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\begin{document}

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\section*{(infinity,1)-category of (infinity,1)-sheaves}

\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{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{LocAndTopology}{Localizations and Grothendieck topology}\dotfill \pageref*{LocAndTopology} \linebreak
\noindent\hyperlink{OverParacompactSpaces}{Over paracompact topological spaces}\dotfill \pageref*{OverParacompactSpaces} \linebreak
\noindent\hyperlink{DiffToOthers}{Difference to more general $(\infty,1)$-toposes}\dotfill \pageref*{DiffToOthers} \linebreak
\noindent\hyperlink{universal_property}{Universal property}\dotfill \pageref*{universal_property} \linebreak
\noindent\hyperlink{compact_generation}{Compact generation}\dotfill \pageref*{compact_generation} \linebreak
\noindent\hyperlink{NonabelianCohomology}{Nonabelian cohomology}\dotfill \pageref*{NonabelianCohomology} \linebreak
\noindent\hyperlink{models}{Models}\dotfill \pageref*{models} \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 notion of $(\infty,1)$-category of $(\infty,1)$-sheaves is the generalization of the notion of [[category of sheaves]] from [[category theory]] to the [[higher category theory]] of [[(∞,1)-categories]].

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

\begin{defn}
\label{CategoryOfSheaves}\hypertarget{CategoryOfSheaves}{}
An \textbf{$(\infty,1)$-category of $(\infty,1)$-sheaves} is a [[reflective sub-(∞,1)-category]]

\begin{displaymath}
Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow}
  PSh(C)
\end{displaymath}
of an [[(∞,1)-category of (∞,1)-presheaves]] such that the following equivalent conditions hold

\begin{itemize}%
\item $L$ is a [[topological localization]];


\item there is the structure of an [[(∞,1)-site]] on $C$ such that the objects of $Sh(C)$ are precisely those [[(∞,1)-presheaves]] $A$ that are [[local objects]] with respect to the \emph{covering} [[monomorphism in an (∞,1)-category|monomorphisms]] $p : U \to j(c)$ in $PSh(C)$ in that

\begin{equation}
A(c) \simeq PSh(j(c),A) \stackrel{PSh(p,A)}{\to} PSh(U,A)
\label{DescentCondition}\end{equation}
is an [[(∞,1)-equivalence]] in [[∞Grpd]].



\end{itemize}
\end{defn}
This is [[Higher Topos Theory|HTT, def. 6.2.2.6]].

An $(\infty,1)$-category of $(\infty,1)$-sheaves is an \textbf{[[(∞,1)-topos]]}.

\begin{remark}
\label{}\hypertarget{}{}
Equivalence \eqref{DescentCondition} is the \emph{[[descent]]} condition and the presheaves satisfying it are the \textbf{[[(∞,1)-sheaves]]} .

Typically $U$ here is the [[Cech nerve]]

\begin{displaymath}
C(\{U_i\}) = \lim_{\to_{[n]}} U_{i_0, \cdots U_{i_n}}
\end{displaymath}
of a [[covering]] family $\{U_i \to c\}$ (where the colimit is the [[limit in a quasi-category|(∞,1)-categorical colimit]] or [[homotopy colimit]]) so that the above [[descent]] condition becomes

\begin{displaymath}
A(c) \simeq PSh(\lim_\to U_\cdots, A)
 \simeq
  \lim_{\leftarrow} A(U_\cdots)
 =
 \lim_{\leftarrow}
 \left(
    \cdots
    \stackrel{\to}{\stackrel{\to}{\to}}
    \prod_{i,j} A(U_i) \times_{A(c)} A(U_j) 
    \stackrel{\to}{\to}\prod_i A(U_i)
 \right)
 \,.
\end{displaymath}
\end{remark}
\begin{remark}
\label{}\hypertarget{}{}
Sometimes [[(∞,1)-sheaves]] are called \textbf{[[∞-stacks]]}, though sometimes the latter term is reserved for [[hypercomplete]] $(\infty,1)$-sheaves and at other times again it refers to [[(∞,2)-sheaves]].

