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\begin{document}

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\section*{presentations of (infinity,1)-sheaf (infinity,1)-toposes}

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

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

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

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

\noindent\hyperlink{contents}{Contents}\dotfill \pageref*{contents} \linebreak
\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{plan}{Plan}\dotfill \pageref*{plan} \linebreak
\noindent\hyperlink{sheaf_toposes}{Sheaf toposes}\dotfill \pageref*{sheaf_toposes} \linebreak
\noindent\hyperlink{presheaves}{Presheaves}\dotfill \pageref*{presheaves} \linebreak
\noindent\hyperlink{sheaves}{Sheaves}\dotfill \pageref*{sheaves} \linebreak
\noindent\hyperlink{categories_and_their_presentation}{$(\infty,1)$-categories and their presentation}\dotfill \pageref*{categories_and_their_presentation} \linebreak
\noindent\hyperlink{presentations}{Presentations}\dotfill \pageref*{presentations} \linebreak
\noindent\hyperlink{sheaf_toposes_2}{$(\infty,1)$-Sheaf $(\infty,1)$-toposes}\dotfill \pageref*{sheaf_toposes_2} \linebreak
\noindent\hyperlink{presheaves_2}{$(\infty,1)$-Presheaves}\dotfill \pageref*{presheaves_2} \linebreak
\noindent\hyperlink{sheaves_2}{$(\infty,1)$-sheaves}\dotfill \pageref*{sheaves_2} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{abelian_sheaf_cohomology_as_special_case_of_stackification}{Abelian sheaf cohomology as special case of $\infty$-stackification}\dotfill \pageref*{abelian_sheaf_cohomology_as_special_case_of_stackification} \linebreak


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

An [[(∞,1)-sheaf]] -- often called a [[∞-stack]] -- is the [[(infinity,1)-category|(∞,1)-categorical]] analog of a [[sheaf]]. Just like a [[category of sheaves]] is a [[topos]], an [[(∞,1)-category of (∞,1)-sheaves]] is an [[(∞,1)-topos]].

There is good [[motivation for sheaves, cohomology and higher stacks]].

Here we recall basic definitions and then concentrate on [[model category|1-categorical models]] that [[presentable (infinity,1)-category|present]] [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-categories of ∞-stacks]].

What we describe is effectively the old theory of the [[model structure on simplicial presheaves]] seen in the new light of [[Higher Topos Theory]].

\hypertarget{plan}{}\section*{{Plan}}\label{plan}

We proceed as follow.

\begin{itemize}%
\item As a preparation,

\begin{itemize}%
\item we recall 1-categorical [[presheaf|presheaves]]


\item and how a [[category of sheaves]] may equivalently be thought of as

\begin{itemize}%
\item a category [[geometric embedding|geometrically embedded]] into that of presheaves


\item or equivalently as the [[localization]] of presheaves at [[local isomorphisms]].



\end{itemize}


\end{itemize}

\item Then in the main part, after recalling

\begin{itemize}%
\item the definition of [[(∞,1)-category]]

\begin{itemize}%
\item and the [[presentable (infinity,1)-category|presentation]] of $(\infty,1)$-categories by [[model category|1-categorical models]]

\end{itemize}


\end{itemize}
we give

\begin{itemize}%
\item the general definition of [[(∞,1)-presheaves]];

\begin{itemize}%
\item and their [[presentable (infinity,1)-category|presentation]] by the [[global model structure on simplicial presheaves]]

\end{itemize}


\end{itemize}
and finally

\begin{itemize}%
\item the definition of the [[(∞,1)-category of (∞,1)-sheaves]] by [[reflective (infinity,1)-subcategory|(∞,1)-geometric embedding]];
\item and the [[model category|1-categorical model]] of this by [[Bousfield localization]] to the [[local model structure on simplicial presheaves]].

\end{itemize}


\end{itemize}
\hypertarget{sheaf_toposes}{}\section*{{Sheaf toposes}}\label{sheaf_toposes}

It is helpful to briefly recall the story that we want to tell in the [[category theory]] context, because in the full [[higher category theory]] context it will be \textbf{literally the same} with all notions such as [[adjoint functor]], [[exact functor]] etc suitably regarded in the context of [[(∞,1)-functors]].

