\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{\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}, year={2012}, publisher={OUP} } @article{craw2007explicit, title={Explicit methods for derived categories of sheaves}, author={Craw, Alastair}, publisher={Citeseer} } @article{weinstein1971symplectic, title={Symplectic manifolds and their {L}agrangian submanifolds}, author={Weinstein, Alan}, journal={Advances in Mathematics}, volume={6}, number={3}, pages={329--346}, year={1971}, publisher={Academic Press} } @article{hanlon2022aspects, title={Aspects of functoriality in homological mirror symmetry for toric varieties}, author={Hanlon, A and Hicks, J}, journal={Advances in Mathematics}, volume={401}, pages={108317}, year={2022}, publisher={Elsevier} } @article{biran2013lagrangian, title={{L}agrangian cobordism. {I}}, author={Biran, Paul and Cornea, Octav}, journal={Journal of the American Mathematical Society}, volume={26}, number={2}, pages={295--340}, year={2013} } @article{tanaka2016fukaya, title={The Fukaya category pairs with Lagrangian cobordisms}, author={Tanaka, Hiro Lee}, journal={arXiv preprint arXiv:1607.04976}, year={2016} } @book{seidel2008fukaya, title={Fukaya categories and Picard-Lefschetz theory}, author={Seidel, Paul}, volume={10}, year={2008}, publisher={European Mathematical Society} } @article{seidel2003long, title={A long exact sequence for symplectic {F}loer cohomology}, author={Seidel, Paul}, journal={Topology}, volume={42}, pages={1003--1063}, year={2003} } @article{da2001lectures, title={Lectures on symplectic geometry}, author={da Silva, Ana Cannas}, journal={Lecture Notes in Mathematics}, volume={1764}, year={2001}, publisher={Springer} } @article{polterovich1991surgery, title={The surgery of {L}agrange submanifolds}, author={Polterovich, Leonid}, journal={Geometric \& Functional Analysis GAFA}, volume={1}, number={2}, pages={198--210}, year={1991}, publisher={Springer} } @misc{perutz2008handleslide, doi = {10.48550/ARXIV.0801.0564}, url = {https://arxiv.org/abs/0801.0564}, author = {Perutz, Timothy}, keywords = {Symplectic Geometry (math.SG), Geometric Topology (math.GT), FOS: Mathematics, FOS: Mathematics, 53D12; 53D40; 57M27; 32U40}, title = {Hamiltonian handleslides for {H}eegaard {F}loer homology}, publisher = {arXiv}, year = {2008}, copyright = {Assumed arXiv.org perpetual, non-exclusive license to distribute this article for submissions made before January 2004} } @incollection{audin1994symplectic, title={Symplectic rigidity: {L}agrangian submanifolds}, author={Audin, Mich{\`e}le and Lalonde, Fran{\c{c}}ois and Polterovich, Leonid}, booktitle={Holomorphic curves in symplectic geometry}, pages={271--321}, year={1994}, publisher={Springer} } @phdthesis{oancea2003suite, title={La suite spectrale de {L}eray-{S}erre en homologie de {F}loer des vari{\'e}t{\'e}s symplectiques compactes {\`a} bord de type contact}, author={Oancea, Alexandru}, year={2003}, school={Universit{\'e} Paris Sud-Paris XI} } @article{abouzaid2010geometric, title={A geometric criterion for generating the {F}ukaya category}, author={Abouzaid, Mohammed}, journal={Publications Math{\'e}matiques de l'IH{\'E}S}, volume={112}, pages={191--240}, year={2010} } @article{viterbo1999functors, title={Functors and computations in {F}loer homology with applications, I}, author={Viterbo, Claude}, journal={Geometric \& Functional Analysis GAFA}, volume={9}, number={5}, pages={985--1033}, year={1999}, publisher={Springer} } @misc{stacks-project, author = {The {Stacks project authors}}, title = {The Stacks project}, howpublished = {\url{https://stacks.math.columbia.edu}}, year = {2022}, } @article{wendlbeginner, title={A beginner’s overview of symplectic homology}, author={Wendl, Chris}, journal={Preprint. www. mathematik. hu-berlin. de/wendl/pub/SH. pdf} } @article{seidel2006biased, title={A biased view of symplectic cohomology}, author={Seidel, Paul}, journal={Current developments in mathematics}, volume={2006}, number={1}, pages={211--254}, year={2006}, publisher={International Press of Boston} } @article{arnol1980lagrange, title={{L}agrange and {L}egendre cobordisms. I}, author={Arnol'd, Vladimir Igorevich}, journal={Funktsional'nyi Analiz i ego