\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}