\hypertarget{presheaves}{}\subsection*{{Presheaves}}\label{presheaves}

Consider a [[category]] $C$ that we want to think as a category of ``test spaces''. Classical choices would be $C =$ [[Top]], the category of [[topological spaces]], $C =$ [[Diff]], the category of smooth manifolds or $C = Op(X)$, the [[category of open subsets]] of some [[topological space]] $X$.

Let [[Set]] be the category of sets. We write

\begin{displaymath}
PSh(C) := [C^{op}, Set] := Func(C^{op}, Set)
\end{displaymath}
for the category of [[presheaf|presheaves]] on $C$. This is like a category of [[space and quantity|very general spaces]] modeled on $C$ as described at [[motivation for sheaves, cohomology and higher stacks]].

\hypertarget{sheaves}{}\subsection*{{Sheaves}}\label{sheaves}

In fact, this is a bit too general for most purposes: the objects of $PSh(C)$ may be very non-local in that they don't respect the way test objects in $C$ are supposed to glue together. The full subcategory on those presheaves that do respect some kind of gluing of test objects is the [[category of sheaves]].

\begin{udefn}
A [[category of sheaves]] on $C$ is a category $Sh(C)$ equipped with a [[geometric embedding]] into $PSh(C)$

\begin{displaymath}
Sh(C) \stackrel{\leftarrow}{\to} PSh(C)
  \,.
\end{displaymath}
\end{udefn}
Recall that this means that

\begin{itemize}%
\item $Sh(C) \hookrightarrow PSh(C)$ is a [[full and faithful functor]];


\item $\bar {(\cdot)} : PSh(C) \to Sh(C)$ is a [[exact functor|left exact]] [[left adjoint]].



\end{itemize}
in other words that

\begin{itemize}%
\item $Sh(C) \hookrightarrow PSh(C)$ is the inclusion of a [[reflective subcategory]];


\item with the special property that the [[left adjoint]] to the inclusion is [[exact functor|left exact]] (i.e. preserves finite [[limits]]).



\end{itemize}
In view of our models for $\infty$-sheaves it is of importance that this implies an equivalence characterization

\begin{uprop}
The category $Sh(C)$ is equivalent to the full [[subcategory]] of $S$-[[local object|local]] presheaves, where $S$ is the set of [[local isomorphisms]].

\end{uprop}
Another useful kind of [[geometric embeddings]] is that of the point:

let ${*}$ be the category with a single morphism (the identity on a single object). Then $PSh({*}) \simeq Sh({*}) \simeq Set$. Geometric embeddings

\begin{displaymath}
x : Sh({*}) \stackrel{\leftarrow}{\to} Sh(C)
\end{displaymath}
are called [[point of a topos|points]] of $Sh(C)$. We say that $Sh(C)$ has \emph{enough points} if isomorphisms of sheaves can be tested on points

\begin{displaymath}
(f : A \stackrel{\simeq}{\to} B)\in Sh(C) 
  \;\;
  \Leftrightarrow
  \;\;
  \forall x : (x^* f : x^* A \stackrel{\simeq}{\to} x^* B)
  \,.
\end{displaymath}
This is the situation we shall concentrate on here.

\begin{itemize}%
\item The topos $Sh(Diff)$ has enough points, one for every $n \in \mathbb{N}$.


\item The topos $Sh(Op(X))$ has enough points: one for every ordinary point of $X$.



\end{itemize}
If $Sh(C)$ has enough points, we may characterize sheaves in yet another way, which is the one that directly suggests the [[local model structure on simplicial presheaves]] discussed below:

\begin{uprop}
Let $S \subset Mor(PSh(C))$ be the set of \emph{[[stalk]]wise isomorphisms}, i.e. those morphisms $f : A \to B$ of presheaves such that for all [[point of a topos|points]] $x$ the morphism $x^* f : x^* A \to x^* B$ is an [[isomorphism]] (of [[sets]]).

If $Sh(C)$ has [[point of a topos|enough points]], then $Sh(C)$ is equivalent to the full [[subcategory]] of $S$-local presheaves.

\end{uprop}
The [[local model structure on simplicial presheaves]] that we are going to describe is obtained from this description of sheaves by

\begin{itemize}%
\item replacing [[sets]] by [[simplicial set]]


\item replacing [[stalk]]wise [[isomorphism]] of sets by [[stalk]]wise [[model structure on simplicial sets|weak homotopy equivalences]] of simplicial sets.



\end{itemize}
So the model structures we shall encounter are plausible guesses. What is less trivial is that this plausible structure indeed [[presentable (infinity,1)-category|presents]] the fully general notion of [[(∞,1)-sheaf]]/[[∞-stack]].