The [[(n,1)-category|(n,1)-categorical]] counting is:

\begin{itemize}%
\item [[sheaf]] = 0-stack = 0-[[truncated]] $(\infty,1)$-sheaf


\item $(2,1)$-sheaf = [[stack]] = 1-truncated $(\infty,1)$-sheaf


\item $(3,1)$-sheaf = 2-stack = 2-truncated $(\infty,1)$-sheaf


\item etc.


\item $(\infty,1)$-sheaf = [[∞-stack]] (or = [[hypercomplete]] $(\infty,1)$-sheaf).



\end{itemize}
\end{remark}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{LocAndTopology}{}\subsubsection*{{Localizations and Grothendieck topology}}\label{LocAndTopology}

We reproduce the proof that the two characterization in def. \ref{CategoryOfSheaves} above are indeed equivalent.

\begin{prop}
\label{}\hypertarget{}{}
For $C$ an [[(∞,1)-site]], the full [[sub-(∞,1)-category]] of $PSh(C)$ on [[local objects]] with respect to the covering monomorphisms in $PSh(C)$ is indeed a [[topological localization]], and hence $Sh(C)$ is indeed an exact [[reflective sub-(∞,1)-category]] of $PSh(C)$ and hence an [[(∞,1)-topos]].

\end{prop}
This is [[Higher Topos Theory|HTT, Prop. 6.2.2.7]]

\begin{proof}
We must prove that the [[(∞,1)-sheafification]] functor $L \colon PSh(C)\to Sh(C)$ preserves [[finite (∞,1)-limits]]. To do so we give an explicit construction of $L$. Given a presheaf $F\in PSh(C)$, define a new presheaf $F^+$ by the formula

\begin{displaymath}
F^+(c)={\lim_{\rightarrow}}_U {\lim_{\leftarrow}}_{u\in U} F(u)
\end{displaymath}
where the colimit is taken over all covering sieves $U$ of $c$; this is called the \emph{[[plus construction]]}. It defines a functor $PSh(C)\to PSh(C)$ and there is an obvious morphism $F\to F^+$ natural in $F$.

It is clear that the construction $F\mapsto F^+$ preserves [[finite (∞,1)-limits]], since [[filtered (∞,1)-colimits]] do, and it is easy to see that the map $F\to F^+$ becomes an [[equivalence in an (∞,1)-category|equivalence]] in $Sh(C)$. Given an [[ordinal]] $\alpha$, let $F^{(\alpha)}$ be the $\alpha$-iteration of the [[plus construction]] applied to the presheaf $F$. Then the functor $F\mapsto F^{(\alpha)}$ preserves finite limits and the canonical map $F\to F^{(\alpha)}$ becomes an equivalence in $Sh(C)$. In particular, if $F^{(\alpha)}$ is a sheaf, then $F^{(\alpha)}\simeq L(F)$. Thus, it suffices to show that there exists an ordinal $\alpha$ such that, for every $F\in PSh(C)$, $F^{(\alpha)}$ is a sheaf.

Fix $c\in C$ and a covering sieve $U$ of $C$. Given a presheaf $G\in PSh(C/c)$, we define an auxiliary presheaf $Match(U,G)\in PSh(C/c)$ by the formula

\begin{displaymath}
Match(U,G)(f: d\to c)={\lim_{\leftarrow}}_{u\in f^\ast U}G(u).
\end{displaymath}
Restriction maps induce a morphism $\theta_G: G\to Match(U,G)$. Since we clearly have $G(u)\stackrel{\sim}{\to} Match(U,G)(u)$ for $u\in U$, the functor $Match(U,-)$ is \emph{idempotent} in the sense that $Match(U,\theta_G)$ and $\theta_{Match(U,G)}$ are (equivalent) equivalences.

By definition, $F\in PSh(C)$ is a sheaf if and only if $F(c)\stackrel{\sim}{\to} Match(U,F|_{C/c})(c)$ for every $c\in C$ and every covering sieve $U$ of $c$. Our goal is therefore to find an ordinal $\alpha$ (depending only on the (∞,1)-site $C$) such that, for every $F\in PSh(C)$, the map

\begin{displaymath}
F^{(\alpha)}(c) \to \Match(U,F^{(\alpha)}|_{C/c})(c)
\end{displaymath}
is an equivalence.

