow{t_0} \\mathbf{F}_m^j \\otimes \\mcM_{\\mcA_m} \\left( X^0 (E) \\right) \\]\n\tis the inclusion. \n\tNow observe that the following diagram of chain complexes commutes \n\t\\[\\begin{tikzcd}\n\t\\mcA \\left( X^0 (E), X^j \\right) \\ar[r, \"t_0\"] & \\mathbf{F}_m^j \\otimes \\mcM_{\\mcA_m} \\left( X^0 (E) \\right) \\\\\n\t\\mathbf{F}_m^j \\otimes \\mcA_m \\left( X^0 (E), X^j \\right) \\ar[u, equal] \\ar[r, equal] & \\mathbf{F}_m^j \\otimes \\mcA_m \\left( X^0 (E), X^j \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tThe inclusion $\\mcA_m \\left( X^0 (E) , X^j \\right) \\hookrightarrow \\mcM_{\\mcA_m} \\left( X^0 (E) \\right)$ is a quasi-isomorphism according to Lemma \\ref{lemma first result for specific module} (observe that it is important here that $j$ is \\emph{strictly} greater than $0$). Therefore the map $t_0 : \\mcA \\left( X^0 (E), X^j \\right) \\to \\mathbf{F}_m^j \\otimes \\mcM_{\\mcA_m} \\left( X^0 (E) \\right)$ is a quasi-isomorphism, which is what we needed to prove. \n\t\n\\end{proof}\n\n\\begin{lemma}\\label{lemma relation G-A}\n\t\n\tFor every $E \\in \\mcE$ and $j \\geq 1$, the map\n\t\\[\\widetilde{\\Psi} : \\mcH \\left( X_{\\bullet}^0 (E), X_{\\bullet}^0 \\right)^{*,j} \\to \\left( \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\right) \\left( X^0 (E) , X^0 \\right)^{*,j} = \\mathbf{F}_m^j \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\left( X^0 (E) , X^0 \\right) \\]\n\tis a quasi-isomorphism.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tUsing the first part of Lemma \\ref{lemma G-module map for relation G-A} and Proposition \\ref{prop quasi-isomorphism between localizations}, we know that there exists a chain map $u_m : \\, _{W_{\\mcG_m}^{-1}} (\\mcM_{\\mcG})_m \\left( X_{\\bullet}^0 (E) \\right) \\to \\, _{W_{\\mcA_m}^{-1}} \\mcM_{\\mcA_m} \\left( X^0 \\right)$ such that the following diagram of chain complexes commutes\n\t\\[\\begin{tikzcd}\n\t\\mcH_m \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right) \\ar[d, \"_{W_{\\mcG_m}^{-1}} t_{\\mcG_m}\"] \\ar[r, \"\\widetilde{\\Psi}_m\"] & \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\left( X^0 (E) , X^0 \\right) \\ar[d, \"_{W_{\\mcA_m}^{-1}} t_{\\mcA_m}^0\"] \\\\\n\t_{W_{\\mcG_m}^{-1}} (\\mcM_{\\mcG})_m \\left( X_{\\bullet}^0 (E) \\right) \\ar[r, \"u_m\"] & _{W_{\\mcA_m}^{-1}} \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\\\\n\t(\\mcM_{\\mcG})_m \\left( X_{\\bullet}^0 (E) \\right) \\ar[u, hook] \\ar[r, \"(t_0)_m\"] & \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tObserve that \n\t\\[\\left\\{\n\t\\begin{array}{lll}\n\t\\mcH_m \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right) & = & \\mcH \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right)_m \\\\\n\t_{W_{\\mcG_m}^{-1}} (\\mcM_{\\mcG})_m \\left( X_{\\bullet}^0 (E) \\right) & = & _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)_m \\\\\n\t(\\mcM_{\\mcG})_m \\left( X_{\\bullet}^0 (E) \\right) & = & \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)_m .