This fully general notion we introduce now.

\hypertarget{categories_and_their_presentation}{}\section*{{$(\infty,1)$-categories and their presentation}}\label{categories_and_their_presentation}

An ordinary [[locally small category]] is a [[enriched category|category enriched]] over the category [[Set]] of [[sets]].

An [[(∞,0)-category]] is an [[∞-groupoid]] which we think of as modeled by a [[simplicial set]] that is a [[Kan complex]].

Recall that there is a notion of [[nerve and realization]]

\begin{displaymath}
N : SSet\text{-}Cat \stackrel{\leftarrow}{\to} SSet : |-|
\end{displaymath}
for [[SSet]]-[[enriched category|enriched categories]] induced by a [[simplicial object|cosimplicial]] [[simplicially enriched category]]

\begin{displaymath}
\Delta_{SSet-Cat} : \Delta \to SSet-Cat
\end{displaymath}
where the [[nerve]] operation $N$ is called the [[homotopy coherent nerve]] of [[simplicially enriched category|simplicially enriched categories]].

\begin{udefn}
An [[(∞,1)-category]] is a [[enriched category|category enriched]] over $\infty$-groupoids, i.e. an [[SSet]]-[[enriched category]] all whose [[hom-objects]] happen to be [[Kan complexes]].

Given two $(\infty,1)$-categories $\mathbf{C}$ and $\mathbf{D}$ the [[(∞,1)-functor]] $(\infty,1)$-category is

\begin{displaymath}
Func(\mathbf{C}, \mathbf{D}) :=
  |SSet(N(\mathbf{C}), N(\mathbf{D}))|
  \,.
\end{displaymath}
This is indeed itself an $(\infty,1)$-category ([[Higher Topos Theory|HTT, prop 1.2.7.3]]).

The [[(∞,1)-category of (∞,1)-categories]] $(\infty,1)Cat$ is that whose

\begin{itemize}%
\item objects are $(\infty,1)$-categories;


\item for $\mathbf{C}$ and $\mathbf{D}$ two $(\infty,1)$-categories the $\infty$-groupoid $(\infty,1)Cat(\mathbf{C}, \mathbf{D})$ is the maximal [[Kan complex]] inside the [[simplicial set]] of maps between the [[homotopy coherent nerves]]

\begin{displaymath}
(\infty,1)Cat(\mathbf{C}, \mathbf{D})
  :=
  Core( SSet(N(\mathbf{C}), N(\mathbf{D})) )
  \,.
\end{displaymath}


\end{itemize}
\end{udefn}
\textbf{Examples}

\begin{itemize}%
\item Using the [[monoidal category|monoidal]] embedding $const : Set \hookrightarrow \infty Grpdf \subset SSet$ every ordinary category is an $(\infty,1)$-category.


\item The $(\infty,1)$-category $\infty Grpd$ ([[∞Grpd]]) is the full [[SSet]-subcategory of [[SSet]] on [[Kan complexes]].



\end{itemize}
\begin{udefn}
The [[simplicial homotopy groups|simplicial connected components]] functor

\begin{displaymath}
\pi_0 : SSet \to Set
\end{displaymath}
is strong monoidal and hence induces a functor

\begin{displaymath}
H : (\infty,1)Cat \to Cat
  \,.
\end{displaymath}
The image $H(\mathbf{C})$ of an $(\infty,1)$-category $\mathbf{C}$ with $H(\mathbf{C})(x,y) = \pi_0(\mathbf{C}(x,y))$ is the [[homotopy category of an (∞,1)-category]].

Two $(\infty,1)$-categories $\mathbf{C}$ and $\mathbf{D}$ are \textbf{equivalent} if they are isomorphic in $H((\infty,1)Cat)$

\begin{displaymath}
(f : \mathbf{C} \to \mathbf{D} \;is equivalence)
  \;\;
  \Leftrightarrow
  \;\;
  (H_{(\infty,1)Cat}(f) : \mathbf{C} \to \mathbf{D}
  \;
  is isomorphism)
  \,.
\end{displaymath}
\end{udefn}
\hypertarget{presentations}{}\subsection*{{Presentations}}\label{presentations}

It is often convenient to [[presentable (infinity,1)-category|present]] $(\infty,1)$-categories by 1-categorical [[model category|models]].

\begin{udefn}
For $\mathbf{A}$ a [[combinatorial simplicial model category]], the $(\infty,1)$-category [[presentable (infinity,1)-category|presented]] by it is the full subcategory $\mathbf{A}^\circ \subset \mathbf{A}$ on objects that are both cofibrant and fibrant.

