\documentclass[11 pt]{article} \usepackage{amsmath,amsthm,amsfonts,amssymb,titlesec} \usepackage{hyperref} \usepackage{tikz} \usepackage{verbatim} \usepackage{accents} \usepackage[citestyle=alphabetic,bibstyle=alphabetic,backend=bibtex]{biblatex} \usepackage{todonotes} \usepackage[american]{babel} \usepackage{fancyhdr} \hypersetup{colorlinks=false} \usetikzlibrary{calc, decorations.pathreplacing,shapes.misc} \usetikzlibrary{decorations.pathmorphing} \usepackage[left=1in,top=1in,right=1in]{geometry} \usepackage[capitalize]{cleveref} \newcommand{\mathcolorbox}[2]{\colorbox{#1}{$\displaystyle #2$}} \newcommand{\xxx}{T base with combinatorial potential data } \newcommand{\Xxx}{T base with combinatorial potential data } \newcommand{\xxxc}{combinatorial potential stratified space } \newcommand{\Xxxc}{combinatorial potential stratified space } \newcommand{\argument}{symplectic character } \newcommand{\arguments}{symplectic characters } \newcommand{\snip}[2]{#1} 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\DeclareMathOperator{\sgn}{sgn} \DeclareMathOperator{\Ext}{Ext} \DeclareMathOperator{\TropB}{TropB} \DeclareMathOperator{\weight}{wt} \DeclareMathOperator{\Span}{span} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Fuk}{Fuk} \DeclareMathOperator{\str}{star} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\Coh}{Coh} \DeclareMathOperator{\CritVal}{CritV} \DeclareMathOperator{\Sing}{Sing} \DeclareMathOperator{\FS}{FS} \DeclareMathOperator{\Vect}{Vect} \DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\Supp}{Supp} \DeclareMathOperator{\Bl}{Bl} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Tw}{Tw} \DeclareMathOperator{\Int}{Int} \DeclareMathOperator{\Arg}{\mathbf{M}}\begin{filecontents}{references.bib} @article{ballard2012hochschild, title={Hochschild dimensions of tilting objects}, author={Ballard, Matthew and Favero, David}, journal={International Mathematics Research Notices}, volume={2012}, number={11}, pages={2607--2645}, 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This means describing the objects, morphisms, and compositions of this category. \begin{definition} \label{def:moduleOveraInfinityCategory} Let $\mathcal A$ be an \emph{$A_\infty$} category; denote the product structure by $m^k_\mathcal A$. A right $\mathcal A$ module (denoted by $M\in \text{Mod}-\mathcal A$) is a \begin{itemize} \item assignment $M(A)$ of a graded module to each $A\in \text{Ob}(\mathcal A$) and, \item for each sequence $A_1, \ldots, A_k$ of objects in $\mathcal A$, a composition map \[ m^{1|{k-1}}_{M|\mathcal A}:M(A_{k-1})\otimes \hom(A_{k-2}, A_{k-1})\otimes \cdots \otimes \hom(A_1, A_0)\to M(A_k)[2-k]. \] \end{itemize} These are required to satisfy the quadratic $A_\infty$ relationships: for every sequence $A_0, \ldots, A_k$ of objects in $A$, \begin{align*} 0=&\sum_{\substack{ j_1+j+j_2=k\\j_1=0 }} (-1)^{\clubsuit_k} m_{M|\mathcal A}^{1|j_2} \circ (m_{A|M}^{1|j}\otimes \operatorname{id}_A^{\otimes j_2})\\ &+\sum_{\substack{ j_1+j+j_2=k\\j_1>0 }} (-1)^{\clubsuit_k} m_{M|{\mathcal A}}^{1|k-j}\circ ( \operatorname{id}_M\otimes \operatorname{id}_A^{\otimes j_1}\otimes m^{j}_A \otimes \operatorname{id}^{\otimes j_2}) \end{align*} \end{definition} The sign $\clubsuit$ follows the \underline{\href{https://jeffhicks.net/snippets/index.php?tag=def:aInfinityAlgebra}{ $A_\infty$ algebra sign convention}}: \[\clubsuit(\underline x,k_1):= k_1+\sum_{j=1}^{k_1} \deg(x_j).\] \begin{example} \label{exm:ringModule} The name module comes from the simplest example. Let $R$ be a ring. Now consider the $A_\infty$ category $\mathcal A$ which only contains one object $A$, and $\hom(A, A)=R$. Let $M$ be an $R$-module. We obtain a $\text{mod}-\mathcal A$ with the assignment $M(A)=M$, and whose product $m^{1|k}:M(A)\tensor A^{\tensor k}\to M(A)$ is \[ m^{1|1}(x,r)=x\cdot r \] and vanishes if $k\neq 1$. The $A_\infty$ module relations state \[m^{1|1}(m^{1|1}(x, r_1),r_2)-m^{1|1}(x, m^2(r_1, r_2))=(x\cdot r_1)\cdot r_2 - x\cdot (r_1\cdot r_2)=0\] which is guaranteed by associativity of the product. Given any chain complex of $R$-modules $M^\bullet$, we similarly obtain a right $\mathcal A$ module by taking $M^\bullet(A)=M^A$; the $A_\infty$ module relations now state that $m^{1|0}\circ m^{1|0}= d_M\circ d_M=0$, and that $d_M$ is a morphism of $R$-modules. There are right $\mathcal A$-modules beyond chain complexes. However, given any right $\mathcal A$-module, the homology of the complex $H^\bullet(M(A))$ is a graded $R$-module.