The morphism $G\to G^+$ in $PSh(C/c)$ factors as

\begin{displaymath}
G\to Match(U,G)\to G^+.
\end{displaymath}
Applying $Match(U,-)$ to this factorization, we get a commutative diagram

\begin{displaymath}
\itexarray{
     G &\to& Match(U,G) &\to& G^+
     \\
     \downarrow^{\mathrlap{\theta_G}} && \downarrow^{\mathrlap{\theta_{Match(U,G)}}} && \downarrow^{\mathrlap{\theta_{G^+}}}
     \\
     Match(U,G) &\to& Match(U,Match(U,G)) &\to& Match(U,G^+)
  }
\end{displaymath}
in which the map $\theta_{Match(U,G)}$ is an equivalence since $Match(U,-)$ is idempotent. By cofinality, the colimit of the maps $\theta_{G^{(n)}}$ as $n\to\infty$ is an equivalence. Applying this to $G=F|_{C/c}$, we get

\begin{displaymath}
F^{(\omega)}(c)\stackrel{\sim}{\to} {\lim_{\rightarrow}}_{n\to\infty} Match(U,F^{(n)}|_{C/c})(c).
\end{displaymath}
This \emph{almost} means that $F^{(\omega)}$ is a sheaf. The problem is that the filtered colimit on the right-hand side need not commute with the limit appearing in the definition of $Match(U,-)$, that is, the canonical map

\begin{displaymath}
{\lim_{\rightarrow}}_{\alpha \lt \omega} Match(U,F^{(\alpha)}|_{C/c})(c) \to \Match(U,F^{(\omega)}|_{C/c})(c)
\end{displaymath}
need not be an equivalence. To solve this problem, we choose a cardinal $\kappa$ such that for every $c\in C$ and every covering sieve $U$ of $c$, the functor $Match(U,(-)|_{C/c})(c):Psh(C)\to \infty Grpd$ preserves $\kappa$-filtered colimits. This is possible because $C$ is small and each of these functors, being the composition of the restriction functor $PSh(C)\to PSh(U)$ and the limit functor $PSh(U)\to\infty Grpd$, has a [[left adjoint|left]] [[adjoint (∞,1)-functor]] and is therefore accessible (see [[HTT|HTT Prop. 5.4.7.7]]). Then the above map with $\omega$ replaced by $\kappa$ is an equivalence. For every ordinal $\alpha\lt\kappa$, applying the above to $F^{(\alpha)}$ shows that

\begin{displaymath}
F^{(\alpha+\omega)}(c)\stackrel{\sim}{\to} {\lim_{\rightarrow}}_{n\to\infty} Match(U,F^{(\alpha+n)}|_{C/c})(c),
\end{displaymath}
Since $\kappa$ is a limit ordinal, we deduce that $F^{(\kappa)}$ is a sheaf by cofinality.

\end{proof}
And conversely:

\begin{prop}
\label{}\hypertarget{}{}
\textbf{(equivalence of site structures and categories of sheaves)}

For $C$ a [[small (∞,1)-category]], there is a bijective correspondence between structure of an [[(∞,1)-site]] on $C$ and equivalence classes of [[topological localization]]s of $PSh(C)$.

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 6.2.2.9]].

\begin{lemma}
\label{}\hypertarget{}{}
For $C$ a small [[(∞,1)-site]] and $Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C)$ the corressponding reflective inclusion of [[(∞,1)-sheaves]] into [[(∞,1)-presheaves]] on $C$ we have that the image under $L$ of a sub-$(\infty,1)$-functor $p : U \to j(c)$ of a [[representable functor|representable]] $j(c)$ is covering precisely if $L(p)$ is an equivalence.

\end{lemma}
This is [[Higher Topos Theory|HTT, lemma 6.2.2.8]].

\begin{proof}
Since $Sh(C)$ is the reflectuive localization of $PSh(C)$ at covering monomorphisms, it is clear that if $p : U \to j(c)$ is covering, then $L(p)$ is an equivalence.

To see the converse, form the 0-truncation of $L i$ and conclude as for ordinary sheaves on the [[homotopy category of an (infinity,1)-category|homotopy catgegory]] of $C$.

\ldots{}

\end{proof}
\begin{proof}
We have seen in (\ldots{}) that for every structure of an $(\infty,1)$-site on $C$ we obtain a topological localization of the presheaf category, and that this is an injective map from site structures to equivalence classes of sheaf categories. It remains to show that it is also a surjective map, i.e. that every [[topological localization]] of $PSh(C)$ comes from the structure of an [[(∞,1)-site]] on $C$.

So consider $S \subset Mor(PSh(C))$ a strongly saturated class of morphisms which s topological (closed under pullbacks). Write $S_0 \subset S$ for the subcalss of those that are [[monomorphism in an (infinity,1)-category|monomorphisms]] of the form $U \to j(c)$.

Observe that then $S$ is indeed generated by (is the smallest strongly saturated class containing) $S_0$: since by the [[co-Yoneda lemma]] every object $X \in PSh(C)$ is a colimit $x \simeq {\lim_\to}_k j(\Xi_k)$ over representables. It follows that every monomorphism $f : Y \to X$ is a colimit (in $Func(\Delta[1],PSh(C))$) of those of the form $U \to j(c)$: for consider the pullback diagram

\begin{displaymath}
\itexarray{
     f^* ({\lim_\to}_k \Xi_k) &\to& Y 
     \\
     \downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}}
     \\
     {\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X
  }
  \;\;\;\;\;
  \simeq
  \;\;\;\;\;
  \itexarray{
     ({\lim_\to}_k f^* \Xi_k) &\to& Y 
     \\
     \downarrow^{\mathrlap{\simeq f}} && \downarrow^{\mathrlap{f}}
     \\
     {\lim_\to}_k \Xi_k &\stackrel{\simeq}{\to}& X
  }
\end{displaymath}
where the equivalence is due to the fact that we have [[universal colimits]] in $PSh(C)$. This realizes $f$ as a colimit over morphisms of the form $f^* j(\Xi_k) \to j(\Xi_k)$ that are each a pullback of a monomorphism. Since monomorphisms are stable under pullback (see [[monomorphism in an (∞,1)-category]] for details), all these component morphisms are themselves monomorphisms.

So every monomorphism in $S$ is generated from $S_0$, but by the assumption that $S$ is topological, it is itself entirely generated from monomorphisms, hence is generated from $S_0$.

So far this establishes that evry topological localization of $PSh(C)$ is a localization at a collection of sieves/ subfunctors $U \to j(c)$ of representables. It remains to show that this collection of subfunctors is indeed an Grothendieck topology and hence exhibits on $C$ the structure of an [[(∞,1)-site]]. We check the necessary three axioms:

\begin{enumerate}%
\item \emph{equivalences cover} -- The equivalences $j(c) \stackrel{\simeq}{\to} j(c)$ belong to $S$ and are monomorphisms, hence belong to $S_0$.


\item \emph{pullback of a cover is covering} - Since monomorphisms are stable under pullback, we haave for every $p : U \to j(c)$ in $S$ and every $j(f) : j(d) \to j(c)$ that also the pullback $f^* p$

\begin{displaymath}
\itexarray{
     f^* U &\to& U
     \\
     \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}}
     \\
     j(d) &\stackrel{f}{\to}& j(c)
  }
\end{displaymath}
is a monomorphism and in $S$, hence in $S_0$.


\item \emph{if restriction of a sieve to a cover is covering, then the sieve is covering} -- Let $p : U \to j(c)$ be an arbitrary monomorphism and $f : X \to j(d)$ in $S_0$. Write $X \simeq {\lim_\to}_k \Xi_k$ and consider the pullback

\begin{displaymath}
\itexarray{
    {\lim_\to}_k p^* \Xi_k &\stackrel{p^* f}{\to}&  U
    \\
    \downarrow^{{\lim_\to}_k f_k^* p} && \downarrow^{\mathrlap{p}}
    \\
    {\lim_\to}_k \Xi_k &\stackrel{f}{\to}& j(c)
  }
  \,,
\end{displaymath}
where again we made use of the [[universal colimits]] in $PSh(C)$. Now notice that

\begin{enumerate}%
\item $f$ is in $S$ by assumption;


\item $p^* f$ is by pullback stability of $S$;


\item each of the $f_k p$ is in $S$ by assumption, hence ${\lim_k f_k^* p}$ is by the fact that $S$ is strongly saturated.


\item so by commutativity $p \circ p^*f$ is in $S$.


\item finally by 2-out-of-3 this means that $p$ is in $S$.



