%Strg c t p fuer pdflatex \documentclass[12pt,german,xcolor=dvipsnames]{beamer} \usepackage{graphicx} \usepackage{amsmath} \usepackage{rotating} \usepackage[german]{babel} \usepackage{enumerate} \usepackage{xspace} %\usepackage{showkeys} \usepackage{young} \usepackage{makeidx} \usepackage{newlfont} \usepackage{amssymb} \usepackage{amsxtra} \usepackage{amsfonts} \usepackage{amscd} \usepackage{mathrsfs} \usepackage{stmaryrd} \usepackage{psfrag} \usepackage{color} \usepackage[latin1]{inputenc} \usepackage{times} \usepackage[mathscr]{euscript} \usepackage[all]{xy} \usepackage[active]{srcltx} \input epsf \DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it} \setlength{\arraycolsep}{2pt} \input{newcommands} \newcommand{\wt}{\tilde} \newcommand{\wh}{\hat} \newcommand{\td}{\tilde} \newcommand{\eps}{\varepsilon} \title[]{~\\[2ex] About the six functors\\ Trying to understand Grothendieck\\ with the help of H\"ormann} \author[]{Wolfgang Soergel} \institute[]{\inst{} Mathematisches Institut\\ Universit\"at Freiburg\\[4ex] \vspace*{.9cm} %\textcolor{red}{\\ allgemeine Fragen/Hinweise hier positionieren} } \date[]{\small \hbox{Juni 2017}} \beamersetuncovermixins{\opaqueness<1>{25}}{\opaqueness<2->{15}} % ================================================== % ================================================== \begin{document} % ================================================== % ================================================== \titlepage \begin{frame} Grothendieck's six functors $$f_*,f^*,f_!, f^!,\otimes, \mathcal{H}{\op{om}}$$ \begin{itemize} \\[2mm]\item%\pause To $X$ a topological space associate a triangulated category $\op{D}(X)\pdef \op{Der} (\op{Ab}_{/X})$ \\[2mm]\item%\pause For any two objects $\mathcal F, \mathcal G\in \op{D}(X)$ construct functorially new objects $\mathcal F\otimes \mathcal G$ and $\mathcal{H}{\op{om}}(\mathcal F, \mathcal G) \in \op{D}(X)$ \\[2mm]\item%\pause For $f:X\ra Y$ a continous map construct triangulated functor $f^{-1}= f^\ast:\op{D} (Y) \ra \op{D} (X)$ and its right adjoint $f_*={\op{R}}f_{(*)}$ \\[2mm]\item%\pause For $f:X\ra Y$ a more special continous map construct triangulated functor $ f_!:\op{D} (X) \ra \op{D} (Y)$ and its right adjoint $f^!$. \end{itemize} \end{frame} \begin{frame} {\Huge Lots of isomorphisms!} \\[4mm] \begin{itemize} \\[2mm]\item Associators and commutators and unit conditions making $\op{D}(X)$ a monoidal category \\[2mm]\item Identifications $f^*g^*\siRa (gf)^*$, $g_!f_!\siRa (gf)_!$ etc \\[2mm]\item Base change $f_!p^*\siRa q^*g_!$ in a cartesian diagram $gp=qf$ \\[2mm]\item Projection formula $ (\pi_{!} \mathcal F) \otimes \mathcal G \sira \pi_{!} (\mathcal F \otimes \pi^\ast \mathcal G)$ \\[2mm]\item more \end{itemize} \\[4mm] {\Huge Compatibilities?} \end{frame} \begin{frame} \begin{itemize} \\[2mm]\item {\large H"ormann: Fibered multicategory over correspondences} \\[2mm]\item He even discusses a more general derivator formalism \\[2mm]\item Today, explain the formalism and sketch how to get there in the topological context \\[2mm]\item Need only discuss $f^*, f_!,\otimes$ \\[2mm]\item We construct the dual cofibration. The direct image functor along a multicorrespondence will be \begin{displaymath} \xymatrix{ &&& \ar[dl]^-{g_{r}}\ar[dlll]_-{g_{1}} K \ar[dr]^-{f}&\\ X_{1}&\ldots&X_{r} && Y } \end{displaymath} $$\hspace{1.8cm}\mathcal F_1\curlywedge\ldots\curlywedge \mathcal F_r\mapsto f_!