Prilozheniya}, volume={14}, number={3}, pages={1--13}, year={1980}, publisher={Russian Academy of Sciences} } @article{fukaya2007lagrangian, title={{L}agrangian intersection {F}loer theory-anomaly and obstruction, chapter 10}, author={Fukaya, K and Oh, YG and Ohta, H and Ono, K}, journal={Preprint, can be found at http://www. math. kyoto-u. ac. jp/\~{} fukaya/Chapter10071117. pdf}, year={2007} } @article{biran2014lagrangian, title={Lagrangian cobordism and Fukaya categories}, author={Biran, Paul and Cornea, Octav}, journal={Geometric and functional analysis}, volume={24}, number={6}, pages={1731--1830}, year={2014}, publisher={Springer} } @article{bourgeois2009symplectic, title={Symplectic homology, autonomous {H}amiltonians, and {M}orse-{B}ott moduli spaces}, author={Bourgeois, Fr{\'e}d{\'e}ric and Oancea, Alexandru}, journal={Duke mathematical journal}, volume={146}, number={1}, pages={71--174}, year={2009}, publisher={Duke University Press} } @incollection{auroux2014beginner, title={A beginner’s introduction to {F}ukaya categories}, author={Auroux, Denis}, booktitle={Contact and symplectic topology}, pages={85--136}, year={2014}, publisher={Springer} } @article{singer1933three, title={Three-dimensional manifolds and their {H}eegaard diagrams}, author={Singer, James}, journal={Transactions of the American Mathematical Society}, volume={35}, number={1}, pages={88--111}, year={1933}, publisher={JSTOR} } @article{ozsvath2004holomorphic, title={Holomorphic disks and three-manifold invariants: properties and applications}, author={Ozsv{\'a}th, Peter and Szab{\'o}, Zolt{\'a}n}, journal={Annals of Mathematics}, pages={1159--1245}, year={2004}, publisher={JSTOR} } @article{ozsvath2004introduction, title={An introduction to {H}eegaard {F}loer homology}, author={Ozsv{\'a}th, Peter and Szab{\'o}, Zolt{\'a}n}, journal={{F}loer homology, gauge theory, and low-dimensional topology}, volume={5}, pages={3--27}, year={2004} } @article{fet1952variational, title={Variational problems on closed manifolds}, author={Fet, Abram Il'ich}, journal={Matematicheskii Sbornik}, volume={72}, number={2}, pages={271--316}, year={1952}, publisher={Russian Academy of Sciences, Steklov Mathematical Institute of Russian~…} }\end{filecontents} \addbibresource{references.bib}\begin{document} \title{the category of twisted complexes} \maketitle \thispagestyle{firstpage} One viewpoint on mapping cones of cochain complexes is that they give \emph{deformations} of (direct sums of) objects of our categories. Given a map of cochain complexes $f: A\to B$, the differential on $\cone(f)$ has the form \[d_{\cone(f)}=\begin{pmatrix} d_A & 0 \\ 0 & d_B \end{pmatrix} + \begin{pmatrix} 0 & 0\\ f & 0\end{pmatrix}\] where the first term is the differential on $A\oplus B[1]$, and the second term ``deforms'' the differential on this chain complex. Twisted complexes extend this story in several directions: firstly, we expand the set of deformations so that the objects we consider are chain complexes up to homotopy, and we allow deformations of the product (and not only differential) structure. \begin{definition} \label{def:twistedComplex} Let $\mathcal C$ be an $A_\infty$ category. A twisted complex $(E, \delta_E)$ consists of: \begin{itemize} \item $E$, a formal direct sum of shifts of objects \[E:=\bigoplus_{i=1}^N E_i[k_i]\] where $E_{i}\in Ob(\mathcal C)$, and $k_i\in \ZZ$. \item A differential $\delta_E$, which can be written as a matrix of degree 1 maps \[\delta^{ij}_E: E_i[k_i]\to E_j[k_j +1] .\] These maps must satisfy the following conditions: \begin{itemize} \item the matrix $\delta_E$ is strictly upper triangular and; \item They satisfy the Maurer-Cartan relation: \[\sum_{k\geq 1} m^k (\delta_E\otimes\cdots \otimes \delta_E) =0.