\input{"preamble.tex"} \addbibresource{Stacks.bib} \let\Begin\begin \let\End\end \newcommand\wrapenv[1]{#1} \makeatletter \def\ScaleWidthIfNeeded{% \ifdim\Gin@nat@width>\linewidth \linewidth \else \Gin@nat@width \fi } \def\ScaleHeightIfNeeded{% \ifdim\Gin@nat@height>0.9\textheight 0.9\textheight \else \Gin@nat@width \fi } \makeatother \setkeys{Gin}{width=\ScaleWidthIfNeeded,height=\ScaleHeightIfNeeded,keepaspectratio}% \title{ \rule{\linewidth}{1pt} \\ \textbf{ Introduction to Stacks and Moduli } \\ {\normalsize Lectures by Jarod Alper. University of Washington, Spring 2021} \\ \rule{\linewidth}{2pt} } \titlehead{ \begin{center} \includegraphics[width=\linewidth,height=0.45\textheight,keepaspectratio]{figures/cover.png} \end{center} \begin{minipage}{.35\linewidth} \begin{flushleft} \vspace{2em} {\fontsize{6pt}{2pt} \textit{Notes: These are notes on an online graduate course in stacks by Jarod Alper in Spring 2021. As such, any errors or inaccuracies are almost certainly my own. } } \\ \end{flushleft} \end{minipage} \hfill \begin{minipage}{.65\linewidth} \end{minipage} } \begin{document} \date{} \maketitle \begin{flushleft} \textit{D. Zack Garza} \\ \textit{University of Georgia} \\ \textit{\href{mailto: dzackgarza@gmail.com}{dzackgarza@gmail.com}} \\ {\tiny \textit{Last updated:} 2021-09-06 } \end{flushleft} \newpage % Note: addsec only in KomaScript \addsec{Table of Contents} \tableofcontents \newpage \hypertarget{friday-july-30}{% \section{Friday, July 30}\label{friday-july-30}} References: \begin{itemize} \tightlist \item Course website: \url{https://sites.math.washington.edu/~jarod/math582C.html} \item \href{https://arxiv.org/pdf/math/9911199.pdf}{Gómez 99: Expository article on algebraic stacks} \end{itemize} \begin{remark} Stated goal of the course: prove that the moduli space \(\mkern 1.5mu\overline{\mkern-1.5mu{ \mathcal{M}_{g} }\mkern-1.5mu}\mkern 1.5mu\) of stable curves (for \(g\geq 2\)) is a smooth, proper, irreducible Deligne-Mumford stack of dimension \(3g-3\). Moreover, it admits a projective coarse moduli space. In the process we'll define \textbf{algebraic spaces} and \textbf{stacks}. Prerequisites: \begin{itemize} \tightlist \item Schemes \item Existence of Hilbert schemes \item Artin approximation \item Resolution of singularities for surfaces \item Deformation theory \end{itemize} \end{remark} \hypertarget{lecture-3-groupoids-and-prestacks-monday-september-06}{% \section{Lecture 3: Groupoids and Prestacks (Monday, September 06)}\label{lecture-3-groupoids-and-prestacks-monday-september-06}} \hypertarget{groupoids}{% \subsection{Groupoids}\label{groupoids}} \begin{remark} Last time: functors, sheaves on sites, descent, and Artin approximation. Today: groupoids and stacks. Recall that a \textbf{site} \(\mathsf{S}\) is a category such that for all \(U\in {\operatorname{Ob}}(\mathsf{S})\), there exists a set \({\mathsf{Cov}}(U) \coloneqq\left\{{U_i \to U}\right\}_{i\in I}\) (a \emph{covering family}) such that \begin{itemize} \tightlist \item \(\operatorname{id}_U \in {\mathsf{Cov}}(U)\), \item \({\mathsf{Cov}}(U)\) is closed under composition. \item \({\mathsf{Cov}}(U)\) is closed under pullbacks: \end{itemize} \begin{center} \begin{tikzcd} {\exists U_i{ \underset{\scriptscriptstyle {U} }{\times} }V} && {U_i} \\ \\ V && U \arrow["{\in {\mathsf{Cov}}(U)}", from=1-3, to=3-3] \arrow[from=3-1, to=3-3] \arrow[dashed, from=1-1, to=1-3] \arrow[dashed, from=1-1, to=3-1] \arrow["\lrcorner"{anchor=center, pos=0.025}, draw=none, from=1-1, to=3-3] \arrow["{\in{\mathsf{Cov}}(U)}"{description}, curve={height=-12pt}, dashed, from=1-1, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMiwyLCJVIl0sWzAsMiwiViJdLFsyLDAsIlVfaSJdLFswLDAsIlxcZXhpc3RzIFVfaVxcZmliZXJwcm9ke1V9ViJdLFsyLDAsIlxcaW4gXFxDb3YoVSkiXSxbMSwwXSxbMywyLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMywxLCIiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMywwLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XSxbMywwLCJcXGluXFxDb3YoVSkiLDEseyJjdXJ2ZSI6LTIsInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==}{Link