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\begin{document}

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\section*{stack}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory}

[[!include (infinity,1)-topos - contents]]

\hypertarget{locality_and_descent}{}\paragraph*{{Locality and descent}}\label{locality_and_descent}

[[!include descent and locality - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{provisional_discussion}{Provisional discussion}\dotfill \pageref*{provisional_discussion} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The term \emph{stack}, is a traditional synonym for \emph{[[2-sheaf]]} or often just \emph{[[(2,1)-sheaf]]} (see there for more details).

It is also often used more restrictively as a synonym for \emph{[[(2,1)-sheaf]].}

This is part of a whole hierarchy of [[higher category theory|higher categorical]] generalizations of the notion of \emph{[[sheaf]]}. A [[(2,1)-sheaf]] / stack is equivalently a [[1-truncated]] [[(∞,1)-sheaf]]/[[∞-stack]].

Generally, then, \emph{[[n-stack]]} is a synonym for \emph{[[n-sheaf|(n+1)-sheaf]]}, or more restrictively for \emph{[[(n,1)-sheaf|(n+1,1)-sheaf]]}.

More concretely this means that a 1-stack on a [[site]] $S$ (or more generally [[(2,1)-site]] or even [[(2,2)-site]] $S$) is

\begin{itemize}%
\item a ([[pseudofunctor|pseudo]]-)[[functor]] from $S^{op}$ to the 2-category [[Cat]] of categories,


\item that satisfies [[descent]] for all covers.



\end{itemize}
If the pseudofunctor takes values in the 2-subcategory [[Grpd]]$\subset Cat$ of groupoids, the stack is sometimes referred to as a stack of groupoids. This is the more commonly occurring case so the term `stack' has come to mean `stack of groupoids' in much of the literature.

In some circles the notion of a stack as a generalized [[groupoid]] is almost more familiar than the notion of sheaf as a [[space and quantity|generalized space]]. For instance [[differentiable stacks]] have attracted much attention in the study of [[Lie groupoids]] and [[orbifolds]], while [[diffeological space]]s are only beginning to be investigated more in [[Lie theory]]. Groupoid stacks are closely related to internal groupoids \href{https://mathoverflow.net/questions/93948/link-between-internal-groupoids-and-stacks-on-a-topos}{MO}.

An [[algebraic stack]], [[differentiable stack]] etc. is a stack over a site of schemes or differentiable manifolds with additional representability conditions.

\hypertarget{provisional_discussion}{}\subsection*{{Provisional discussion}}\label{provisional_discussion}

The following is ``provisional'' material on stacks that [[Todd Trimble]] wrote in the course of a discussion with [[Urs Schreiber|Urs]]. Somebody should turn this here into a coherent entry on stacks.


\vspace{.5em} \hrule \vspace{.5em}
(Todd speaking.) I don't really speak ``stacks'', but in an effort to build a bridge between sheaves and stacks, I'll write down what I thought I understood, and ask someone such as Urs to come in and check. (Warning: I'm treating this edit box almost as a sandbox, in that what I say below is all a bit provisional until we get some discussion going.)

Hi Todd, thanks for this. I started making some remarks on the relation between descent $\infty$-categories and pseudofunctors from [[covers]] regarded as [[sieves]] (hence as presheaves) at [[descent and codescent]] in the section titled \emph{Descent in terms of pseudo-functors}.

At the simplest level, let $C$ be a category. As we know, a presheaf on $C$ is just a functor $X: C^{op} \to Set$.

Now let's categorify just once: regard a category $C$ as a bicategory whose local hom-categories are discrete. What I'll call a ``pre-stack'' is then a homomorphism of bicategories $X: C^{op} \to Cat$. Here I'm following Street's terminology: a homomorphism of bicategories is the ``pseudo'' version of a weak map of bicategories, as opposed to the ``lax'' version. So, we have given coherent isomorphisms $X(f)X(g) \to X(f g)$, and so on.

Now suppose that $C$ also comes equipped with a topology $J$, and let $F$ be a $J$-covering sieve for $c$, so that in particular it's a subfunctor $i: F \hookrightarrow \hom(-, c)$. We want to build a (truncated) simplicial object out of this, and to this end I'll use some yoga which was basically developed in my Cafe post \href{http://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html}{on the bar construction} perhaps this may go partway to addressing your most recent \href{http://golem.ph.utexas.edu/category/2007/05/on_the_bar_construction.html#c021027}{query} there, Urs.

Namely, there is a canonical way of presenting $F$ as a colimit of representables. Officially, it's given by a coend formula, but it's probably more illuminating to think of it in terms of tensor products over $C$:

\begin{displaymath}
\hom_C(-, -) \otimes_C F(-) \cong F(-)
\end{displaymath}
In the long-winded version, this says that $F$ is the coequalizer of a diagram having the form

\begin{displaymath}
\sum_{c, d} \hom_C(-, c) \times \hom_C(c, d) \times F(d) \stackrel{\to}{\to} \sum_c \hom_C(-, c) \times F(c) \to F(-)
\end{displaymath}
where the more visible one of the two parallel arrows involves the contravariant action of $C$ on $F$:

\begin{displaymath}
\hom(c, d) \times F(d) \to F(c)
\end{displaymath}
and the less visible one uses $C$ acting on itself:

\begin{displaymath}
\hom(-, c) \times \hom(c, d) \times F(d) \to hom(-, d) \times F(d)
\end{displaymath}
The point now is that this coequalizer diagram represents the tail end of a simplicial object (with $F(-)$ appearing in dimension -1), which in the notation of the bar construction one could call $B(C, C, F)$. Let me explain this last bit.