\end{enumerate}


\end{enumerate}
\end{proof}
\hypertarget{OverParacompactSpaces}{}\subsubsection*{{Over paracompact topological spaces}}\label{OverParacompactSpaces}

We discuss how $(\infty,1)$-sheaves over a [[paracompact topological space]] are equivalent to topological spaces [[overcategory|over]] $X$. This is the analogue of the 1-categorical statement that [[sheaves]] on $X$ are equivalent to [[etale space]]s over $X$: an etale space over $X$ is one whose [[fiber]]s are [[discrete topological space]], hence 0-[[truncated]] spaces. The [[n-category]] analogy has [[homotopy n-type]]s as fibers.

\begin{defn}
\label{}\hypertarget{}{}
For $Y \to X$ a [[morphism]] in [[Top]], and $U \in Op(X)$ an [[open subset]] of $X$, write

\begin{displaymath}
Sing_X(Y,U) := Hom_X(U \times \Delta^\bullet, X)
\end{displaymath}
for the [[simplicial set]] (in fact a [[Kan complex]]) of [[continuous map]]s

\begin{displaymath}
\itexarray{
    U \times \Delta^k && \to && Y
    \\
    & \searrow && \swarrow
    \\
    && X
  }
\end{displaymath}
from $U$ times the topological $k$-[[simplex]] $\Delta^k$ into $Y$, that are [[section]]s of $Y \to X$.

\end{defn}
This is a relative version of the [[singular simplicial complex]] functor.

\begin{prop}
\label{OverTopModelStructure}\hypertarget{OverTopModelStructure}{}
Let $(X, \mathcal{B})$ be a [[topological space]] equipped with a [[base for the topology]] $\mathcal{B}$.

There is a [[model category]] structure on the [[over category]] $Top/X$ with weak equivalences and fibration precisely those morphisms $Y \to Z$ over $X$ such that for each $U \in \mathcal{B}$ the induced morphism $Sing_X(Y,U) \to Sing_X(Z,U)$ is a weak equivalence or fibration, respectively, in the standard [[model structure on simplicial sets]].

\end{prop}
This is [[Higher Topos Theory|HTT, prop 7.1.2.1]].

Write $(Top/X)^\circ$ for the [[(∞,1)-category]] [[locally presentable (∞,1)-category|presented]] by this model structure.

\begin{prop}
\label{}\hypertarget{}{}
Let $X$ be a [[paracompact topological space]] and write as usual $Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$ for the $(\infty,1)$-category of $(\infty,1)$-sheaves on the [[category of open subsets]] of $X$; equipped with the canonical structure of a [[site]].

Let $\mathcal{B}$ be the set of \textbf{$F_\sigma$-open subsets} of $X$. This are those [[open subset]]s that are countable unions of [[closed subset]]s, equivalently the 0-sets of [[continuous function]]s $X \to [0,1]$.

Let $Top/X^\circ$ be the corresponding $(\infty,1)$-categoty according to the \hyperlink{OverTopModelStructure}{above proposition}. Then $Sing_X(-,-)$ constitutes an [[equivalence of (∞,1)-categories]]

\begin{displaymath}
Top/X^\circ \simeq Sh_{(\infty,1)}(X)
  \,.
\end{displaymath}
\end{prop}
This is [[Higher Topos Theory|HTT, corollary 7.1.4.4]].

\hypertarget{DiffToOthers}{}\subsubsection*{{Difference to more general $(\infty,1)$-toposes}}\label{DiffToOthers}

The [[(∞,1)-topos]]es that are $(\infty,1)$-categories of sheaves, i.e. that arise by [[topological localization]] from an [[(∞,1)-category of (∞,1)-presheaves]], enjoy a number of special properties over other classes of $(\infty,1)$-toposes, such as notably [[hypercomplete (∞,1)-topos]]es.

The following lists these properties. ([[Higher Topos Theory|HTT, section 6.5.4]].)

\hypertarget{universal_property}{}\paragraph*{{Universal property}}\label{universal_property}

The construction of [[(∞,1)-sheaf]] [[(∞,1)-topos]]es on a given [[locale]] is singled out over the construction of other kinds of $(\infty,1)$-toposes (such as [[hypercomplete (∞,1)-topos]]es) by the following universal property:

forming $(\infty,1)$-sheaves is, roughly, [[right adjoint]] to the functor $\tau_{\leq -1}$ that sends each $(\infty,1)$-topos to its underlying [[locale]] of [[subobject]]s of the [[terminal object]].

