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\begin{document}

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\section*{2-sheaf}

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

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

[[!include descent and locality - contents]]

\hypertarget{topos_theory}{}\paragraph*{{$(\infty,2)$-Topos theory}}\label{topos_theory}

[[!include (infinity,2)-topos theory - contents]]

\hypertarget{2category_theory}{}\paragraph*{{2-Category theory}}\label{2category_theory}

[[!include 2-category theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{characterization_of_over_sites}{Characterization of over $(n,r)$-sites}\dotfill \pageref*{characterization_of_over_sites} \linebreak
\noindent\hyperlink{over_a_1site}{Over a 1-site}\dotfill \pageref*{over_a_1site} \linebreak
\noindent\hyperlink{over_a_site__as_internal_categories}{Over a $(2,1)$-site -- As internal categories}\dotfill \pageref*{over_a_site__as_internal_categories} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{codomain_fibrations__sheaves_of_modules}{Codomain fibrations / sheaves of modules}\dotfill \pageref*{codomain_fibrations__sheaves_of_modules} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{ReferencesInTermsOfInternalCategories}{In terms of categories internal to sheaf toposes}\dotfill \pageref*{ReferencesInTermsOfInternalCategories} \linebreak
\noindent\hyperlink{InTermsOfFiberedCategories}{In terms of fibered categories}\dotfill \pageref*{InTermsOfFiberedCategories} \linebreak
\noindent\hyperlink{2sites}{2-Sites}\dotfill \pageref*{2sites} \linebreak


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

The notion of \emph{2-sheaf} is the generalization of the notion of [[sheaf]] to the [[higher category theory]] of [[2-categories]]/[[bicategories]]. A 2-category of 2-sheaves forms a [[2-topos]].

\begin{remark}
\label{}\hypertarget{}{}
A \emph{2-sheaf} is a higher sheaf of [[categories]]. More restrictive than this is a higher sheaf with values in [[groupoids]], which would be a \emph{[[(2,1)-sheaf]]}. Both these notions are often referred to as \textbf{[[stack]]}, or sometimes ``stack of groupoids'' and ``stack of categories'' for definiteness. But moreover, traditionally a [[stack]] (in either flavor) is considered only over a [[1-site]], whereas it makes sense to consider [[(2,1)-sheaves]] more generally over [[(2,1)-sites]] and 2-sheaves over [[2-sites]].

Therefore, saying ``2-sheaf'' serves to indicate the full generality of the notion of higher sheaves in [[2-category theory]], as opposed to various special cases of this general notion which have traditionally been considered.

\end{remark}
\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $C$ be a [[2-site]] having finite [[2-limit]]s (for convenience). For a covering family $(f_i:U_i\to U)_i$ we have the comma objects

We also have the [[nlab:double comma object|double comma objects]] $(f_i/f_j/f_k) = (f_i/f_j)\times_{U_j} (f_j/f_k)$ with projections $r_{i j   k}:(f_i/f_j/f_k)\to (f_i/f_j)$, $s_{i j k}:(f_i/f_j/f_k)\to (f_j/f_k)$, and $t_{i j k}:(f_i/f_j/f_k)\to (f_i/f_k)$.

Now, a functor $X:C^{op} \to Cat$ is called a \textbf{2-presheaf}. It is \textbf{1-separated} if

\begin{itemize}%
\item For any covering family $(f_i:U_i\to U)_i$ and any $x,y\in X(U)$ and $a,b: x\to y$, if $X(f_i)(a) = X(f_i)(b)$ for all $i$, then $a=b$.

\end{itemize}
It is \textbf{2-separated} if it is 1-separated and

\begin{itemize}%
\item For any covering family $(f_i:U_i\to U)_i$ and any $x,y\in X(U)$, given $b_i:X(f_i)(x) \to X(f_i)(y)$ such that $\mu_{i j}(y) \circ X(p_{i j})(b_i) = X(q_{i j})(b_i) \circ \mu_{i j}(x)$, there exists a (necessarily unique) $b:x\to y$ such that $b_i = X(f_i)(b)$.

\end{itemize}
It is a \textbf{2-sheaf} if it is 2-separated and

\begin{itemize}%
\item For any covering family $(f_i:U_i\to U)_i$ and any $x_i\in X(U_i)$ together with morphisms $\zeta_{i j}:X(p_{i j})(x_i) \to X(q_{i j})(x_j)$ such that the following diagram commutes:\begin{displaymath}
\itexarray{X(r_{i j k})X(p_{i j})(x_i) &
\overset{X(r_{i j k})(\zeta_{i j})}{\to} &
X(r_{i j k})X(q_{i j})(x_j) & \overset{\cong}{\to} &
X(s_{i j k})X(p_{j k})(x_j)\\
^\cong \downarrow && && \downarrow ^{X(s_{i j k})(\zeta_{j k})}\\
X(t_{i j k}) X(p_{i k})(x_i) &
\underset{X(t_{i j k})(\zeta_{i k})}{\to} &
X(t_{i j k}) X(q_{i k})(x_k) & \underset{\cong}{\to} &
X(s_{i j k}) X(q_{j k})(x_k)}
\end{displaymath}
there exists an object $x\in X(U)$ and isomorphisms $X(f_i)(x)\cong x_i$ such that for all $i,j$ the following square commutes:

\begin{displaymath}
\itexarray{
X(p_{i j})X(f_i)(X) & \overset{\cong}{\to}
&     X(p_{i j})(x_i)\\
^{X(\mu_{i j})}\downarrow && \downarrow^{\zeta_{i j}}\\
X(q_{i j})X(f_j)(x)  & \underset{\cong}{\to} &
X(q_{i j})(x_j).}
\end{displaymath}


\end{itemize}
A 2-sheaf, especially on a 1-site, is frequently called a \textbf{[[stack]]}. However, this has the unfortunate consequence that a 3-sheaf is then called a 2-stack, and so on with the numbering all offset by one. Also, it can be helpful to use a new term because of the notable differences between 2-sheaves on 2-sites and 2-sheaves on 1-sites. The main novelty is that $\mu_{i j}$ and $\zeta_{i j}$ \emph{need not be invertible}.

Note, though, they must be invertible as soon as $C$ is (2,1)-site: $\mu_{i j}$ by definition and $\zeta_{i j}$ since an inverse is provided by $\iota_{i j}^*(\zeta_{i j})$, where $\iota_{i j}\mapsto (f_i/f_j) \to (f_j/f_i)$ is the symmetry equivalence.

If $C$ lacks finite limits, then in the definitions of ``2-separated'' and ``2-sheaf'' instead of the comma objects $(f_i/f_j)$, we need to use arbitrary objects $V$ equipped with maps $p:V\to U_i$, $q:V\to U_j$, and a 2-cell $f_i p \to f_j q$. We leave the precise definition to the reader.

A 2-site is said to be \textbf{subcanonical} if for any $U\in C$, the representable functor $C(-,U)$ is a 2-sheaf. When $C$ has finite limits, it is easy to verify that this is true precisely when every covering family is a (necessarily pullback-stable) quotient of its kernel [[2-polycongruence]]. In particular, the regular coverage on a regular 2-category is subcanonical, as is the coherent coverage on a coherent 2-category.

The 2-category $2Sh(C)$ of 2-sheaves on a small 2-site $C$ is, by definition, a [[Grothendieck 2-topos]].

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{characterization_of_over_sites}{}\subsubsection*{{Characterization of over $(n,r)$-sites}}\label{characterization_of_over_sites}

If the underlying [[2-site]] happens to be an [[(n,r)-site]] for $n$ and/or $r$ lower than 2, there may be other equivalent ways to think of 2-sheaves.

A [[2-topos]] with a [[2-site]] of definition that happens to be just a 1-site or [[(2,1)-site]] is \emph{1-localic} or \emph{(2,1)-localic}.

\hypertarget{over_a_1site}{}\paragraph*{{Over a 1-site}}\label{over_a_1site}

Over a 1-site, the [[Grothendieck construction]] says that [[2-functors]] on the site are equivalent to [[fibered categories]] over the site. Hence in this case the theory of 2-sheaves can be entirely formulated in terms of fibered categories. See \emph{\hyperlink{InTermsOfFiberedCategories}{References -- In terms of fibered categories}}.

Also, over a 1-site a 2-sheaf is essentially a \emph{[[indexed category]]}. Therefore stacks over 1-sites can also be discussed in this language, see notably the work (\hyperlink{BungePare}{Bunge-Pare}).

In particular, if the 1-site $C$ is a [[topos]], then every topos \emph{over} $C$ as its [[base topos]] (a $C$-topos) induces an [[indexed category]].

\begin{prop}
\label{}\hypertarget{}{}
If $C$ is a topos and $E$ is a $C$-topos, then (the [[indexed category]] corresponding to) $E$ is a 2-sheaf on $C$ with respect to the [[canonical topology]].

\end{prop}
This appears as (\hyperlink{BungePare}{Bunge-Pare, corollary 2.6}).

Moreover, over a [[1-site]] the [[2-topos]] of 2-sheaves ought to be equivalent to the (suitably defined) [[2-category]] of [[internal categories]] in the underlying [[1-topos]]. See \emph{\hyperlink{ReferencesInTermsOfInternalCategories}{References -- In terms of internal categories}}.