to Diagram} \end{quote} \end{remark} \begin{example}[The big étale site] Take \(\mathsf{S} \coloneqq{\mathsf{Sch}}_{\text{Ét}}\) to be the big étale site: the category of all schemes, with covering families given by étale morphisms \(\left\{{U_i\to U}\right\}_{i\in I}\) such that \(\displaystyle\coprod_i U_i \twoheadrightarrow U\). Note that there is a special covering family given by \emph{surjective} etale morphisms. \todo[inline]{Reducing to case of single surjective etale cover somehow?} \end{example} \begin{definition}[Sheaves on sites] Let \(\mathsf{C}\) be a category (e.g.~\(\mathsf{C} \coloneqq{\mathsf{Set}}\)) and recall that a \emph{presheaf} on a category \(\mathsf{S}\) is a contravariant functor \(\mathsf{S}\to \mathsf{C}\). A \(\mathsf{C}{\hbox{-}}\)valued \textbf{sheaf} on a site \(\mathsf{S}\) is a presheaf \begin{align*} {\mathcal{F}}:\mathsf{S} \to \mathsf{C} \end{align*} such that for all \(U_i, U_j\in {\mathsf{Cov}}(U)\), the following equalizer diagram is exact in \(\mathsf{C}\) \begin{center} \begin{tikzcd} 0 \stackarr{1}[r] & F(U) \stackarr{3}[r] & \prod\limits_{i} F(U_i) \stackarr{5}[r] & \prod\limits_{i, j} F(U_i { \underset{\scriptscriptstyle {U} }{\times} } U_j) \end{tikzcd} \end{center} \end{definition} \begin{exercise}[Criterion for sheaves on the big etale site] Show that a presheaf \(F\) is a sheaf on \({\mathsf{Sch}}_\text{Ét}\) iff \begin{itemize} \tightlist \item \(F\) is a sheaf on \({\mathsf{Sch}}_{\mathrm{Zar}}\) and \item For all etale surjections \(U' \twoheadrightarrow_{\text{ét}} U\) of affines, the equalizer diagram is exact. \end{itemize} \end{exercise} \begin{proposition}[Yoneda] For \(X\in {\mathsf{Sch}}\), the presheaf \begin{align*} h_X \coloneqq\operatorname{Mor}({-}, X): {\mathsf{Sch}}\to {\mathsf{Set}} \end{align*} is a sheaf on \({\mathsf{Sch}}_{\text{Ét}}\). \end{proposition} \begin{remark} We'll often consider \emph{moduli functors}: functors \(F: {\mathsf{Sch}}\to {\mathsf{Set}}\) where \(F(S)\) is a family of objects over \(S\). Then \(F\) will be a sheaf iff families glue uniquely in the étale topology, and representability of such functors will imply they are sheaves. \end{remark} \begin{example}[A non-sheaf] Consider the following moduli functor: \begin{figure} \centering \resizebox{\columnwidth}{!}{% \begin{tikzpicture} \node {% \(\begin{aligned} F_{{\mathsf{Alg}}}: {\mathsf{Sch}}&\to {\mathsf{Set}}\\ S &\mapsto \left\{ \begin{tikzcd} \mathcal{C} \ar[d] \\ S \end{tikzcd} \right. \begin{aligned} \text{Smooth families of}\\ \text{genus $g$ curves.} \end{aligned} \end{aligned}\) }; \end{tikzpicture} } \end{figure} This is \emph{not} representable by a scheme and not a sheaf. \end{example} \begin{remark} Why care about representability? Suppose there were a scheme \(M\), so \begin{align*} F_{{\mathsf{Alg}}}(S) \simeq \operatorname{Mor}(S, M) .\end{align*} Then taking \(\operatorname{id}_M \in \operatorname{Mor}(M, M)\) should yield a universal family \({\mathcal{U}}\to M\): \includegraphics{figures/2021-09-06_14-50-50.png} Then the points of \(M\) would correspond to isomorphism classes of curves, and every family of curves would be a pullback of this. For any \(S\in{\mathsf{Sch}}\) and a family \({\mathcal{C}}\xrightarrow{f} S\), the fiber \(f^{-1}(s)\in{\mathcal{C}}\) is a curve for any \(s\in S\), so one could define a map \begin{align*} g: S &\to M \\ s &\mapsto [s] ,\end{align*} where we send a curve to its isomorphism class. Then \({\mathcal{C}}\) would fit into a pullback diagram: \begin{center} \begin{tikzcd} {\mathcal{C}}&& {\mathcal{U}}\\ \\ S && M \arrow[from=1-3, to=3-3] \arrow[from=3-1, to=3-3] \arrow[from=1-1, to=3-1] \arrow[dashed, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1jYyJdLFsyLDAsIlxcbWN1Il0sWzAsMiwiUyJdLFsyLDIsIk0iXSxbMSwzXSxbMiwzXSxbMCwyXSxbMCwxLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=}{Link to Diagram} \end{quote} If \(S\) was itself a curve, then \(g: S\to M\) would be a path in \(M\) deforming a base curve. \end{remark} \hypertarget{groupoids-1}{% \subsection{Groupoids}\label{groupoids-1}} \begin{remark} Recall that a \textbf{groupoid} is a category where every morphism is an isomorphism. Morphisms of groupoids are functors, and isomorphisms of groupoids are equivalences of categories. \end{remark} \begin{example}[Groupoid of a set] A basic example is the category of sets where \begin{align*} \operatorname{Mor}(A, B) \coloneqq \begin{cases} \operatorname{id}_A & A=B \\ \emptyset & \text{else}. \end{cases} \end{align*} A similar construction: for any set \(\Sigma\), one can form a groupoid \({\mathcal{C}}_\Sigma\): \begin{itemize} \tightlist \item Object: Elements \(x\in \Sigma\). \item Morphisms: \(\operatorname{id}_x\) \end{itemize} \end{example} \begin{example}[Moduli of curves] Define a category \({ \mathcal{M}_{g} }({\mathbb{C}})\): \begin{itemize} \tightlist \item Objects: smooth projective curves over \({\mathbb{C}}\) of genus \(g\). \item Morphisms: \begin{align*} \operatorname{Mor}(C, C') = \mathop{\mathrm{Isom}}_{{\mathsf{Sch}}_{/ {{\mathbb{C}}}} }(C, C') \subseteq \operatorname{Mor}_{{\mathsf{Sch}}_{/ {{\mathbb{C}}}} }(C, C') .\end{align*} \end{itemize} \end{example} \begin{example}[Equivalence of groupoids] Groupoids are equivalent iff they are equivalent as categories. The following is an example of mapping the quotient groupoid \([C_2/C_4]\) to \({\mathsf{B}}C_2\): \includegraphics{figures/2021-09-06_19-13-21.png} \end{example} \begin{example}[Groupoids equivalent to sets] If a groupoid \({\mathfrak{X}}\) is equivalent to \(\mathsf{C}_{\Sigma}\) for any \(\Sigma \in {\mathsf{Set}}\), we say \({\mathfrak{X}}\) is \textbf{equivalent to a set}. For example, the following groupoid is equivalent to a 2-element set: \includegraphics{figures/2021-09-06_19-15-23.png} \end{example} \begin{example}[Quotient groupoids] For \(G\curvearrowright\Sigma\) a group acting on any set, define the \textbf{quotient groupoid} \([\Sigma/G]\) in the following way: \begin{itemize} \tightlist \item Objects: \(x\in \Sigma\), i.e.~one object for each element of the set \(\Sigma\). \item Morphisms: \(\operatorname{Mor}(x, x') = \left\{{g\in G {~\mathrel{\Big|}~}gx' = x}\right\}\). \end{itemize} \end{example} \begin{exercise}[Groupoids equivalent to sets] Show that \([\Sigma/G]\) is equivalent to a set iff \(G\curvearrowright\Sigma\) is a free action. \end{exercise} \begin{example}[Classifying stacks] For \(\Sigma = \left\{{{\operatorname{pt}}}\right\}\), we obtain \begin{align*} {\mathsf{B}}G \coloneqq[{\operatorname{pt}}/ G] ,\end{align*} where there is one object \({\operatorname{pt}}\) and \(\operatorname{Mor}({\operatorname{pt}}, {\operatorname{pt}}) = G\). \end{example} \begin{example}[from representation stability] Define \({\mathsf{FinSet}}\) to be the category of finite sets where the morphisms are set bijections. Then \({\mathsf{FinSet}}= \displaystyle\coprod_{n\in {\mathbb{Z}}_{\geq 0}} {\mathsf{B}}S_n\) for \(S_n\) the symmetric group. \end{example} \begin{definition}[Fiber products of groupoids] For \(C, D' \to D\) morphisms of groupoids, we can construct their \textbf{fiber product} as the cartesian diagram: \begin{center} \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{C{ \underset{\scriptscriptstyle {D} }{\times} }D'} && {D'} \\ \\ C && D \arrow["f"', from=3-1, to=3-3] \arrow["{g}", from=1-3, to=3-3] \arrow["{{\operatorname{pr}}_1}"', from=1-1, to=3-1] \arrow["{{\operatorname{pr}}_2}", from=1-1, to=1-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwyLCJDIl0sWzIsMiwiRCJdLFsyLDAsIkQnIl0sWzAsMCwiQ1xcZmliZXJwcm9ke0R9RCciLFsyNDAsNjAsNjAsMV1dLFswLDEsImYiLDJdLFsyLDEsImYnIl0sWzMsMCwiXFxwcl8xIiwyXSxbMywyLCJcXHByXzIiXV0=}{Link to Diagram} \end{quote} It can be constructed as the following category: \begin{align*} {\operatorname{Ob}}(C{ \underset{\scriptscriptstyle {D} }{\times} } D') \coloneqq \left\{ \begin{array}{l} (c, d', \alpha) \end{array} \middle\vert \begin{array}{l} c\in C, d'\in D', \\ \\ \alpha: f(c) \xrightarrow{\sim} g(d') \end{array} \right\} \end{align*} \includegraphics{figures/BigDiagram1.pdf} \end{definition} \begin{exercise}[Universal property of pullbacks in Groupoids] Describe the universal property of the pullback in the 2-category of groupoids. \end{exercise} \begin{example}[$G$ is a pullback of $\B G$] \(G\) regarded as a groupoid is the pullback over inclusions of points into \({\mathsf{B}}G\): \begin{center} \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{G} && {\operatorname{pt}}\\ \\ {\operatorname{pt}}&& {{\mathsf{B}}G} \arrow[from=3-1, to=3-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=3-1] \arrow[from=1-3, to=3-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJHIixbMjQwLDYwLDYwLDFdXSxbMiwwLCJcXHB0Il0sWzAsMiwiXFxwdCJdLFsyLDIsIlxcQiBHIl0sWzIsM10sWzAsMiwiIiwwLHsiY29sb3VyIjpbMjQwLDYwLDYwXX1dLFsxLDNdLFswLDEsIiIsMix7ImNvbG91ciI6WzI0MCw2MCw2MF19XSxbMCwzLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=}{Link to Diagram} \end{quote} \end{example} \begin{example}[Orbit/Stabilizer] Let \(G\curvearrowright\Sigma\) and \(x\in \Sigma\), and let \(Gx\) be the orbit and \(G_x\) be the stabilizer. Then there is a morphism of groupoids \(f \in \operatorname{Mor}({\mathsf{B}}G_x, [\Sigma/G])\) inducing a pullback: \begin{center} \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{G_x} & {} & \Sigma \\ \\ {{\mathsf{B}}G_x} && {[\Sigma/G]} \\ {\operatorname{pt}}&& x \arrow["{\exists f}", from=3-1, to=3-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=3-1] \arrow[from=1-3, to=3-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, color={rgb,255:red,92;green,92;blue,214}, draw=none, from=1-1, to=3-3] \arrow[maps to, from=4-1, to=4-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNyxbMCwwLCJHX3giLFsyNDAsNjAsNjAsMV1dLFsxLDBdLFsyLDAsIlxcU2lnbWEiXSxbMCwyLCJcXEIgR194Il0sWzIsMiwiW1xcU2lnbWEvR10iXSxbMCwzLCJcXHB0Il0sWzIsMywieCJdLFszLDQsIlxcZXhpc3RzIGYiXSxbMCwzLCIiLDAseyJjb2xvdXIiOlsyNDAsNjAsNjBdfV0sWzIsNF0sWzAsMiwiIiwyLHsiY29sb3VyIjpbMjQwLDYwLDYwXX1dLFswLDQsIiIsMSx7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dLFs1LDYsIiIsmfx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1hcHMgdG8ifX19XV0=}{Link to Diagram} \end{quote} \end{example} \hypertarget{prestacks}{% \subsection{Prestacks}\label{prestacks}} \begin{remark} Motivation: to specify a moduli functor, we'll need the data of \begin{itemize} \tightlist \item Families over \(S\), \item How to pull back families under morphisms, and \item \emph{How} objects are isomorphic. \end{itemize} As a first attempt, we might try to define a 2-functor \(F: {\mathsf{Sch}}\to {\mathsf{Grpd}}\) between 2-categories, where the latter is the category of groupoids. For this, we need the following data: \begin{itemize} \tightlist \item For all \(S\in {\mathsf{Sch}}\), an assignment of a groupoid \(F(S)\), \item For all morphisms \(f\in \operatorname{Mor}_{{\mathsf{Sch}}}(S, T)\), an assignment of morphisms of groupoids \begin{align*} f^* \in \operatorname{Mor}_{{\mathsf{Grpd}}}(F(T), F(S)) .\end{align*} \item For compositions of morphisms of schemes \(S \xrightarrow{f} T \xrightarrow{g} U\), an isomorphism of functors \begin{align*} \psi_{fg}: g^* \circ f^* \xrightarrow{\sim} (g \circ f)^* .