See [[Higher Topos Theory|HTT, item 1) of section 6.5.4]].

For $X,Y$ two $(\infty,1)$-toposes, write $Geom(X,Y) \subset Func(X,Y)$ for the full [[sub-(∞,1)-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those that are [[geometric morphism]]s.

\begin{lemma}
\label{}\hypertarget{}{}
For $C$ an [[essentially small (∞,1)-category|small]] [[(n,1)-category]] with [[finite (∞,1)-limits]] and equipped with the structure of an [[(∞,1)-site]] and for $Y$ an [[(∞,1)-topos]], the [[truncated|truncation functor]]

\begin{displaymath}
\tau_{\leq n-1} : 
  Geom(Y, Sh(C)) \to Geom(\tau_{\leq n-1} Y, \tau_{\leq n-1} Sh(C))
\end{displaymath}
is an [[equivalence in a quasi-category|equivalence]] (of [[(∞,1)-categories]]).

\end{lemma}
This is [[Higher Topos Theory|HTT, lemma 6.4.5.6]].

See also [[n-localic (∞,1)-topos]].

\hypertarget{compact_generation}{}\paragraph*{{Compact generation}}\label{compact_generation}

\begin{prop}
\label{}\hypertarget{}{}
Let $X$ be a [[coherent topological space]] and let $Op(X)$ be its [[category of open subsets]] with the standard structure of an [[(∞,1)-site]].

Then $Sh_{(\infty,1)}(X) := Sh_{(\infty,1)}(Op(X))$ is \emph{compactly generated} in that it is generated by [[filtered colimit]]s of [[compact object]]s.

Moreover, the compact objects of $Sh_{(\infty,1)}(X)$ are those that are [[stalk]]wise compact objects in [[∞Grpd]] and [[locally constant ∞-stack|locally constant]] along a suitable [[stratification]] of $X$.

\end{prop}
This is [[Higher Topos Theory|HTT, prop. 6.5.4.4]].

This statement is false for the [[hypercompletion]] of $Sh_{(\infty,1)}(X)$, in general.

\hypertarget{NonabelianCohomology}{}\paragraph*{{Nonabelian cohomology}}\label{NonabelianCohomology}

For $X$ a [[topological space]], let

\begin{displaymath}
(LConst \dashv \Gamma) :  Sh_{(\infty,1)}(X)
  \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
\end{displaymath}
be the [[global section]]s terminal [[geometric morphism]].

For $A \in \infty Grpd$, the ([[nonabelian cohomology|nonabelian]]) [[cohomology]] of $X$ with coefficients in $A$ is usually defined in [[∞Grpd]] as

\begin{displaymath}
H(X,A) := \pi_0 Func(Sing X, A)
  \,,
\end{displaymath}
where $Sing X$ is the [[fundamental ∞-groupoid]] of $X$. On the other hand, if we send $A$ into $Sh_{(\infty,1)}(X)$ via $LConst$, the there is the \emph{intrinsic} [[cohomology]] of the $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$

\begin{displaymath}
H'(X,A) := \pi_0 Sh_{(\infty,1)}(X)(X, LConst A)
  \,.
\end{displaymath}
Noticing that $X$ is in fact the [[terminal object]] of $Sh_{(\infty,1)}(X)$ and that $Sh_{(\infty,1)}(X)(X,-)$ is in fact that [[global section]]s functor, this is equivalently

\begin{displaymath}
\cdots \simeq \pi_0 \Gamma LConst A
  \,.
\end{displaymath}
\begin{theorem}
\label{}\hypertarget{}{}
If $X$ is a [[paracompact space]], then these two definitions of [[nonabelian cohomology]] of $X$ with [[constant ∞-stack|constant coefficients]] $A \in \infty Grpd$ agree:

\begin{displaymath}
H(X,A) := \pi_0 \infty Grpd(Sing X,A)  \simeq Sh_{(\infty,1)}(X)(X,LConst A)
  \,.
\end{displaymath}
\end{theorem}
This is [[Higher Topos Theory|HTT, theorem 7.1.0.1]].