(g_1^*\mathcal F_1\otimes\ldots\otimes g_r^*\mathcal F_r)$$ \end{itemize} \end{frame} \begin{frame} \\[2mm] % Fibered {\color{Red}multicategory} over correspondences \begin{itemize} \\[2mm]\item Basic example of multicategory: vector spaces, multilinear maps, can be multicomposed. Formally: \\[2mm]\item Let $\mathcal M$ be a set of ``objects'' \\[2mm]\item Let $\mathcal M^\curlywedge$ be the set of finite words $M=M_1\curlywedge\ldots\curlywedge M_r$ in $\mathcal M$ for $r\geq 0$ \\[2mm]\item Assume given a category $\mathcal M^\curlywedge$ with objects $\mathcal M^\curlywedge$ \\[2mm]\item Assume given a functor $\mathcal M^\curlywedge\ra \op{Set}$ mapping each word to its index set $\{1,\ldots,r\}$ \\[2mm]\item For any map $f$ of the index set $\{1,\ldots,r\}$ of a word $M$ to the index set $\{1,\ldots,s\}$ of a word $N$ assume given a bijection $$\mathcal M^\curlywedge_f(M,N)\sira\prod_{1\leq j\leq s}\mathcal M^\curlywedge(M|_{f^{-1}(j)},N_j)$$ \\[2mm]\item Assume compatibility with composition \end{itemize} \end{frame} \begin{frame} % Fibered {\color{Red}multicategory} over correspondences \begin{itemize} \\[2mm]\item Define universal multimorphisms $M_1\curlywedge\ldots\curlywedge M_r\ra M_1\otimes\ldots\otimes M_r$ and $\curlywedge \ra \mathbb I$ \\[2mm]\item If they always exist and the multicomposition of universal multimorphisms is again universal, this leads to a symmetric monoidal category \end{itemize} \end{frame} \begin{frame} \\[2mm] % {\color{Red}Fibered} multicategory over correspondences \begin{itemize} \\[2mm]\item Given a functor $p:\mathscr C\ra \mathscr B$, the {\bf fibre} $\mathscr C_X$ over $X\in \mathscr B$ is $(p^{-1}(X),p^{-1}(\op{id}_X))$ \\[2mm]\item A morphism $\varphi:\mathcal F\ra \mathcal G$ in $\mathscr C$ over say $f:X\ra Y$ in $\mathscr B$ is {\bf cartesian}, if every morphism $\psi:\mathcal E\ra \mathcal G$ over $f$ uniquely factors through $\mathcal F$ with a first morphism over $\op{id}_X$, so \begin{displaymath} \xymatrix{ \mathcal E \ar[drr]\ar@{-->}_{\exists !}[d]&& \\ \mathcal F \ar[rr] && \mathcal G\\ X \ar[rr] && Y } \end{displaymath} \\[2mm]\item A functor is a {\bf fibration}, if for all $f:X\ra Y$ and $\mathcal G\in\mathscr C_Y$ there is a cartesian lift $\mathcal F\ra \mathcal G$, and if the composition of cartesian lifts is cartesian. \end{itemize} \end{frame} \begin{frame} \\[2mm] % {\color{Red}Fibered} multicategory over correspondences \begin{itemize} \\[2mm]\item $\op{Ab}_{/\op{Top}}$ abelian sheaves on topological spaces Given $\mathcal F\in \op{Ab}_{/X}$ and $\mathcal G\in \op{Ab}_{/Y}$ a morphism is a pair $(\varphi, f)$ such that with \'etale spaces commutes \begin{displaymath} \xymatrix{ \bar{\mathcal F} \ar[d]\ar[r]&\bar{\mathcal G} \ar[d] \\ X \ar[r]& Y } \end{displaymath} The obvious functor $\op{Ab}_{/\op{Top}}\ra \op{Top}$ is a fibration. Cartesian morphisms are those with cartesian diagrams. \end{itemize} \end{frame} \begin{frame} \\[2mm] % {\color{Red}Fibered} multicategory over correspondences \begin{itemize} \\[2mm]\item $\op{Ab}_{\sslash\op{Top}}$ abelian sheaves on topological spaces again. Given $\mathcal F\in \op{Ab}_{\sslash X}$ and $\mathcal G\in \op{Ab}_{\sslash Y}$ a morphism is a pair $(\varphi, f)$ with $$\varphi:\mathcal G (V)\ra \mathcal F (U)$$ whenever $f(U)\subset V$, in a compatible way. We say $\varphi$ is a {\bf comorphism} $\mathcal G\ra \mathcal F$ or an {\bf opcomorphism $\mathcal F\ra \mathcal G$ over $f$}. Again $\op{Ab}_{\sslash\op{Top}}\ra \op{Top}$ is a fibration. Cartesian morphisms are the obvious opcomorphisms $f^\ast \mathcal G\ra \mathcal G$. For the fibres we have $$\op{Ab}_{\sslash X}=(\op{Ab}_{/ X})^{\op{opp}}$$ \end{itemize} \end{frame} \begin{frame} \\[2mm] % {\color{Red}Fibered} multicategory over correspondences \begin{itemize} \\[2mm]\item Given a fibration $p:\mathscr C\ra\mathscr B$ we have pullback functors $f^*$ unique up to unique isomorphisms and the identifications $$f^*g^*\siRa (gf)^*\quad\text{and}\quad {\op{id}}_X^*\siRa \op{Id}$$ are given automatically with all kinds of compatibilities \end{itemize} \end{frame} \begin{frame} \\[2mm] {\color{Red}Fibered} multicategory over correspondences \begin{itemize} \\[2mm]\item Any category $\mathscr T$ gives rise to a multicategory $\mathscr T$, the multimorphisms $$X_1\curlywedge \ldots \curlywedge X_r\ra Y$$ being tupels $(g_1,\ldots,g_r)$ with $g_i:X_i\ra Y$ \end{itemize} \end{frame} \begin{frame} \\[2mm] % {\color{Red}Fibered} multicategory over correspondences \begin{itemize} \\[2mm]\item We construct a multifunctor $$\op{Ab}_{\sslash\op{Top}}^{\op{opp}}\ra \op{Top}^{\op{opp}}$$ A multicomorphism $$\varphi:\mathcal F_1\curlywedge \ldots \curlywedge\mathcal F_1\ra \mathcal G$$ over $g_i^\circ:X_i\ra Y$ alias continous maps $g_i:Y\ra X_i$ is a collection of multilinear maps $$\mathcal F_1(U_1)\times \ldots \times\mathcal F_r(U_r)\ra \mathcal G(V)$$ whenever $g_i(V)\subset U_i\;\forall i$, compatible with restrictions. \end{itemize} \end{frame} \begin{frame} \\[2mm] % {\color{Red}Fibered} multicategory over correspondences \begin{itemize} \\[2mm]\item The multifunctor $$\op{Ab}_{\sslash\op{Top}}^{\op{opp}}\ra \op{Top}^{\op{opp}}$$ induces a cofibration (dual to fibration) on the categories of words. $$\mathcal F\curlywedge \mathcal G\ra \mathcal F\otimes \mathcal G$$ is cocartesian over $(\op{id}^\circ,\op{id}^\circ):X \curlywedge X\ra X$. $$\mathcal F\ra g^\ast \mathcal F$$ is cocartesian over $g^\circ:X \ra Y$ alias a continous map $g:Y \ra X$. The isomorphisms $$g^*\mathcal F\otimes g^*\mathcal G\sira g^*(\mathcal F\otimes \mathcal G)$$ express the equality of the pushforwards along $Y \curlywedge Y\ra X \curlywedge X\ra X$ and $Y \curlywedge Y\ra Y\ra X$. \end{itemize} \end{frame} \begin{frame} \\[2mm] % {\color{Red}Fibered} multicategory over correspondences \begin{itemize} \\[2mm]\item We define an opcomorphism $\varphi^\circ:\mathcal F\ra\mathcal G$ over $f:X\ra Y$ to be {\bf proper}, if for all $V\co Y$ and $s\in \mathcal G(V)$ and $\varphi(s)\in \mathcal F(f^{-1}V)$ the map $f:\op{supp}\varphi(s)\ra V$ is proper. \\[2mm]\item Equivalent: the induced sheaf homomorphism $\mathcal G\ra f_\ast \mathcal F$ factors over $f_! \mathcal F\subset f_* \mathcal F$ \\[2mm]\item The composition of proper opcomorphisms is proper. The pullback of proper opcomorphisms is proper. \\[2mm]\item Restricting to locally compact Hausdorff spaces or more generally ``locally proper (eigentlich) separated'' maps, we get a cofibration $$\op{Ab}^!_{\sslash \op{Top}^{\op{les}}}\ra \op{Top}^{\op{les}}$$ with direct image functors $f_!$ \end{itemize} \end{frame} \begin{frame} Now consider a correspondence $k=f\bar g$ of topological spaces \begin{displaymath} \xymatrix{ & \ar[dl]_-{g} K \ar[dr]^-{f}&\\ X&& Y } \end{displaymath} and for $\mathcal F\in\op{Ab}_{\sslash X}$ and $\mathcal G\in\op{Ab}_{\sslash Y}$ set $$\op{Ab}_{\sslash k}^!(\mathcal F,\mathcal G)\pdef \op{Ab}_{\sslash f}^!(g^*\mathcal F,\mathcal G)$$ {\Large Category of correspondences?} \end{frame} \begin{frame} \begin{itemize} \\[2mm]\item In a category composition needs to be strictly associative. \\[2mm]\item Define morphisms in the category of correspondences to be zigzags. \\[2mm]\item Make it a two-category, by defining morphisms between correspondences to be proper morphisms between the limit objects of the corresponding zizag diagrams, compatible with their morphisms to $X$ and $Y$. \\[2mm]\item Define morphisms over correspondences of abelian sheaves as before with the limit object $K$ of the correspondence \end{itemize} \end{frame} \begin{frame}\begin{itemize} \\[2mm]\item Restricting to locally compact Hausdorff spaces, get cofibration and twofunctor $$\op{Ab}^!_{\sslash \op{Top}^{\op{lch}}} \ra \text{correspondences in }\op{Top}^{\op{lch}}$$ with direct image $f_!g^*$ along $f\bar g$. This englobes base change and its compatibilities. Any two-morphism has exactly one lift for a given end. \\[2mm] \item Put everything together, localize carefully, get the final statement \end{itemize} \end{frame} \begin{frame} {\bf Thanks!} \end{frame} \end{document} %%% Local Variables: %%% mode: pdflatex %%% End: