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\begin{document}

\title{Stacks}

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\tableofcontents

\section{Introduction}
\label{section-introduction}

\noindent
Stacks are defined in this document. See \cite{DM}.

\section{Definition}
\label{section-definition}

\noindent
Let $\mathcal{C}$ be a site. The $2$-category of stacks over
$\mathcal{C}$ will be a full sub-$2$-category of the $2$-category
of categories over $\mathcal{C}$, see Categories,
\hyperref[categories-definition-categories-over-C]%
{Definition~\ref*{categories-definition-categories-over-C}}. 
Thus a stack will be given by a functor of categories
$p : \mathcal{S} \to \mathcal{C}$ which satisfies the following
conditions:
\begin{enumerate}
\item $p : \mathcal{S} \to \mathcal{C}$ is a category fibred
in groupoids, see Categories,
\hyperref[categories-definition-fibred-groupoids]%
{Definition~\ref*{categories-definition-fibred-groupoids}},
\item descent for morphisms holds, and
\item descent data for objects are effective.
\end{enumerate}

\subsection{Explanation}
\label{subsection-definition-explanation}

\noindent
To explain this, we choose a collection of pullback functors as in
Categories, \hyperref[categories-lemma-fibred-groupoids]%
{Lemma~\ref*{categories-lemma-fibred-groupoids}}. Another approach is to use
Categories, \hyperref[categories-lemma-fibred-strict]%
{Lemma~\ref*{categories-lemma-fibred-strict}}.

\smallskip\noindent
First, suppose that $x,y\in \text{Ob}(\mathcal{S}_U)$ are
objects in the fibre category over $U$. We are going to define
a contravariant functor
$$
\text{Isom}(x,y) : \mathcal{C}/U \longrightarrow \text{Sets}.
$$
In other words this will be a presheaf on $\mathcal{C}/U$, see
Sites, \hyperref[sites-definition-presheaf]%
{Definition~\ref*{sites-definition-presheaf}}. Namely, for 
$f : V \to U$ we set 
$$
\text{Isom}(x,y)(f:V\to U) = 
\text{Mor}_{\mathcal{S}_V}(f^\ast x, f^\ast y).
$$
We also have to define the restriction map corresponding to a
morphism $(g,\text{id}_U) : (f' : V' \to U)  \to (f : V\to U)$ 
in $\mathcal{C}/U$ (in other words $f' = f \circ g$). This will be a map
$$
\text{Mor}_{\mathcal{S}_V}(f^\ast x, f^\ast y) \longrightarrow
\text{Mor}_{\mathcal{S}_{V'}}({f'}^\ast x, {f'}^\ast y).
$$
This map will basically be $g^\ast$, except that this transforms
an element $\phi$ of the left hand side into an element 
$g^\ast \phi$
of $\text{Mor}_{\mathcal{S}_{V'}}(g^\ast f^\ast x, g^\ast f^\ast y)$.
At this point we use the transformation $t$ of 
Categories, \hyperref[categories-lemma-fibred-groupoids]%
{Lemma~\ref*{categories-lemma-fibred-groupoids}}.
In a formula, the restriction map maps $\phi$ to
$$
t_y \circ g^\ast \phi \circ (t_x)^{-1}.
$$

\begin{lemma}
\label{lemma-painfull}
This actually does give a presheaf.
\end{lemma}

\begin{proof}
Let $(g',\text{id}_U) : (f' : V' \to U)  \to (f : V\to U)$ and
$(g'',\text{id}_U) : (f'' : V'' \to U)  \to (f' : V' \to U)$ be morphisms in
$\mathcal{C}/U$, and let $\phi \in {Mor}_{\mathcal{S}_V}(f^\ast x, f^\ast y)$.
It suffices to show that 
$$
[(\text{Isom}(x,y))(g' \circ g'')](\phi) = 
[(\text{Isom}(x,y))(g'')]([(\text{Isom}(x,y))(g')](\phi))
$$
By \hyperref[categories-lemma-fibred-groupoids]%
{Lemma~\ref*{categories-lemma-fibred-groupoids}} there are pullback functors 
$r: (g' \circ g'')^\ast f'^\ast \longrightarrow f''^\ast$, 
$t': g'^\ast f^\ast \longrightarrow f'^\ast$, 
$t'': g''^\ast f'^\ast \longrightarrow f''^\ast$ and
$u: g''^\ast g'^\ast \longrightarrow (g' \circ g'')^\ast$.  It follows from the
uniqueness part of the lemma that 
$t''_{y} \circ g''^\ast t'_{y} = r_{y} \circ u_{f^\ast y}$ and 
$g''^\ast t'^{-1}_{x} \circ t''^{-1}_{x} = u^{-1}_{f^\ast x} \circ r^{-1}_{x}$.
We now have that 
\begin{eqnarray*}
[(\text{Isom}(x,y))(g' \circ g'')](\phi) & = & 
r_{y} \circ (g' \circ g'')^\ast \phi \circ r^{-1}_{x}\\
& = & r_{y} \circ u_{f^\ast y} \circ 
g''^\ast g'^\ast \phi \circ u^{-1}_{f^\ast x} \circ r^{-1}_{x} \\
& = & t''_{y} \circ g''^\ast t'_{y} \circ 
g''^\ast g'^\ast \phi \circ g''^\ast t'^{-1}_{x} \circ t''^{-1}_{x} \\
& = & t''_{y} \circ g''^\ast (t'_{y} \circ 
g'^\ast \phi \circ t'^{-1}_{x}) \circ t''^{-1}_{x} \\
& = & [(\text{Isom}(x,y))(g'')]([(\text{Isom}(x,y))(g')](\phi)).
\end{eqnarray*}

\noindent
Alternatively we can argue with
\hyperref[categories-lemma-fibred-strict]{Lemma~\ref*{categories-lemma-fibred-strict}}
which says that $\mathcal{S}\to\mathcal{C}$ is equivalent to the
category assoicated to a contravariant function $F\colon
\mathcal{C}\to \text{Groupoids}$.  Then $t_y$ is the identity
transformation for all $y$ so the restriction map is $g\mapsto g^*\phi
= F(g)\phi$ which is clearly functorial making $\text{Isom}(x,y)$ a
presheaf.
\end{proof}

\smallskip\noindent
OK, so the second condition listed in Section \ref{section-definition}
is simply the condition that $\text{Isom}(x,y)$ is a sheaf on the site
$\mathcal{C}/U$! 

\smallskip\noindent
In order to explain the meaning of the third condition, we must define a
descent datum. First, we introduce some notation. Suppose that
$\{f_i : U_i \to U\}_{i\in I}$ is a covering in the site $\mathcal{C}$.
We will be looking at fibre products $U_{ij} = U_i \times_U U_j$ and
$U_{ijk} = U_i \times_U U_j \times_U U_k$. It is very important to allow
$i=j$ and even $i=j=k$ in the following. The projection maps from
$U_{ij}$ to $U_i$, resp.\ $U_j$ are denoted $\text{pr}_1$, resp.\ 
$\text{pr}_2$. The projection maps from $U_{ijk}$ to the twofold fibre
products are $\text{pr}_{12} : U_{ijk} \to U_{ij}$, 
$\text{pr}_{13} : U_{ijk} \to U_{ik}$, and  
$\text{pr}_{23} : U_{ijk} \to U_{jk}$. The projection maps from 
$U_{ijk}$ to $U_{i}$, $U_{j}$, and $U_{k}$ are $\text{pr}_1$, 
$\text{pr}_2$, and $\text{pr}_3$, respectively. 

\smallskip\noindent
This notation is potentially ambiguous, but it is standard in the
literature. For an example of the ambiguity: note the relations
$\text{pr}_1 = \text{pr}_{1} \circ \text{pr}_{12} =
\text{pr}_1 \circ \text{pr}_{13}$, $\text{pr}_2 =
\text{pr}_2 \circ \text{pr}_{12} = \text{pr}_1 \circ \text{pr}_{23}$,
and $\text{pr}_3 = \text{pr}_2 \circ \text{pr}_{13} =
\text{pr}_3 \circ \text{pr}_{23}$.

\smallskip\noindent
Let $x_i \in \text{Ob}(\mathcal{S}_{U_i})$, $i\in I$ be a collection of
objects, and let
$$
\phi_{ij} : \text{pr}_{1}^\ast x_i \longrightarrow
\text{pr}_{2}^\ast x_j \qquad (i,j \in I)
$$
be a collection of morphisms in the fibre categories
$\mathcal{S}_{U_{ij}}$. In the definition that follows we will
identify, for instance, $\text{pr}_1^\ast x_i$ on $U_{ijk}$ with both 
$\text{pr}_{12}^\ast \text{pr}_{1}^\ast x_i$ and 
$\text{pr}_{13}^\ast \text{pr}_{1}^\ast x_i$.

\begin{definition}
\label{definition-descent-data}
The collection $\{x_i, \phi_{ij}\}$ is a descent datum if
the following cocycle condition is satisfied: For every
triple $(i,j,k)\in I^3$ the diagram
$$
\xymatrix{
\text{pr}_1^\ast x_i 
	\ar[rr]^{\text{pr}_{13}^\ast \phi_{ik}}
	\ar[rd]_{\text{pr}_{12}^\ast \phi_{ij}}
& & 
\text{pr}_3^\ast x_k \\
& \text{pr}_2^\ast x_j \ar[ru]_{\text{pr}_{23}^\ast \phi_{jk}}
}
$$
in the fibre category $\mathcal{S}_{U_{ijk}}$ is commutative.
\end{definition}

\begin{remarks}
\label{remarks-definition-descent-datum}
Two remarks about this definition are in order.  

\smallskip\noindent
(1) There is a diagonal morphism $\Delta_i : U_i \to U_{ii}$. We can pull back
$\phi_{ii}$ via this morphism to get an automorphism 
${\Delta_i}^\ast \phi_{ii} \in \text{Aut}_{U_i}(x_i)$.
On pulling back the cocycle condition for the triple $(i,i,i)$ 
by $\Delta_{123} : U_i \to U_{iii}$ we deduce that
${\Delta_i}^\ast \phi_{ii} \circ {\Delta_i}^\ast \phi_{ii} =
{\Delta_i}^\ast \phi_{ii}$; thus ${\Delta_i}^\ast \phi_{ii} =
\text{id}_{x_i}$.

\smallskip\noindent
(2) There is a morphism
$\Delta_{13}: U_{ij} \to U_{iji}$ and we can pull back the
cocycle condition for the triple $(i,j,i)$ to get the
identity $(\sigma^\ast \phi_{ji}) \circ \phi_{ij} = 
\text{id}_{\text{pr}_{1/2}^\ast x_i}$, where $\sigma: U_{ij} \to U_{ji}$ is the
switching morphism.
\end{remarks}

\smallskip\noindent
A morphism of descent data 
$\alpha : \{x_i, \phi_{ij}\} \rightarrow \{y_i, \psi_{ij}\}$ is given by a 
collection of morphisms $\alpha_i : x_i \to y_i$ in the fibre category
over $U_i$ such that the following diagrams
$$
\xymatrix{
\text{pr}_1^\ast x_i
	\ar[r]^{\text{pr}_1^\ast \alpha_i}
	\ar[d]_{\phi_{ij}}
&
\text{pr}_1^\ast y_i
	\ar[d]^{\psi_{ij}}
\\
\text{pr}_2^\ast x_j
	\ar[r]_{\text{pr}_2^\ast \alpha_j}
&
\text{pr}_2^\ast y_j
}
$$
commute. Note that every morphism of descent data is an isomorphism.