\hypertarget{models}{}\subsection*{{Models}}\label{models}

The topological localizations of an [[(∞,1)-category of (∞,1)-presheaves]] are [[presentable (∞,1)-category|presented]] by the [[Bousfield localization of model categories|left Bousfield localization]] of the global [[model structure on simplicial presheaves]] at the set of [[Cech cover]]s.

The [[hypercompletion|hypercomplete]] $(\infty,1)$-sheaf toposes are [[presentable (infinity,1)-category|presented]] by the local Joyal-Jardine [[model structure on simplicial presheaves]].

Detailed discussion of this [[model category]] presentation is at

\begin{itemize}%
\item [[models for infinity-stack (infinity,1)-toposes|models for (infinity,1)-category of (infinity,1)-sheaves]] .

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

\begin{itemize}%
\item [[geometric homotopy type theory]]

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

The study of simplicial presheaves apparently goes back to

\begin{itemize}%
\item K. Brown, [[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]

\end{itemize}
which considers locally [[Kan complex|Kan]] [[simplicial presheaf|simplicial presheaves]] as a [[category of fibrant objects]].

This was later conceived in terms of a [[model structure on simplicial presheaves]] and on simplicial sheaves by Joyal and Jardine. To\"e{}n summarizes the situation and emphasizes the interpretation in terms of [[∞-stacks]] living in $(\infty,1)$-categories for instance in

B. To\"e{}n, \emph{Higher and derived stacks: a global overview} (\href{http://arxiv.org/abs/math/0604504}{arXiv}) .

This concerns mostly [[hypercomplete]] $(\infty,1)$-sheaves, though.

The full picture in terms of Grothendieck-[[(∞,1)-topos]]es of [[(∞,1)-sheaves]] is the topic of

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Higher Topos Theory]]} .

\begin{itemize}%
\item localization $(\infty,1)$-functors ($(\infty,1)$-sheafification for the present purpose) are discussed in section 5.2.7;


\item local objects ($(\infty,1)$-sheaves for the present purpose) and [[local isomorphism]]s are discussed in section 5.5.4;


\item the definition of $(\infty,1)$-topoi of $(\infty,1)$-sheaves is then definition 6.1.0.4 in section 6.1;


\item the characterization of $(\infty,1)$-sheaves in terms of [[descent]] is in section 6.1.3


\item the relation between the [[model structure on simplicial presheaves|Brown?Joyal?Jardine model]] and the general story is discussed at length in section 6.5.4



\end{itemize}


\end{itemize}
An overview is in

\begin{itemize}%
\item [[Denis-Charles Cisinski]], \emph{Cat\'e{}gories sup\'e{}rieures et th\'e{}orie des topos}, S\'e{}minaire Bourbaki, 21.3.2015, \href{http://www.math.univ-toulouse.fr/~dcisinsk/1097.pdf}{pdf}.

\end{itemize}
[[!redirects (infinity,1)-category of (infinity,1)-sheaves]] [[!redirects (∞,1)-category of (∞,1)-sheaves]] [[!redirects (infinity,1)-categories of (infinity,1)-sheaves]] [[!redirects (∞,1)-categories of (∞,1)-sheaves]]

[[!redirects (∞,1)-sheaf (∞,1)-topos]] [[!redirects (∞,1)-sheaf (∞,1)-toposes]] [[!redirects (∞,1)-sheaf (∞,1)-topoi]]

[[!redirects ∞-stack (∞,1)-topos]] [[!redirects ∞-stack (∞,1)-toposes]] [[!redirects ∞-stack (∞,1)-topoi]]

[[!redirects (∞,1)-sheaf (∞,1)-category]] [[!redirects (∞,1)-sheaf (∞,1)-categories]]

[[!redirects (infinity,1)-sheaf (infinity,1)-category]] [[!redirects (infinity,1)-sheaf (infinity,1)-categories]]

[[!redirects (∞,1)-topos of (∞,1)-sheaves]] [[!redirects (∞,1)-toposes of (∞,1)-sheaves]] [[!redirects (infinity,1)-topos of (infinity,1)-sheaves]] [[!redirects (infinity,1)-toposes of (infinity,1)-sheaves]]

[[!redirects (∞,1)-sheaf ∞-topos]] [[!redirects (∞,1)-sheaf ∞-toposes]]



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