\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} } 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For example, in the setting of abelian groups, we can associate to every morphism $f: A\to B$ a kernel, image, and cokernel. Because this language is so useful, there is a whole language of \emph{abelian categories} whose morphisms come with the data of kernels, images, and cokernels. In many categories there is not a reasonable way to construct the ``kernel'' of a morphism. For example, given a continuous map $f: X\to Y$ between two topological spaces, we have no reasonable definition of a kernel of a continuous map. There is, however, the notion of a cone $\operatorname{cone}(f)$, which remembers which points in $Y$ came from the image of $f$. \begin{definition} \label{def:topologicalMappingCone} Given \(f: X\to Y\) a continuous map, the \emph{cone} of \(f\) is the space \[\text{cone}(f):= (X\times I)\cup Y / \sim,\] where the equivalence identifies the points $(x_1, 0)\sim (x_2, 0)$ and $(x, 1)\sim f(x)$. We have continuous maps \(X\xrightarrow{f} Y \xrightarrow{i} \text{cone}(f)\). \end{definition} In the homotopy category of pointed spaces, the mapping cone takes a special meaning. Let $f: (X,x)\to (Y, y)$ be a pointed map. From this we can form $(Z, z):=(\cone(f), y)$ which is again a pointed space. We now address some relations between spaces $(X, x), (Y, y)$ and $(Z, z)$. \begin{itemize} \item Observe that the composition $i\circ f: (X, x)\to (Z, x))$ is homotopic to the constant map. In the homotopy category of pointed spaces, we can therefore write $i\circ f \sim 0$, where $0: (X, x)\to (Z, x)$ is the map factoring through a point. \item We can additionally look at the cone \[(Y, y)\xrightarrow{i}(Z, z).\] This second cone can be rewritten in terms of the data $f, X$ and $Y$ as \[ (Y\times J)\cup ((X\times I)\cup Y / \sim\] where the relations are \[(x_1, 0)\sim (x_2, 0) , (x, 1)\sim f(x)) , (y_1, 0)\sim (y_2, 0), (y, 1)\sim y.\] This is homotopic to the \underline{\href{https://jeffhicks.net/snippets/index.php?tag=def:suspension}{ suspension}} $\Sigma X$ allowing us to write the ``long exact sequence'' \[(X, x)\to (Y, y)\to (Z, z)\to (\Sigma X, x)\to (\Sigma Y, y)\to \cdots\] in the homotopy category of pointed spaces. \end{itemize} A triangulated category is a set of axioms for a category which encapsulates many of the properties of the cone construction. \begin{definition} \label{def:triangulatedCategory} A triangulated category is an \emph{additive} category $\mathcal C$, along with the structure of \begin{itemize} \item an additive automorphism $\Sigma: \mathcal C\to \mathcal C$, called the \emph{shift functor} and \item a collection of \emph{triangles}, which are triples of objects and morphisms written as \[A\xrightarrow{f} B \xrightarrow{g} C\xrightarrow{h} \Sigma A.\] \end{itemize} Denote by $X[n]=\Sigma^nX$. This data is required to satisfy the axioms for a triangulated category, \begin{description} \item[TR1], concerning which triangles must exist: \begin{itemize} \item The triangle $X\xrightarrow{\id} X\to 0 \to \Sigma X$ is an exact triangle \item For every morphism $f:X\to Y$ there exists an object (called the cone) so that $X\to Y \to \cone(f)$ is an exact triangle \item Every triangle which is isomorphic to an exact triangle is exact. \end{itemize} \item[TR2], concerning the interchange between exact triangles and suspension. If $X\xrightarrow{f} Y \xrightarrow{g} Z\xrightarrow{h} X[1]$ is an exact triangle, then so are $Y\to Z\to X[1]\to Y[1]$ and $Z[-1]\to X\to Y\to Z$. \item[TR3] Given a commutative square, if we complete the rows to exact triangles, then there exists a morphism between the third objects making everything commute. \item[TR4] The octahedral axiom, which states that given exact triangles \begin{align*} X\xrightarrow{f} Y \xrightarrow{g} Z'\xrightarrow{h} X[1]\\ Y\xrightarrow{i} Z \xrightarrow{j} X'\xrightarrow{k} Y[1]\\ X\xrightarrow{i\circ f} Z \xrightarrow{l} Y'\xrightarrow{m} Z[1] \end{align*} There exists a triangle $Z'\to Y'\to X'\to Z'[1]$. making the diagram of these triangles commute. \end{description} \end{definition} \begin{definition} \label{def:mappingConeOfCochainComplexes} Given two cochain complexes $M^\bullet$ and $N^\bullet$ and a chain map $f: M^\bullet\to N^\bullet$, the \emph{cone of $f$} is a new chain complex \[\text{cone}(f):=\left(M^{i+1}\oplus N^i, d=\begin{pmatrix}- d_M^{i+1} & 0 \\ - f^{i+1} & d_N^i\end{pmatrix}\right).\] \end{definition} We observe that when $X, Y$ are simplicial spaces and $f:X\to Y$ is a simplicial map that simplicial cochains topological mapping cone (\cref{def:topologicalMappingCone}) are the cone-cochains, \[C^\bullet(\cone(f))=\cone(f^*:C^\bullet(X)\to C^\bullet(Y)).\] This does not end up giving a triangulated category. However, the category of chain complexes with morphisms modulo chain homotopies is a triangulated category. This is called the \emph{homotopy category}. Triangulated categories capture many important aspects of homological algebra for chain complexes through the study of cohomological functors. \begin{definition} \label{def:cohomologicalFunctor} Let $\mathcal C$ be a triangulated category, and $\mathcal A$ be an abelian category. A \emph{cohomological functor} is a functor $F: \mathcal C\to \mathcal A$ which sends exact triangles \[A\to B\to C\to A[1]\] to exact sequences \[F(A)\to F(B)\to F(C).\] \end{definition} From this, we see that cohomological functors associate to exact triangles in $\mathcal C$ a \emph{long exact sequence} \[\cdots F(C[-1])\to F(A)\to F(B)\to F(C)\to F(A[1])\to \cdots \] \begin{proposition} \label{prp:homologyIsCohomologicalFunctor} Let $\mathcal A$ be an abelian category, and $\mathcal K(\mathcal A)$ the homotopy category, the functor \[H^0: \mathcal K(\mathcal A)\to \mathcal A\] is an example of a cohomological functor. \end{proposition} \begin{proof} \label{prf:homologyIsCohomologicalFunctor} The idea of proof is to observe that every exact triangle in the homotopy category is homotopic to one of the form $A\xrightarrow{f} B \to \cone(f)$, which is an exact sequence of chain complexes. The long exact sequence of cohomology groups arising from the snake lemma then proves the proposition. \end{proof} \printbibliography \end{document}