The point is that any category $C$ can be regarded as a monad in the bicategory of spans. The underlying span is of course

\begin{displaymath}
C_0 \stackrel{dom}{\leftarrow} C_1 \stackrel{cod}{\to} C_0
\end{displaymath}
and a presheaf $F$ on $C$, as a discrete op-fibration, has an underlying span

\begin{displaymath}
C_0 \leftarrow F \to 1
\end{displaymath}
and is precisely an algebra over the monad $C$. Then, given the data of a monad and an algebra over that monad, one proceeds to build the bar construction as a simplicial object, and I think this is probably the simplicial thingy we want to base the category of descent data on (given a pre-stack $X$).

In fact, if memory serves the category of descent data can be efficiently expressed in bicategorical language as follows. The covering sieve $F$ becomes a homomorphism of bicategories by changing base from $Set$ to $Cat$:

\begin{displaymath}
C^{op} \stackrel{F}{\to} Set \stackrel{discrete}{\to} Cat
\end{displaymath}
and, abbreviating $discrete$ to $d$, it turns out that

\begin{displaymath}
Desc_F(X) \simeq Nat(d F, X)
\end{displaymath}
where the thing on the right side is the category of strong (i.e., pseudo) natural transformations between the indicated bicategory homomorphisms.

In that case, the stack condition on $X$ becomes the statement that the canonical functor

\begin{displaymath}
X(c) \stackrel{Yoneda}{\cong} Nat(d \hom(-, c), X) \to Nat(d F, X)
\end{displaymath}
(where the first equivalence comes from the bicategorical Yoneda lemma, and the second functor is induced from the subfunctor $i: F \to \hom(-, c)$) is an equivalence for all $J$-covering sieves $F$. This formulation connects up nicely, that is, is a straight categorification of what was put down in the entry [[sheaf]].

\hypertarget{examples}{}\subsubsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The stack of $BG$ is a functor $Mfd^{op} \to Gpd$, sending $U\in Mfd$ to the groupoid of $G$-principal bundles over $U$ with $G$-equivariant morphisms; sending $U\xrightarrow{f} V$ to the functor induced by pullbacks of principal bundles via $f$. Then $BG$ comes from the prestack $BG^p: Mfd^{op} \to Gpd$ sending $U\in Mfd$ to the groupoid of trivial principal bundles over $U$ with $G$-equivariant morphisms (then it is just a $G$-valued function $U\xrightarrow{g} G$; sending $U\xrightarrow{f}V$ to the functor induced by pullbacks of principal bundles via $f$. Then sheafification or stackification will give us $BG$ back.

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[sheaf]]


\item [[presheaf of groupoids]]


\item [[group stack]]


\item [[2-sheaf]] / [[(2,1)-sheaf]] / \textbf{stack} / [[model structure for (2,1)-sheaves]]

\begin{itemize}%
\item [[representable morphism of stacks]]


\item [[mapping stack]], [[quotient stack]]


\item [[geometric stack]]

\begin{itemize}%
\item [[differentiable stack]], [[Deligne-Mumford stack]]

\end{itemize}

\item [[rigidification of a stack]]


\item [[moduli stack]]


\item [[Picard stack]]



\end{itemize}

\item [[(∞,1)-sheaf]] / [[∞-stack]]



\end{itemize}
Special kinds of stacks include

\begin{itemize}%
\item [[geometric stacks]];


\item [[gerbes]].



\end{itemize}
[[!include homotopy n-types - table]]

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Ieke Moerdijk]], \emph{Introduction to the language of stacks and gerbes} (\href{http://arxiv.org/abs/math/0212266}{arXiv})
\item [[Jochen Heinloth]], \emph{Some notes on differentiable stacks} (\href{http://www.uni-due.de/~hm0002/stacks.pdf}{pdf})

\end{itemize}
The article

\begin{itemize}%
\item [[Angelo Vistoli]], \emph{Grothendieck topologies, fibered categories and descent theory} \href{http://www.ams.org/mathscinet-getitem?mr=2223406}{MR2223406}; \href{http://arxiv.org/abs/math/0412512}{math.AG/0412512} pp. 1--104 in Barbara Fantechi, Lothar G\"o{}ttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, Angelo Vistoli, \emph{Fundamental algebraic geometry. Grothendieck's [[FGA explained]]}, Mathematical Surveys and Monographs \textbf{123}, Amer. Math. Soc. 2005. x+339 pp. \href{http://www.ams.org/mathscinet-getitem?mr=2007f:14001}{MR2007f:14001}

\end{itemize}
discusses stacks focusing on their dual incarnation as [[Grothendieck fibration]]s.

A [[model category]] structure on [[presheaves of groupoids]], presenting stack the way the [[model structure on simplicial presheaves]] models [[∞-stacks]] is discussed in

\begin{itemize}%
\item [[Sharon Hollander]], \emph{A homotopy theory for stacks}, Israel Journal of Mathematics, 163(1), 93-124 (\href{http://arxiv.org/abs/math.AT/0110247}{arXiv:math.AT/0110247})

\end{itemize}
[[!redirects stack\emph{]] [[!redirects stacks]]}



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