\n\t\\end{array}\n\t\\right. \\]\n\tApplying the adjunction of Definition \\ref{definition adjunction} to the last diagram, we get the following commutative diagram of Adams-graded chain complexes\n\t\\[\\begin{tikzcd}\n\t\\mcH \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right) \\ar[d, \"_{W_{\\mcG}^{-1}} t_{\\mcG}\"] \\ar[r, \"\\widetilde{\\Psi}\"] & \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\left( X^0 (E) , X^0 \\right) \\ar[d, \"\\mathrm{id} \\otimes _{W_{\\mcA_m}^{-1}} t_{\\mcA_m}^0\"] \\\\\n\t_{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right) \\ar[r, \"u\"] & \\mathbf{F} \\left[ t_m \\right] \\otimes _{W_{\\mcA_m}^{-1}} \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\\\\n\t\\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right) \\ar[u, hook] \\ar[r, \"t_0\"] & \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tSpecializing to the components of fixed Adams degree $j \\geq 1$, we get the following commutative diagram of chain complexes \n\t\\[\\begin{tikzcd}\n\t\\mcH \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right)^{*,j} \\ar[d, \"_{W_{\\mcG}^{-1}} t_{\\mcG}\"] \\ar[r, \"\\widetilde{\\Psi}\"] & \\mathbf{F}_m^j \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\left( X^0 (E) , X^0 \\right) \\ar[d, \"\\mathrm{id} \\otimes _{W_{\\mcA_m}^{-1}} t_{\\mcA_m}^0\"] \\\\\n\t_{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)^{*,j} \\ar[r, \"u\"] & \\mathbf{F}_m^j \\otimes _{W_{\\mcA_m}^{-1}} \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\\\\n\t\\mcM_{\\mcG} \\left( X_{\\bullet}^0 (E) \\right)^{*,j} \\ar[u, hook] \\ar[r, \"t_0\"] & \\mathbf{F}_m^j \\otimes \\mcM_{\\mcA_m} \\left( \\Psi_m X \\right) \\ar[u, hook] .\n\t\\end{tikzcd} \\]\n\tUsing Lemmas \\ref{lemma properties of modules}, \\ref{lemma properties of modules bis}, and \\cite[Lemma 3.13]{GPS20}, we know that all the vertical maps on the left are quasi-isomorphisms. \n\tSimilarly, using Lemmas \\ref{lemma second result for specific module}, \\ref{lemma third result for specific module}, and \\cite[Lemma 3.13]{GPS20}, we know that all the vertical maps on the right are quasi-isomorphisms. \n\tMoreover, the second part of Lemma \\ref{lemma G-module map for relation G-A} states that the lowest horizontal map is a quasi-isomorphism. \n\tThus, the chain map $\\widetilde{\\Psi} : \\mcH \\left( X_{\\bullet}^0 (E) , X_{\\bullet}^0 \\right)^{*,j} \\to \\mathbf{F}_m^j \\otimes \\mcA_m [ W_{\\mcA_m}^{-1} ] \\left( X^0 (E) , X^0 \\right)$ is a quasi-isomorphism. \n\t\n\\end{proof}\n\n\\subsubsection{End of the proof}\n\nWe end the section with the proof of Theorem \\ref{thm mapping torus in weak situation}.\nNow that we have proved Lemma \\ref{lemma relation G-A} which takes care of the \\emph{positive} Adams-degrees, we have to treat the zero Adams-degree part (recall that $\\mcH$ is concentrated in non-negative Adams-degree because $\\mcA$ is assumed to be weakly directed).\n \nLet $\\mcI$ be the (non full) $A_{\\infty}$-subcategory of $\\mcH$ with \n\\[\\mathrm{ob} \\left( \\mcI \\right) = \\left\\{ X_{\\bullet}^0 (E) \\mid E \\in \\mcE \\right\\} \\text{ and } \\mcI \\left( X, Y \\right) = \\mcG \\left( X, Y \\right) \\oplus \\left( \\bigoplus \\limits_{j \\geq 1} \\mcH \\left( X, Y \\right)^{*,j} \\right) \\]\n(recall that if $V$ is an Adams-graded vector space, we denote by $V^{*,j}$ its component of Adams-degree $j$). \n\n\\begin{lemma}\n\t\n\tThe inclusion $\\mcI \\hookrightarrow \\mcH$ is a quasi-equivalence.\n\t\n\\end{lemma}\n\n\\begin{proof}\n\t\n\tObserve that the inclusion $\\mcI \\hookrightarrow \\mcH$ is cohomologically essentially surjective because every object of $\\mcH$ can be related to one of $\\mcI$ by a zigzag of morphisms in $W_{\\mcG}$, which are quasi-isomorphisms in $\\mcH$ (see \\cite[Lemma 3.12]{GPS20}). \n\tTherefore, it suffices to show that the inclusion \n\t\\[\\mcG \\left( X_{\\bullet}^0 \\left( E \\right), X_{\\bullet}^0 \\left( E_{\\diamond} \\right) \\right) \\hookrightarrow \\mcH \\left( X_{\\bullet}^0 \\left( E \\right), X_{\\bullet}^0 \\left( E_{\\diamond} \\right) \\right)^{*,0} \\]\n\tis a quasi-isomorphism for every $E, E_{\\diamond} \\in \\mcE$.