\end{align*} \item Compatibility of these isomorphisms on chains of compositions \(S \to T \to U \to V \to \cdots\). \footnote{This leads to the notion of \textbf{lax} or \textbf{pseudofunctors}.} \end{itemize} This is a lot of data to track, so instead we'll construct a large category \({\mathfrak{X}}\) that encodes all of this, along with a fibration \begin{center} \begin{tikzcd} {\mathfrak{X}}\coloneqq\displaystyle\coprod_{S\in {\mathsf{Sch}}} F(S) \ar[d, "p"] & (S, \alpha \in F(S)) \ar[d, maps to] \\ {\mathsf{Sch}}& S \end{tikzcd} \end{center} Here \(S \in {\mathsf{Sch}}\) and \(F(S) \in {\mathsf{Grpd}}\), so the ``fibers'' above \(S\) are groupoids. \end{remark} \begin{definition}[Prestack] Let \(p:{\mathfrak{X}}\to \mathsf{C}\) be a functor between two 1-categories, so we have the following data: \begin{center} \begin{tikzcd} {\mathfrak{X}}&& a && b & {\in {\operatorname{Ob}}({\mathfrak{X}})} \\ \\ \mathsf{C} && S && T & {\in {\operatorname{Ob}}(\mathsf{C})} \arrow["f", from=3-3, to=3-5] \arrow["p"', from=1-1, to=3-1] \arrow[maps to, from=1-3, to=3-3] \arrow[maps to, from=1-5, to=3-5] \arrow["\alpha", from=1-3, to=1-5] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsOCxbMCwwLCJcXG1meCJdLFswLDIsIlMiXSxbMiwwLCJhIl0sWzQsMCwiYiJdLFsyLDIsIlMiXSxbNCwyLCJUIl0sWzUsMCwiXFxpbiBcXE9iKFxcbWZ4KSJdLFs1LDIsIlxcaW4gXFxPYihTKSJdLFs0LDUsImYiXSxbMCwxLCJwIiwyXSxbMiw0LCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJtYXBzIHRvIn19fV0sWzMsNSwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dLFsyLDNdXQ==}{Link to Diagram} \end{quote} Then \({\mathfrak{X}}, p\) define a \textbf{prestack} over \(\mathsf{C}\) iff \begin{itemize} \tightlist \item Pullbacks exist: for \(S \xrightarrow{f} T\), there exists a (not necessarily unique) map \(f^*b\), sometimes denoted \({ \left.{{b}} \right|_{{f}} }\), yielding a cartesian square: \end{itemize} \begin{center} \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{\exists a} && b \\ \\ S && T \arrow[from=3-1, to=3-3] \arrow[from=1-3, to=3-3] \arrow["{f^* b = { \left.{{b}} \right|_{{f}} }}", color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=1-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=3-1] \arrow["\lrcorner"{anchor=center, pos=0.125}, color={rgb,255:red,92;green,92;blue,214}, draw=none, from=1-1, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXGV4aXN0cyBhIixbMjQwLDYwLDYwLDFdXSxbMiwwLCJiIl0sWzAsMiwiUyJdLFsyLDIsIlQiXSxbMiwzXSxbMSwzXSxbMCwxLCJmXiogYiA9IFxccm97Yn17Zn0iLDAseyJjb2xvdXIiOlsyNDAsNjAsNjBdLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19LFsyNDAsNjAsNjAsMV1dLFswLDIsIiIsMCx7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDMsIiIsMSx7ImNvbG91ciI6WzI0MCw2MCw2MF0sInN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==}{Link to Diagram} \end{quote} \begin{itemize} \tightlist \item A universal property making \({\mathfrak{X}}\) a \emph{fibered category}: every arrow in \({\mathfrak{X}}\) is a pullback, so there are always lifts of the following form: \end{itemize} \begin{center} \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{a} && b && c \\ \\ R && S && R \arrow[from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow[maps to, from=1-3, to=3-3] \arrow[maps to, from=1-5, to=3-5] \arrow[from=1-3, to=1-5] \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, maps to, from=1-1, to=3-1] \arrow["{\exists !}", color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=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}{Link to Diagram} \end{quote} \end{definition} \begin{slogan} An alternative definition: a prestack is a category \emph{fibered in groupoids}. \end{slogan} \begin{warnings} We often conflate \({\mathfrak{X}}\) and the functor \({\mathfrak{X}}\xrightarrow{p} S\), and don't spell out the composition law in \({\mathfrak{X}}\). Moreover, we write \(f^*b\) or \({ \left.