\] \end{itemize} \end{itemize} \end{definition} The condition that the matrix $\delta_E$ is strictly upper triangular is to ensure that the sum in the Maurer-Cartan relation converges. One can also ask that there exists a filtration on $E$, the formal direct sum of shifts of objects, and that the differential $\delta_E$ respects the filtration (see Section 31 of \cite{seidel2008fukaya}). From this perspective, the twisted complex looks more like a formal deformation of the direct sum. With regards to the first point: Suppose that we have a (not necessarily exact) sequence of chain complexes $A\xrightarrow{f} B \xrightarrow{g} C$. The total complex of this sequence will not be a chain complex (as $g\circ f \neq 0$). However, to build a twisted complex from this data we will only need that $g\circ f$ is homotopic to zero. Suppose that $H:A\to C[1]$ is a homotopy (so that $d_AH+Hd_C=g\circ f$). Then \[\delta = \begin{pmatrix} 0 & 0 & 0\\ f & 0 & 0\\ H & g & 0 \end{pmatrix}\] gives us a twisted complex on $A\oplus B[1]\oplus C[2]$. For the second point: Let $(A, m^k)$ be an $A_\infty$ algebra. There are a particularly nice class of deformations of $A_\infty$ governed by elements $a\in A^1$ satisfying the Maurer-Cartan equation: \[m^1(a)+m^2(a\otimes a)+m^3(a\otimes a \otimes a)+\cdots =0.\] In order for this equation to make sense, one needs show that the sum converges. This is usually achieved by asking that $A$ be filtered and that $m^k(a^{\otimes k})$ lies increasingly positive filtration levels. When one can make sense of this equation, we can define a new $A_\infty$ algebra, $(A, m^k_a)$ whose product is defined by \[m^k_a:=\sum_{n>0}\sum_{j_0+\cdots+j_k=n} m^{k+n}(a^{\otimes j_0}\otimes \id \otimes a^{\otimes j_1}\otimes \id \cdots \otimes a^{\otimes j_{k-1}}\otimes \id \otimes a^{\otimes j_k})\] Now consider the setting where $C$ is a chain complex, and $A=\hom(C, C)$. Then $a\in A^1$ corresponds to a map $a: C\to C[1]$, and the Maurer-Cartan equation has two terms: \begin{itemize} \item The first term $m^1(a) = d_A a + a d_A$. The vanishing of this term states that $a$ is a chain map; \item The vanishing of the second term $m^2(a, a)$ tells us that $a$ squares to zero (so that it gives a differential). \end{itemize} The combination of these two terms checks the condition that $(d_A+a)\circ (d_A -a)=0$; that is that we can deform the differential by $(-1)^k a$. \begin{definition} \label{def:morphismOfTwistedComplexes} Let $(E, \delta_E)$ and $(F, \delta_F)$ be two twisted complexes. A \emph{morphism of twisted complexes} is a collection of morphisms of $\mathcal C$ \[f_{ij}:E_i[k_i]\to E_j[k_j].\] The set of morphisms can therefore be written as $\hom^d((E, \delta_E), (F, \delta_F))=\bigoplus_{i, j}\hom^{d+k^F_j-k^E_i}(E_i, F_j).$ Given a sequence $\{(E_i, \delta_i)\}_{i=0}^k$ of twisted complexes, and $a_i\in\hom^d((E_{i-1}, \delta_{E_{i-1}}), (E_{i}, \delta_{E_{i}}))$, we have a composition \[m^k_{\operatorname{Tw}}(a_{k}\otimes \cdots \otimes a_{1} ):=\sum_{j_0, \ldots, j_k\geq 0} m^k(\delta_k^{\otimes j_k}\otimes a_{k}\otimes \delta_{k-1}^{\otimes j_{k-1}}\otimes a_{k-1}\otimes \cdots \otimes a_1\otimes \delta_0^{\otimes j_{0}}).\] \end{definition} \begin{proposition} \label{prp:categoryOfTwistedComplexes} Let $\mathcal C$ be an $A_\infty$ category. The category of twisted complexes, $\operatorname{Tw}(\mathcal C)$, is the $A_\infty$ category whose objects are twisted complexes, morphisms are morphisms of twisted complexes, and $A_\infty$ compositions are given by $m^k_{\operatorname{Tw}}$. \end{proposition} \begin{theorem} \label{thm:categoryOfTwistedComplexesIsTriangulated} Let $\mathcal C$ be an $A_\infty$ category. The category of twisted complexes, $\operatorname{Tw}(\mathcal C)$ is a triangulated category. There is a fully faithful inclusion $\mathcal A\to \operatorname{Tw}(\mathcal C)$. Furthermore, the image of $\mathcal A$ generated $\operatorname{Tw}(\mathcal C)$. \end{theorem} \begin{proof} \label{prf:categoryOfTwistedComplexesIsTriangulated} To give twisted complexes the structure of a triangulated category, we must specify what the exact triangles are. Given a morphism $f: (E, \delta_E)\to (F, \delta_F)$, we can define the cone of $f$ to be the twisted complex $(E[1]\oplus F, \delta')$ where $\delta'$ is the matrix \[ \left(\begin{array}{c|c} \delta_E &0 \\ \hline f^\delta_F & \delta_F\end{array}\right). \] \end{proof} There exists an inclusion functor $i:\mathcal C\to \Tw(\mathcal C)$. We can therefore declare that the triangle $A\to B\to C$ is exact in $\mathcal A$ is if $C$ is quasi-isomorphic to $\cone(A\to B)$ in the category of twisted complexes. \printbibliography \end{document}