\n\t\n\tLet $E_{\\diamond}$ be an element of $\\mcE$. When we write an object $X_{\\bullet}^n$ without specifying the element of $\\mcE$, we mean $X_{\\bullet}^n (E_{\\diamond})$.\n\tRecall that we introduced a pair $\\left( \\mcM_{\\mcG}, t_{\\mcG} \\right)$ in Definition \\ref{definition module for G}.\n\tAccording to Lemmas \\ref{lemma properties of modules}, \\ref{lemma properties of modules bis} and \\cite[Lemma 3.13]{GPS20}, the inclusion $\\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right) \\hookrightarrow \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right)$ and the map $_{W_{\\mcG}^{-1}} t_{\\mcG} : \\, \\mcH \\left( X^0_{\\bullet} (E) , X^0_{\\bullet} \\right) \\to \\, _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right)$ are quasi-isomorphisms for every $E \\in \\mcE$. Besides, observe that \n\t\\[\\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right)^{*,0} = \\mcG \\left( X^0_{\\bullet} (E),X^0_{\\bullet} \\right) . \\]\n\tThe result then follows from the commutativity of the following diagram\n\t\\[\\begin{tikzcd}\n\t\\mcG \\left( X_{\\bullet}^0 \\left( E \\right), X_{\\bullet}^0 \\right) \\ar[d, equal] \\ar[r, equal] & \\mcG \\left( X_{\\bullet}^0 \\left( E \\right), X_{\\bullet}^0 \\right) \\ar[d, hook] \\ar[r, equal] & \\mcG \\left( X_{\\bullet}^0 \\left( E \\right), X_{\\bullet}^0 \\right) \\ar[d, hook] \\\\\n\t\\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right)^{*,0} \\ar[r, hook, \"\\sim\"] & _{W_{\\mcG}^{-1}} \\mcM_{\\mcG} \\left( X^0_{\\bullet} (E) \\right)^{*,0} & \\mcH \\left( X^0_{\\bullet} (E) , X^0_{\\bullet} \\right)^{*,0} \\ar[l, \"\\sim\" above, \"_{W_{\\mcG}^{-1}} t_{\\mcG}\" below] .\n\t\\end{tikzcd} \\] \n\t\n\\end{proof}\n\nThe following diagram of Adams-graded $A_{\\infty}$-categories is commutative\n\\[ \\begin{tikzcd}\n\\mcH \\ar[r, \"\\widetilde{\\Psi}\"] & \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\\\\n\\mcI \\ar[u, hook, \"\\sim\" right] \\ar[r, \"\\widetilde{\\Psi}\"] & \\mcA_m^0 \\oplus \\left( t \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right]^0 \\right) \\ar[u, hook]\n\\end{tikzcd} \\]\n(recall that if $\\mcC$ is an $A_{\\infty}$-category equipped with a splitting $\\mathrm{ob} \\left( \\mcC \\right) \\simeq \\mathbf{Z} \\times \\mcE$, then we denote by $\\mcC^0$ the full $A_{\\infty}$-subcategory of $\\mcC$ whose set of objects corresponds to $\\{ 0 \\} \\times \\mcE$).\nMoreover, since $\\mcA$ is assumed to be weakly directed with respect to the $\\mathbf{Z}$-splitting of $\\mathrm{ob} \\left( \\mcA \\right)$, Lemma \\ref{lemma relation G-A} implies that the bottom horizontal $A_{\\infty}$-functor is a quasi-equivalence. \nTherefore we have\n\\[\\mcH \\simeq \\mcA_m^0 \\oplus \\left( t \\mathbf{F} \\left[ t_m \\right] \\otimes \\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right]^0 \\right) . \\]\nRecall that $W_{\\mcA_m} = f \\left( \\mathrm{units} \\right) \\cup \\{ \\mathrm{units} \\}$, so that \n\\[\\mcA_m \\left[ W_{\\mcA_m}^{-1} \\right] \\simeq \\mcA_m \\left[ f \\left( \\mathrm{units} \\right)^{-1} \\right] . \\]\nThis concludes the proof of Theorem \\ref{thm mapping torus in weak situation}, since $\\mcH$ is quasi-equivalent to the mapping torus of $\\tau$ (see Lemma \\ref{lemma relation G - mapping torus}).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}