\end{udefn}
\textbf{Remark} The axioms of a simplicial model category ensure that the [[hom-object|hom-simplicial sets]] of $\mathbf{A}^\circ$ are indeed [[Kan complexes]]. (for instance [[Higher Topos Theory|HTT, remark 3.1.8]]).

\begin{uprop}
Let $\mathbf{A}$ and $\mathbf{B}$ be [[combinatorial simplicial model category|combinatorial simplicial model categories]]. Then the corresponding $(\infty,1)$-categories $\mathbf{A}^\circ$ and $\mathbf{B}^\circ$ are equivalent precisely if there is a sequence of [[SSet]]-[[enriched functor|enriched]] [[Quillen equivalences]]

\begin{displaymath}
\mathbf{A}
  \stackrel{\leftarrow}{\to}
  \stackrel{\to}{\leftarrow}
  \stackrel{\leftarrow}{\to}
  \cdots
  \mathbf{B}
  \,.
\end{displaymath}
\end{uprop}
\hypertarget{sheaf_toposes_2}{}\section*{{$(\infty,1)$-Sheaf $(\infty,1)$-toposes}}\label{sheaf_toposes_2}

There is now an obvious definition of $(\infty,1)$-categories of $(\infty,1)$-presheaves and of $(\infty,1)$-sheaves by interpreting the 1-categorical story in the $(\infty,1)$-categorical context.

\hypertarget{presheaves_2}{}\subsection*{{$(\infty,1)$-Presheaves}}\label{presheaves_2}

Now we generalize the above from sheaves to [[(∞,1)-sheaves]] also known as [[∞-stacks]].

\begin{udefn}
The $(\infty,1)$-category of [[(∞,1)-presheaves]] on $C$ is

\begin{displaymath}
PSh_\infty(C) := [C^{op}, \infty Grpd] = 
  Func( C^{op}, \infty Grpd )
  \,.
\end{displaymath}
\end{udefn}
\begin{uprop}
see also [[Higher Topos Theory|HTT, prop. 5.1.1.1]])

The $(\infty,1)$-category presented by the [[global model structure on simplicial presheaves]] $SPSh(C)_{proj}$ on $C$ (either the projective or the injective one) is equivalent to that of $(\infty,1)$-presheaves on $C$:

\begin{displaymath}
(SPSh(C)_{proj})^{\circ}
 \simeq
 (SPSh(C)_{inj})^{\circ}
 \simeq
 PSh_\infty(C)
 \,.
\end{displaymath}
\end{uprop}
\hypertarget{sheaves_2}{}\subsection*{{$(\infty,1)$-sheaves}}\label{sheaves_2}

There are $(\infty,1)$-category analogs of all the familiar notions from [[category theory]], in particular

\begin{itemize}%
\item [[adjoint functor]]


\item [[exact functor]] (preserving finite [[limits]]).



\end{itemize}
Using this we obtain a definition of [[geometric embedding]] of $(\infty,1)$-toposes , i.e. left exaxt [[reflective (∞,1)-subcategories]] by literally copying the 1-categorical definition.

\begin{udefn}
An [[(∞,1)-category of (∞,1)-sheaves]] is a [[geometric embedding]] into an [[(∞,1)-category of (∞,1)-presheaves]]

\begin{displaymath}
Sh_\infty(C) \stackrel{\leftarrow}{\to} PSh_\infty(C)
  \,.
\end{displaymath}
\end{udefn}
\begin{lemma}
\label{}\hypertarget{}{}
Let the [[combinatorial simplicial model category]] $\mathbf{B}$ be a left [[Bousfield localization]] of the [[combinatorial simplicial model category]] $\mathbf{A}$ then

\begin{displaymath}
\mathbf{B}^\circ \stackrel{\leftarrow}{\to}
  \mathbf{A}^\circ
\end{displaymath}
is the inclusion of a [[reflective (∞,1)-subcategory]].

\end{lemma}
\begin{proof}
By [[Higher Topos Theory|HTT, prop A.3.7.4]] every combinatorial simplicial left [[Bousfield localization]] is given by a set $S$ of cofibrations such that

\begin{itemize}%
\item the fibrant objects of $\mathbf{B}$ are precisely the fibrant objects in $\mathbf{A}$ that are $S$-[[local object]];


\item the weak equivalences of $\mathbf{B}$ are the $S$-local morphisms in $\mathbf{A}$.



