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\begin{document}

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\section*{simplicial presheaf}

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

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

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

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

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{interpretation_as_stacks}{Interpretation as $\infty$-stacks}\dotfill \pageref*{interpretation_as_stacks} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\emph{Simplicial presheaves} over some [[site]] $S$ are

\begin{itemize}%
\item [[presheaf|Presheaves]] with values in the category [[SimpSet]] of simplicial sets, i.e., functors $S^{op} \to \Simp\Set$, i.e., functors $S^{op} \to [\Delta^{op}, \Set]$;

\end{itemize}
or equivalently, using the Hom-[[adjoint functor|adjunction]] and symmetry of the [[closed monoidal category|closed monoidal structure]] on [[Cat]]

\begin{itemize}%
\item simplicial objects in the category of presheaves, i.e. functors $\Delta^{op} \to [S^{op},\Set]$.

\end{itemize}
\hypertarget{interpretation_as_stacks}{}\subsection*{{Interpretation as $\infty$-stacks}}\label{interpretation_as_stacks}

Regarding $\Simp\Set$ as a [[model category]] using the standard [[model structure on simplicial sets]] and inducing from that a model structure on $[S^{op}, \Simp\Set]$ makes simplicial presheaves a model for $\infty$-[[infinity-stack|stacks]], as described at [[infinity-stack homotopically]].

In more illustrative language this means that a simplicial presheaf on $S$ can be regarded as an $\infty$-[[infinity-groupoid|groupoid]] (in particular a [[Kan complex]]) whose space of $n$-morphisms is modeled on the objects of $S$ in the sense described at [[space and quantity]].

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

\begin{itemize}%
\item Notice that most definitions of $\infty$-[[infinity-category|category]] the $\infty$-category is itself defined to be a [[simplicial set]] with extra structure (in a [[geometric definition of higher category]]) or gives rise to a simplicial set under taking its [[nerve]] (in an [[algebraic definition of higher category]]). So most notions of presheaves of higher categories will naturally induce presheaves of simplicial sets.


\item In particular, regarding a [[group]] $G$ as a one object category $\mathbf{B}G$ and then taking the nerve $N(\mathbf{B}G) \in \Simp\Set$ of these (the ``classifying simplicial set of the group whose [[geometric realization]] is the [[classifying space]] $\mathcal{B}G$), which is clearly a functorial operation, turns any presheaf with values in groups into a simplicial presheaf.



\end{itemize}
\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item There are various useful [[model category]] structures on the category of simplicial presheaves. See [[model structure on simplicial presheaves]].

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

Here are some basic but useful facts about simplicial presheaves.

\begin{uprop}
Every simplicial presheaf $X$ is a [[homotopy limit|homotopy colimit]] over a [[diagram]] of [[Set]]-valued sheaves regarded as discrete simplicial sheaves.

More precisely, for $X : S^{op} \to SSet$ a simplicial presheaf, let $D_X : \Delta^{op} \to Set \hookrightarrow SSet$ be given by $D_X : [n] \mapsto X_n$. Then there is a weak equivalence

\begin{displaymath}
hocolim_{[n] \in \Delta} D_X([n]) \stackrel{\simeq}{\to} X
  \,.
\end{displaymath}
\end{uprop}
\begin{proof}
See for instance \href{http://www.math.uiuc.edu/K-theory/0563/spre.pdf#page=6}{remark 2.1, p. 6}

\begin{itemize}%
\item Daniel Dugger, Sharon Hollander, Daniel C. Isaksen, \emph{Hypercovers and simplicial presheaves} (\href{http://www.math.uiuc.edu/K-theory/0563/}{web})

\end{itemize}
(which is otherwise about [[descent for simplicial presheaves]]).

\end{proof}
\begin{ucor}
Let $[-,-] : SSet^{S^{op}} \to SSet$ be the canonical $SSet$-enrichment of the category of simplicial presheaves (i.e. the assignment of [[SSet]]-[[enriched functor category|enriched functor categories]]).

It follows in particular from the above that every such [[hom-object]] $[X,A]$ of simplical presheaves can be written as a [[homotopy limit]] (in [[SSet]] for instance realized as a [[weighted limit]], as described there) over evaluations of $X$.

\end{ucor}
\begin{proof}
First the above yields

\begin{displaymath}
\begin{aligned}
     [X, A ]  & \simeq [ hocolim_{[n] \in \Delta} X_n , A ]
  \\
      & holim_{[n] \in \Delta} [X_n, A]
  \end{aligned}
  \,.
\end{displaymath}
Next from the [[co-Yoneda lemma]] we know that the [[Set]]-valued presheaves $X_n$ are in turn colimits over representables in $S$, so that

\begin{displaymath}
\begin{aligned}
     \cdots & \simeq 
     holim_{[n] \in \Delta} 
     [ colim_i U_{i}, A]
     \\
     & \simeq
     holim_{[n] \in \Delta} lim_i
     [  U_{i}, A]       
  \end{aligned}
  \,.
\end{displaymath}
And finally the [[Yoneda lemma]] reduces this to

\begin{displaymath}
\begin{aligned}
     \cdots
      & 
     holim_{[n] \in \Delta} lim_i
     A(U_i)            
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
Notice that these kinds of computations are in particular often used when checking/computing [[descent|descent and codescent]] along a [[cover]] or [[hypercover]]. For more on that in the context of simplicial presheaves see [[descent for simplicial presheaves]].

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[model structure on simplicial presheaves]]


\item [[descent for simplicial presheaves]]


\item [[presheaf of spectra]]



\end{itemize}
Applications appear for instance at

\begin{itemize}%
\item [[geometric infinity-function theory]]

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

The theory of simplicial presheaves and of simplicial sheaves was developed by J. Jardine in a long series of articles, some of which are listed below. It's usage as a model for [[infinity-stack]]s was developed by T\"o{}en as described at [[infinity-stack homotopically]].

\begin{itemize}%
\item \textbf{JardStackSSh} -- J. Jardine, \emph{Stacks and the homotopy theory of simplicial sheaves}, Homology, homotopy and applications, vol. 3(2), 2001 p. 361-284 (\href{http://intlpress.com/HHA/v3/n2/a5/v3n2a5.pdf}{pdf})
\item \textbf{JardSimpSh} -- J. Jardine, \emph{Fields Lectures: Simplicial presheaves} (\href{http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf}{pdf})

\end{itemize}
For their interpretation in the more general context of [[(infinity,1)-category of (infinity,1)-sheaves|(infinity,1)-sheaves]] see \href{}{section 6.5.2} of

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

\end{itemize}
[[!redirects simplicial presheaves]]

[[!redirects category of simplicial presheaves]] [[!redirects categories of simplicial presheaves]]



\end{document}