{{b}} \right|_{{f}} }\) for a \emph{choice} of a pullback. \end{warnings} \begin{definition}[Fiber Categories] For \(p: {\mathfrak{X}}\to \mathsf{C}\) a functor and \(S\in {\operatorname{Ob}}(\mathsf{C})\) any fixed object, the associated \textbf{fiber category over \(S\)}, denoted \({\mathfrak{X}}(S)\), is the subcategory of \({\mathfrak{X}}\) defined by: \begin{itemize} \tightlist \item Objects: \(a\in {\operatorname{Ob}}({\mathfrak{X}})\) such that \(a \xrightarrow{p} S\), \item Morphisms: \(\operatorname{Mor}(a, a')\) are morphisms \(f\in \operatorname{Mor}_{{\mathfrak{X}}}(a, a')\) over \(\operatorname{id}_S\): \end{itemize} \begin{center} \begin{tikzcd} a \ar[rd, ""] \ar[rr, "f"] & & a' \ar[ld, ""] \\ & S & \end{tikzcd} \end{center} \end{definition} \begin{remark} We can now equivalently define presheaves as categories fibered in sets. \end{remark} \begin{exercise}[Justifying 'category fibered in groupoids'] Show that if \({\mathfrak{X}}\to \mathsf{C}\) is a prestack, then for all \(S\in \mathsf{C}\), all maps in \({\mathfrak{X}}(S)\) are invertible. Conclude that the fiber categories \({\mathfrak{X}}(S)\) are all groupoids. \end{exercise} \begin{example}[Presheaves] Every presheaf forms a prestack. Let \(F \in \underset{ \mathsf{pre} } {\mathsf{Sh} }({\mathsf{Sch}}, {\mathsf{Set}})\) be a presheaf of sets, and define \({\mathfrak{X}}_F\) as the following category: \begin{itemize} \tightlist \item Objects: Pairs \((S, a \in F(S))\) where \(S\in {\mathsf{Sch}}\) and \(F(s) \in {\mathsf{Set}}\). \item Morphisms: \begin{align*} \operatorname{Mor}( (S, a), (T, b) ) \coloneqq\left\{{ S \xrightarrow{f} T {~\mathrel{\Big|}~}a = f^* b}\right\} .\end{align*} \end{itemize} Note that we'll often conflate \(F\) and \({\mathfrak{X}}_F\). This yields the fibration \begin{center} \begin{tikzcd} {{\mathfrak{X}}_F} && {(S, a)} \\ \\ {\mathsf{Sch}}&& S \arrow[from=1-1, to=3-1, "p"] \arrow[maps to, from=1-3, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1jeF9GIl0sWzAsMiwiXFxTY2giXSxbMiwwLCIoUywgYSkiXSxbMiwyLCJTIl0sWzAsMV0sWzIsMywiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dXQ==}{Link to Diagram} \end{quote} \end{example} \begin{example}[Schemes] For \(X\in {\mathsf{Sch}}\), take its Yoneda functor \(h_X: {\mathsf{Sch}}\to {\mathsf{Set}}\). Then define the category \({\mathfrak{X}}_X\): \begin{itemize} \tightlist \item Objects: Morphisms \(S\to X\) of schemes. \item Morphisms: \(\operatorname{Mor}(S\to X, T\to X)\) are morphisms over \(X\): \end{itemize} \begin{center} \begin{tikzcd} S \ar[rd, ""] \ar[rr, ""] & & T \ar[ld, ""] \\ & X & \end{tikzcd} \end{center} This yields the fibration \begin{center} \begin{tikzcd} {{\mathfrak{X}}_X} && {(S\to X)} \\ \\ {\mathsf{Sch}}&& S \arrow[from=1-1, to=3-1, "p"] \arrow[maps to, from=1-3, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1jeF9GIl0sWzAsMiwiXFxTY2giXSxbMiwwLCIoUywgYSkiXSxbMiwyLCJTIl0sWzAsMV0sWzIsMywiIiwwLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibWFwcyB0byJ9fX1dXQ==}{Link to Diagram} \end{quote} \end{example} \begin{example}[Moduli of curves] Define \({ \mathcal{M}_{g} }\) as the following category: \begin{itemize} \tightlist \item Objects: families \({\mathcal{C}}\to S\) of smooth genus \(g\) curves, \item Morphisms: \(\operatorname{Mor}({\mathcal{C}}\to S, {\mathcal{C}}'\to S')\): cartesian squares \end{itemize} \begin{center} \begin{tikzcd} {\mathcal{C}}&& {{\mathcal{C}}'} \\ \\ S && S' \arrow[from=3-1, to=3-3] \arrow[from=1-1, to=3-1] \arrow[from=1-3, to=3-3] \arrow[from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJDIl0sWzAsMiwiUyJdLFsyLDIsIlMiXSxbMiwwLCJDJyJdLFsxLDIsIlxcaWRfUyIsMl0sWzAsMV0sWzMsMl0sWzAsM10sWzAsMiwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d}{Link to Diagram} \end{quote} This yields a fibration \begin{center} \begin{tikzcd} {{ \mathcal{M}_{g} }} && {({\mathcal{C}}\to S)} \\ \\ {\mathsf{Sch}}&& S \arrow[from=1-1, to=3-1] \arrow[maps to, from=1-3, to=3-3] \end{tikzcd} \end{center} \end{example} \begin{example}[Bundles] For \(C\) a smooth connected projective curve over \(k\) a field, define \({\mathsf{Bun}}(C)\) as the following category: \begin{itemize} \tightlist \item Objects: pairs \((S, F)\) where \(F\) is a vector bundle over \(C\times S\). \item Morphisms: \begin{align*} \operatorname{Mor}((S, F), (S', F')) = \left\{ \begin{array}{l} f\in \operatorname{Mor}_{{\mathsf{Sch}}}(S, S') \\ \text{and a chosen isomorphism} \\ \alpha: (f\times \operatorname{id})^* \circ F' \xrightarrow{\sim} F \end{array} \right\} .\end{align*} \end{itemize} \begin{remark} A technical point: the choice of pushforward here is not necessarily canonical. However, as part of the data, one can take morphisms \(F' \to (f\times\operatorname{id})_* \circ F\) such that the adjunction yields an isomorphism. \end{remark} \end{example} \begin{example}[Quotient prestack] Let \(X_{/ {S}} \in {\mathsf{Grp}}{\mathsf{Sch}}\) where \(G\curvearrowright X\). Then define a category \([X/G]^{\mathsf{pre}}\): \begin{itemize} \tightlist \item Objects: Morphisms over \(\operatorname{id}_S\): \end{itemize} \begin{center} \begin{tikzcd} T \ar[rd, ""] \ar[rr, ""] & & X \ar[ld, ""] \\ & S & \end{tikzcd} \end{center} \begin{itemize} \tightlist \item Morphisms: \end{itemize} \begin{align*} \operatorname{Mor}(T\to X, T'\to X) \coloneqq \left\{ \begin{array}{l} T\to T' \end{array} \,\, \middle\vert \begin{array}{l} (T \to T' \to X ) = g(T \to X) \\ g\in G(T) \\ G(T) \curvearrowright X(T) \end{array} \right\} .\end{align*} \end{example} \begin{remark} A group scheme can alternatively be thought of as a functor with a factorization through \({\mathsf{Grp}}\). \end{remark} \begin{exercise}[Quotient prestacks and quotient groupoids] Show that for \(T\in {\mathsf{Sch}}\), there is an equivalence \begin{align*} [X/G]^{\mathsf{pre}}(T) \xrightarrow{\sim} [X(T) / G(T)] ,\end{align*} where the left-hand side is a fibered category over \(T\) and the right-hand side is a quotient groupoid. \end{exercise} \hypertarget{morphisms-of-prestacks}{% \subsubsection{Morphisms of Prestacks}\label{morphisms-of-prestacks}} \begin{definition}[Morphisms of prestacks] A \textbf{morphism of prestacks} is a functor \({\mathfrak{X}}\xrightarrow{f} {\mathfrak{X}}'\) such that there is a (strictly) commutative triangle \begin{center} \begin{tikzcd} {\mathfrak{X}}&& {\mathfrak{X}}' \\ \\ & \mathsf{C} \arrow["f", from=1-1, to=1-3] \arrow["{p_X}"', from=1-1, to=3-2] \arrow["{p_Y}", from=1-3, to=3-2] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXG1jeCJdLFsyLDAsIlxcbWN5Il0sWzEsMiwiXFxTY2giXSxbMCwxLCJmIl0sWzAsMiwicF9YIiwyXSxbMSwyLCJwX1kiXV0=}{Link to Diagram} \end{quote} Here we require a strict equality \(p_X(a) = p_Y(f(a))\) for any \(a\in {\mathfrak{X}}\) A \textbf{2-morphism} \(\alpha\) between morphisms \(f, g\) is a natural transformation: \begin{center} \begin{tikzcd} {\mathfrak{X}}&&& {\mathfrak{X}}' \arrow[""{name=0, anchor=center, inner sep=0}, "f", curve={height=-30pt}, from=1-1, to=1-4] \arrow[""{name=1, anchor=center, inner sep=0}, "g"', curve={height=30pt}, from=1-1, to=1-4] \arrow["\alpha", shorten <=8pt, shorten >=8pt, Rightarrow, from=0, to=1] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsMixbMCwwLCJcXG1meCJdLFszLDAsIlxcbWZ5Il0sWzAsMSwiZiIsMCx7ImN1cnZlIjotNX1dLFswLDEsImciLDIseyJjdXJ2ZSI6NX1dLFsyLDMsIlxcYWxwaGEiLDAseyJzaG9ydGVuIjp7InNvdXJjZSI6MjAsInRhcmdldCI6MjB9fV1d}{Link to Diagram} \end{quote} such that for all \(a\in {\mathfrak{X}}\), the following triangle \(\alpha_a\in \operatorname{Mor}_{{\mathfrak{X}}'}(f(a), g(a))\) is a morphisms over \(\operatorname{id}_S\) for any \(S\in \mathsf{C}\): \begin{center} \begin{tikzcd} f(a) \ar[rd, ""] \ar[rr, ""] & & g(a) \ar[ld, ""] \\ & S & \end{tikzcd} \end{center} We define a category \(\operatorname{Mor}({\mathfrak{X}}, {\mathfrak{X}}')\) by: \begin{itemize} \tightlist \item Objects: morphisms of prestacks. \item Morphisms: 2-morphisms of prestacks. \end{itemize} \end{definition} \begin{exercise}[?] Show that \(\operatorname{Mor}({\mathfrak{X}}, {\mathfrak{X}}')\) is a groupoid. \end{exercise} \begin{definition}[2-commutativity] A diagram is \textbf{2-commutative} iff there exists a 2-morphism \(\alpha: g \circ f' \xrightarrow{\sim} f\circ g'\) which is an isomorphism: \begin{center} \begin{tikzcd} {{\mathfrak{X}}{ \underset{\scriptscriptstyle {{\mathfrak{X}}'} }{\times} } {\mathfrak{X}}''} && {{\mathfrak{X}}''} \\ \\ {\mathfrak{X}}&& {\mathfrak{X}}' \arrow["g", from=1-3, to=3-3] \arrow["f"', from=3-1, to=3-3] \arrow["{g'}"', from=1-1, to=3-1] \arrow["{f'}", from=1-1, to=1-3] \arrow["\alpha", Rightarrow, from=3-1, to=1-3] \end{tikzcd} \end{center} \begin{quote} \href{https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXG1meCBcXGZpYmVycHJvZHtcXG1meX0gXFxtZnknIl0sWzIsMCwiXFxtZnknIl0sWzIsMiwiXFxtZnkiXSxbMCwyLCJcXG1meCJdLFsxLDIsImciXSxbMywyLCJmIiwyXSxbMCwzLCJnJyIsMl0sWzAsMSwiZiciXSxbMywxLCJcXGFscGhhIiwwLHsibGV2ZWwiOjJ9XV0=}{Link to Diagram} \end{quote} \end{definition} \begin{definition}[Isomorphisms of prestacks] An \textbf{isomorphism} of prestacks is a 1-isomorphism of prestacks \(f: {\mathfrak{X}}\to {\mathfrak{X}}'\) along with 2-isomorphisms \(g\circ f \xrightarrow{\sim} \operatorname{id}_{{\mathfrak{X}}}\) and \(f\circ g \xrightarrow{\sim} \operatorname{id}_{{\mathfrak{X}}'}\). \end{definition} \begin{exercise}[Isomorphisms of prestacks can be checked on fibers] Show that \({\mathfrak{X}}\to {\mathfrak{X}}'\) is an isomorphism iff \({\mathfrak{X}}(S) \xrightarrow{\sim} {\mathfrak{X}}'(S)\) is an isomorphism on all fibers. \end{exercise} \begin{proposition}[2-Yoneda] If \({\mathfrak{X}}\in {\underset{ \mathsf{pre} } {\mathsf{St} } } {}_{/ {\mathsf{C}}}\) is a prestack over \(\mathsf{C}\), then for any \(S\in {\operatorname{Ob}}(\mathsf{C})\), there is an equivalence of categories induced by the following functor: \begin{align*} \operatorname{Mor}(S, {\mathfrak{X}}) & \xrightarrow{\sim} {\mathfrak{X}}(S) \\ f &\mapsto f_S(\operatorname{id}_S ) .\end{align*} \end{proposition} \begin{remark} For \(S\in {\mathsf{Sch}}\), view \(S\) as a prestack and consider a morphism \(f:S\to {\mathfrak{X}}\). How is this specified? For all \(T\in {\mathsf{Sch}}\), the objects of \(S_{/ {T}}\) are morphisms \begin{align*} f_T: \operatorname{Mor}(T, S) \to {\mathfrak{X}}(T) \end{align*} and if \(T=S\) this sends \(\operatorname{id}_S\) to \(f_S(\operatorname{id}_S)\in {\mathfrak{X}}(S)\). What is the inverse? For \(a\in {\mathfrak{X}}(S)\) and for each \(T \xrightarrow{g} S\), \textbf{choose} a pullback \(g^* a\). Then define \(f: S \to {\mathfrak{X}}\) by \begin{align*} f_T: \operatorname{Mor}(T, S) &\to {\mathfrak{X}}(T) \\ g &\mapsto g^* a .\end{align*} \end{remark} \begin{exercise}[?] Define what this equivalence should do on morphisms. \end{exercise} \begin{remark} Next time: fiber products of prestacks. \end{remark} \addsec{ToDos} \listoftodos[List of Todos] \cleardoublepage % Hook into amsthm environments to list them. \addsec{Definitions} \renewcommand{\listtheoremname}{} \listoftheorems[ignoreall,show={definition}, numwidth=3.5em] \cleardoublepage \addsec{Theorems} \renewcommand{\listtheoremname}{} \listoftheorems[ignoreall,show={theorem,proposition}, numwidth=3.5em] \cleardoublepage \addsec{Exercises} \renewcommand{\listtheoremname}{} \listoftheorems[ignoreall,show={exercise}, numwidth=3.5em] \cleardoublepage \addsec{Figures} \listoffigures \cleardoublepage \printbibliography[title=Bibliography] \end{document}