\end{example} \begin{example} \label{exm:zeroModule} Let $\mathcal A$ be an $A_\infty$ category. We can define the zero module $M$ which has the property that for all $A\in \mathcal A$, our module assigns $M(A)=0$. As a result, the composition maps $m^{1|k}$ all vanish. This trivially satisfies the quadratic $A_\infty$ module relations. While this appears to be a artificial example, it is generally desirable to have a zero object in your category, and there is no reason a priori for your original $A_\infty$ category $\mathcal A$ to have a zero object. For the example we are interested--- the Fukaya category--- there is no ``zero'' Lagrangian submanifold.\end{example} \begin{example} \label{exm:yonedaModule} Let $\mathcal A$ be an $A_\infty$ category. Let $A\in \mathcal A$ be an object. We can associate to $A$ the \emph{Yoneda module}, $\mathcal Y_A$, which on every object $B\in \mathcal A$ assigns the chain complex \[\mathcal Y_A(B):=\hom(B, A),\] and whose product maps are defined by \[m^{1|k-1}(m,a_{k-1}, \ldots, a_0):= m^k(m,a_{k-1}, \ldots, a_0).\] Note that the $A_\infty$ module relations for $Y_A(B)$ are exactly the $A_\infty$ product relations for $\mathcal A$. \label{exm:yonedaModule} \end{example} \begin{definition} \label{def:morphismOfaInfinityModules} Let $M, N$ be $A_\infty$ modules. A \emph{pre-morphism of $A_\infty$ modules} of degree $d$ is a collection of maps $f^{1|k-1}: A^{\otimes k-1}\otimes M\to N[d-k+1]$. The set of pre-morphisms, $\hom^\bullet(M, N)$ is a cochain complex whose differential is (up to sign) \begin{align*} (m^1_{\hom^\bullet(M, N)} f)^{1|k-1}:=&\sum_{j+j_2=k}(-1)^\diamondsuit m^{1|j_2}_{N|\mathcal A} (f^{1|j-1}\tensor \id^{\tensor j_2})\\&+ \sum_{\substack{j_1+j+j_2=k\\j_1>0}} (-1)^\diamondsuit f^{1|j_1+j_2}(\id^{\tensor j_1}\tensor m^j_{\mathcal A}\tensor \id^{\tensor j_1})\\ &+ \sum_{\substack{j_1+j+j_2=k\\j_1=0}} (-1)^\diamondsuit f^{1|j_2}(m^{1|j}_{M|\mathcal A}\tensor \id^{\tensor j_1}). \end{align*} \end{definition} \begin{definition} \label{def:categoryOfAInfinityModules} Let $\mathcal A$ be an \emph{$A_\infty$} category. The \emph{category of right $\mathcal A$- modules}, denoted by $\text{Mod}-\mathcal A$, is the differential graded category whose: \begin{itemize} \item Objects are right $\mathcal A$ modules and \item chain complexes of morphisms are the chain complexes of $\mathcal A$ module pre-morphism, \item composition is given by \[m^2_{\text{mod}-\mathcal A}(f, g)=\sum_{j+j_2=k}-(-1)^\diamondsuit f^{1|j_2}(g^{1|j-1}\tensor \id^{j^2})\] \item higher product vanishing for $k\geq 3.$ \end{itemize} \end{definition} \begin{proposition} \label{prp:aInfinityModulesIsDG} Let $\mathcal A$ be an $A_\infty$ category. The category of modules over $\mathcal A$ has the structure of a dg-category. \end{proposition} One observes that $\text{mod}-\mathcal A$ is in general a ``nicer'' category than $\mathcal A$, as it inherits many of the properties of the category of chain complexes. \begin{proposition} \label{prp:categoryOfAInfinityModulesIsTriangulated} Let $\mathcal A$ be an \emph{$A_\infty$} category. Then $H^0(\text{mod}-\mathcal A)$ is a triangulated category. \end{proposition} \begin{proof} \label{prf:categoryOfAInfinityModulesIsTriangulated} We only describe the exact triangles in the category. Given $f\in \hom(M, N)$ a morphism of right $A_\infty$ modules, we define the cone module to be \[\cone(f)(A):=M(A)\oplus N(A)[1]\] whose $A_\infty$ module structure is given by \[ m^{1|k}_{\cone(f)|\mathcal A}:=m^{1|k}_{M|\mathcal A}\oplus (f^{1|k}+ m^{1|k}_{N|\mathcal A}).\] \end{proof} The Yoneda module construction (\cref{exm:yonedaModule}) gives a fully faithful functor $\mathcal A \to \text{mod}-\mathcal A$. As the category $\text{mod}-\mathcal A$ has mapping cones, this gives a triangulated envelope for $\mathcal A$. We therefore say that $A\to B \to C$ is an exact triangle in $\mathcal C$ if the image under Yoneda embedding is isomorphic to $A\to B \to \cone(f)$. \printbibliography \end{document}