\smallskip\noindent
An object $x \in \text{Ob}(\mathcal{S}_U)$ gives rise to a descent
datum in the following manner. First we set $x_i = f_i^\ast x$.
Then we set $\phi_{ij} = {t_{j}}^{-1} \circ t_{i}$, where 
$t_{i}: \text{pr}_1^\ast f_i^\ast x \to (f_i \circ \text{pr}_1)^\ast x$
and $t_{j}: \text{pr}_2^\ast f_j^\ast x \to (f_j \circ \text{pr}_2)^\ast x$ 
are the canonical isomorphisms guaranteed by Categories, 
\hyperref[categories-lemma-fibred-groupoids]%
{Lemma~\ref*{categories-lemma-fibred-groupoids}}.
The lemma below shows this is a descent datum; we will call
this the {\it canonical descent datum} associated to $x$.

\begin{lemma}
\label{lemma-trivial-cocycle}
The cocycle condition holds for the datum described above.
\end{lemma}

\begin{proof}
First, note that $f_i \circ\text{pr}_1 = f_j \circ \text{pr}_2=
f_k\circ \text{pr}_3$. Then note that $\text{pr}_{13}^\ast \phi_{ik}$,
$\text{pr}_{23}^\ast \phi_{jk}$ and $\text{pr}_{12}^\ast \phi_{ij}$ factor
uniquely through $(f_i\circ\text{pr}_1)^\ast x = 
(f_j \circ\text{pr}_2)^\ast x = (f_k \circ\text{pr}_3)^\ast x$
by Lemma 3.1.3.
\end{proof}

\begin{definition}
\label{definition-effective-descent-datum}
A descent datum $\{x_i,\phi_{ij}\}$ is said to be effective
when there exists an $x\in \text{Ob}(\mathcal{S}_U)$ 
such that $\{x_i,\phi_{ij}\}$ is isomorphic to the
canonical descent datum associated to $x$.
\end{definition}

\noindent
At this point we are ready to give the definition of a
stack. 

\begin{definition}
\label{defintion-stack}
A stack (in groupoids) over a site $\mathcal{C}$ is a 
category $p : \mathcal{S} \to \mathcal{C}$ over $\mathcal{C}$
such that
\begin{enumerate}
\item $p : \mathcal{S} \to \mathcal{C}$ is a category fibred 
in groupoids over $\mathcal{C}$, 
\item for all $U \in \text{Ob}(\mathcal{C})$ and all
$x,y\in \text{Ob}(\mathcal{S}_U)$ the presheaf
$\text{Isom}(x,y)$ is a sheaf on $\mathcal{C}/U$, and
\item for all coverings $\mathcal{U}=\{U_i \to U\}$ in $\mathcal{C}$, 
all descent data $\{x_i,\phi_{ij}\}$ for $\mathcal{U}$ are effective.
\end{enumerate}
\end{definition}

\noindent
Usually the hardest part to check is the third condition.

\begin{lemma}
\label{lemma-2-product-stacks}
Suppose that $f : \mathcal{X} \to \mathcal{S}$ and
$g : \mathcal{Y} \to \mathcal{S}$ are morphisms of stacks over
$\mathcal{C}$. Let $\mathcal{X} \times_\mathcal{S}\mathcal{Y}$, $p$, $q$,
$\psi$ be the explicit 2-fibre product of $f$ and $g$ in the 2-category
of categories over $\mathcal{C}$ described in
\hyperref[categories-lemma-2-product-categories-over-C]%
{Lemma~\ref*{categories-lemma-2-product-categories-over-C}}.
Then $\mathcal{X} \times_\mathcal{S}\mathcal{Y}$ is a stack. In particular
the 2-category of stacks over $\mathcal{C}$ has 2-fibre products (and
they are as described in 
\hyperref[categories-lemma-2-product-categories-over-C]%
{Lemma~\ref*{categories-lemma-2-product-categories-over-C}}).
\end{lemma}

\begin{proof}
FIXME.
\end{proof}

\subsection{Examples}
\label{subsection-examples}

\noindent
FIXME: Here we need lots of examples.

\noindent
FIXME: To be continued.

\smallskip\noindent
To continue reading, 
\begin{enumerate}

\item visit the next section: Stacks and Groupoids,
\autoref{stacks-groupoids-section-introduction}, or 

\item go back to the
table of contents: \url{index.html#contents}.

\end{enumerate}

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