\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
}
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  \ifdim\Gin@nat@height>0.9\textheight
    0.9\textheight
  \else
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  \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}