\end{itemize}
Accordingly $\mathbf{B}^\circ$ is the full $\infty Grpd$-enriched subcategory of $\mathbf{A}^\circ$ on $S$-[[local objects]]. (see also [[Higher Topos Theory|HTT, prop 6.5.2.14]]).

By [[Higher Topos Theory|HTT, prop. 5.5.4.15]] this means that $\mathbf{B}$ is a [[reflective (∞,1)-subcategory]] of $\mathbf{A}$.

\end{proof}
\textbf{Remark} Notice that this does not yet say that the localization is \emph{left exact} .

But this makes at least plausible that the [[local model structure on simplicial presheaves]] is a [[presentable (infinity,1)-category|presentation]] for an [[(∞,1)-category of (∞,1)-sheaves]].

That this is indeed the case is

\begin{lemma}
\label{SizeOfLeftCoset}\hypertarget{SizeOfLeftCoset}{}
The [[local model structure on simplicial presheaves]] $SSh(C)^{l loc}_{proj}$ presents the [[hypercompletion|hypercompleted version]] of the [[(∞,1)-category of (∞,1)-sheaves]] $Sh^{hc}(C)$ on $C$.

\begin{displaymath}
(SPSh(C)_{proj}^{loc})^\circ
  \simeq
  Sh^{hc}(C)
  \,.
\end{displaymath}
\end{lemma}
\textbf{Remark} See the discussion at [[?ech cohomology]] for the role of [[hypercompletion]].

\hypertarget{applications}{}\section*{{Applications}}\label{applications}

\hypertarget{abelian_sheaf_cohomology_as_special_case_of_stackification}{}\subsection*{{Abelian sheaf cohomology as special case of $\infty$-stackification}}\label{abelian_sheaf_cohomology_as_special_case_of_stackification}

The [[nerve]] operation of the [[Dold-Kan correspondence]]

\begin{displaymath}
N : Ch_+ \to SimpAb \subset \infty Grpd
\end{displaymath}
embeds [[sheaf|sheaves]] with values in non-negatively graded [[chain complex]]es of abelian groups into simplicial sheaves as those simplicial sheaves with values in [[Kan complex]]es that carry a struict abelian group structure. This way [[homological algebra]] and [[abelian sheaf cohomology]] are realized as special cases of models for $\infty$-stacks: a complex of abelian sheaves presents a stably abelian $\infty$-stack.

\begin{uprop}
Under the [[Dold-Kan correspondence]] [[abelian sheaf cohomology]] identifies with the [[hom-set]] of the [[homotopy category of an (infinity,1)-category|homotopy category]] corresponding [[infinity-stack]] [[(infinity,1)-topos]].

More precisely, let

\begin{itemize}%
\item the underlying [[site]] be the [[category of open subsets]] $C = Op(X)$ of a [[topological space]] $X$,


\item let $A \in Sh(X)$ be a sheaf with values in abelian groups on $X$;


\item let $\mathbf{B}^n A \in Sh(X,SSet)$ be the image of the complex of sheaves $A[-n]$ concentrated in degree $n$ under the [[Dold-Kan correspondence|Dold-Kan]] [[nerve]];


\item write $X \in Sh(X)$ for the [[terminal object]] sheaf in $Sh(X)$ (the sheaf constant on the singleton set).



\end{itemize}
Then degree $n$ [[abelian sheaf cohomology]] of $X$ with coefficients in $A$ is homotopy classes of maps from $X$ to $\mathbf{B}^n A$:

\begin{displaymath}
H^n(X,A) \simeq Ho_{SSh(X)}(X, \mathbf{B}^n A)
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
The original proof was given in [[BrownAHT]] in terms of the [[category of fibrant objects]] structure on locally Kan simplicial sheaves.

The analogous arguments in terms of the full injective model structure were given by Jardine. See section 6 of his \href{http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf}{lecture notes}.

\end{proof}
[[!redirects presentations of (∞,1)-sheaf (∞,1)-toposes]]

[[!redirects models for infinity-stack (infinity,1)-topoi]] [[!redirects models for ∞-stack (∞,1)-toposes]] [[!redirects models for ∞-stack (∞,1)-topoi]] [[!redirects models for infinity-stack (infinity,1)-toposes]]



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