\hypertarget{over_a_site__as_internal_categories}{}\paragraph*{{Over a $(2,1)$-site -- As internal categories}}\label{over_a_site__as_internal_categories}

Over a [[(2,1)-site]] the [[2-topos]] of 2-sheaves ought to be equivalent to the [[2-category]] of [[internal (infinity,1)-categories]] in the corresponding [[(2,1)-topos]].

This is discussed at \emph{\href{2-topos#InTermsOfInternalCategories}{2-Topos -- In terms of internal categories}}.

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

\hypertarget{codomain_fibrations__sheaves_of_modules}{}\subsubsection*{{Codomain fibrations / sheaves of modules}}\label{codomain_fibrations__sheaves_of_modules}

A classical class of examples for 2-sheaves are [[codomain fibrations]] over suitable sites, or rather their [[tangent categories]]. As discussed there, this includes the case of sheaves of categories of [[modules]] over sites of [[algebra over an algebraic theory|algebras]].

\begin{prop}
\label{}\hypertarget{}{}
For $C$ an [[exact category]] with [[finite limits]], the [[codomain fibration]] $Cod : C^I \to C$ or equivalently (under the [[Grothendieck construction]]), the self-[[indexed category|indexing]] of $C$ is a 2-sheaf with respect to the [[canonical topology]].

\end{prop}
This is for instance (\hyperlink{BungePare}{Bunge-Pare, corollary 2.4}).

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

\begin{itemize}%
\item [[presheaf]] / [[sheaf]] / [[cosheaf]]


\item \textbf{2-sheaf} / [[stack]]


\item [[(∞,1)-sheaf]] / [[∞-stack]]


\item [[(∞,2)-sheaf]]


\item [[(∞,n)-sheaf]]


\item [[descent]]



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

Historically, the original definition of \emph{[[stack]]} included the case of category-valued functors, hence of 2-sheaves, in:

\begin{itemize}%
\item [[J. Giraud]], \emph{Cohomologie non ab\'e{}lienne, Grundlehren number 179, Springer Verlag (1971)}

\end{itemize}
\hypertarget{ReferencesInTermsOfInternalCategories}{}\subsubsection*{{In terms of categories internal to sheaf toposes}}\label{ReferencesInTermsOfInternalCategories}

Category-valued stacks as [[internal categories]] in the underlying [[sheaf topos]] have been considered in

\begin{itemize}%
\item [[Marta Bunge]], [[Robert Pare]], \emph{Stacks and equivalence of indexed categories}, Cahiers de Topologie et G\'e{}om\'e{}trie Diff\'e{}rentielle Cat\'e{}goriques, 20 no.4 (1979) (\href{http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0}{numdam})

\end{itemize}
\begin{itemize}%
\item [[Marta Bunge]], \emph{Stack completions and Morita equivalence for categories in a topos}, Cahiers de topologie et g\'e{}om\'e{}trie diff\'e{}rentielle xx-4, (1979) 401-436, (\href{http://www.ams.org/mathscinet-getitem?mr=558106}{MR558106}, \href{http://www.numdam.org/item?id=CTGDC_1979__20_4_401_0}{numdam})

\end{itemize}
and in section 3 of

\begin{itemize}%
\item [[André Joyal]], [[Myles Tierney]], \emph{Strong stacks and classifying spaces} Category theory ([[Como]], 1990), 213---236, Lecture Notes in Math. 1488, Springer (1991) (\href{http://www.pps.jussieu.fr/~mellies/semantics-operads-categories/joyal-tierney-strong-stacks.pdf}{pdf})

\end{itemize}
\hypertarget{InTermsOfFiberedCategories}{}\subsubsection*{{In terms of fibered categories}}\label{InTermsOfFiberedCategories}

A discussion of stacks over [[1-sites]] in terms of their [[Grothendieck construction|associated]] [[fibered categories]] is in

\begin{itemize}%
\item [[Angelo Vistoli]], \emph{Notes on Grothendieck topologies, fibered categories and descent theory} (\href{http://homepage.sns.it/vistoli/descent.pdf}{pdf})

\end{itemize}
\hypertarget{2sites}{}\subsubsection*{{2-Sites}}\label{2sites}

The above text involves content transferred from

\begin{itemize}%
\item [[Michael Shulman]], \emph{[[michaelshulman:2-site]]}

\end{itemize}
2-sites were earlier considered in

\begin{itemize}%
\item [[Ross Street]], \emph{[[StreetCBS]]}

\end{itemize}
[[!redirects 2-sheaves]]



\end{document}