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\begin{document}

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

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

\hypertarget{category_theory}{}\paragraph*{{$(\infty,1)$-Category theory}}\label{category_theory}

[[!include quasi-category theory 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{Models}{Models}\dotfill \pageref*{Models} \linebreak
\noindent\hyperlink{limits_and_colimits}{Limits and colimits}\dotfill \pageref*{limits_and_colimits} \linebreak
\noindent\hyperlink{FreeColimCompletion}{As the free completion under colimits}\dotfill \pageref*{FreeColimCompletion} \linebreak
\noindent\hyperlink{WithOvercategories}{Relation to slicing}\dotfill \pageref*{WithOvercategories} \linebreak
\noindent\hyperlink{subcategories_of_presheaf_categories}{$(\infty,1)$-subcategories of $(\infty)$-presheaf categories}\dotfill \pageref*{subcategories_of_presheaf_categories} \linebreak
\noindent\hyperlink{locally_presentable_categories}{Locally presentable $(\infty,1)$-categories}\dotfill \pageref*{locally_presentable_categories} \linebreak
\noindent\hyperlink{sheaf_categories}{$(\infty,1)$-Sheaf $(\infty,1)$-categories}\dotfill \pageref*{sheaf_categories} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

For [[∞Grpd]] the [[(∞,1)-category]] of [[∞-groupoids]], and for $S$ a [[(∞,1)-category]] (or in fact any [[simplicial set]]), an \textbf{$(\infty,1)$-presheaf} on $S$ is an $(\infty,1)$-functor

\begin{displaymath}
F : S^{op} \to \infty Grpd
  \,.
\end{displaymath}
The \textbf{$(\infty,1)$-category of $(\infty,1)$-presheaves} is the [[(∞,1)-category of (∞,1)-functors]]

\begin{displaymath}
PSh_{(\infty,1)}(S) := Func(S^{op}, \infinity Grpd)
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{Models}{}\subsubsection*{{Models}}\label{Models}

A [[model category|model]] for an $(\infty,1)$-presheaf categories is the [[model structure on simplicial presheaves]]. See also the discussion at [[models for ∞-stack (∞,1)-toposes]].

\begin{uprop}
For $C$ a [[simplicially enriched category]] with [[Kan complex]]es as hom-objects, write $[C^{op}, sSet_{Quillen}]_{proj}$ and $[C^{op}, sSet_{Quillen}]_{inj}$ for the projective or injective, respectively, gloabl [[model structure on simplicial presheaves]]. Write $(-)^\circ$ for the full [[sSet]]-[[enriched category|enriched]] [[subcategory]] on fibrant-cofibrant objects, and $N(-)$ for the [[homotopy coherent nerve]] that sends a Kan-complex enriched category to a [[quasi-category]].

Then there is an [[equivalence of quasi-categories]]

\begin{displaymath}
PSh(N(C))
  \simeq
  N ([C^{op}, sSet_{Quillen}]_{proj})^\circ
  \,.
\end{displaymath}
Similarly for the injective model structure.

\end{uprop}
\begin{proof}
This is a special case of the more general statement that the [[model structure on functors]] models an [[(∞,1)-category of (∞,1)-functors]]. See there for more details.

\end{proof}
Notice that the result in particular means that any $(\infty,1)$-presheaf -- an ``$\infty$-[[pseudofunctor]]'' -- may be \emph{straightened} or \emph{rectified} to a genuine [[sSet]]-[[enriched functor]], that respects horizontal compositions strictly.

\hypertarget{limits_and_colimits}{}\subsubsection*{{Limits and colimits}}\label{limits_and_colimits}

In an ordinary [[category of presheaves]], [[limit]]s and colimits are computed objectwise, as described at [[limits and colimits by example]]. The analogous statement is true for [[limit in a quasi-category|(∞,1)-limits]] and colimits in an $(\infty,1)$-category of $(\infty,1)$-presheaves.

This is a special case of the general existence of limits and colimits in an [[(∞,1)-category of (∞,1)-functors]]. See there for more details.

\begin{ucor}
For $C$ a small $(\infty,1)$-category, the $(\infty,1)$-category $PSh(C)$ admits all small limits and colimits.

\end{ucor}
See around [[Higher Topos Theory|HTT, cor. 5.1.2.4]].

\hypertarget{FreeColimCompletion}{}\subsubsection*{{As the free completion under colimits}}\label{FreeColimCompletion}

An ordinary [[category of presheaves]] on a small category $C$ is the [[free cocompletion]] of $C$, the free completion under forming colimits.

The analogous result holds for $(\infty,1)$-category of $(\infty,1)$-presheaves.

\begin{ulemma}
Let $C$ be a small [[quasi-category]] and $j : S \to PSh(C)$ the [[(∞,1)-Yoneda embedding]].

The identity [[(∞,1)-functor]] $Id : PSh(C) \to PSh(C)$ is the left [[(∞,1)-Kan extension]] of $j$ along itself.

\end{ulemma}
\begin{proof}
This is [[Higher Topos Theory|HTT, lemma 5.1.5.3]].

\end{proof}
For $D$ a [[quasi-category]] with all small [[limit in a quasi-category|colimits]], write $Func^L(PSh(C),D) \subset Func(PSh(C),D)$ for the full [[sub-quasi-category]] of the [[(∞,1)-category of (∞,1)-functors]] on those that preserve small [[limit in a quasi-category|colimits]].

\begin{ulemma}
Composition with the Yoneda embedding $j : C \to PSh(C)$ induces an [[equivalence of quasi-categories]]

\begin{displaymath}
Func^L(PSh(C),D) \to Func(C,D)
  \,.
\end{displaymath}
\end{ulemma}
\begin{proof}
This is [[Higher Topos Theory|HTT, theorem 5.1.5.6]].

\end{proof}
In terms of the model given by the [[model structure on simplicial presheaves]], this is statement made in

\begin{itemize}%
\item [[Dan Dugger]], \emph{[[Universal homotopy theories]]} ,

\end{itemize}
which gives that article its name.

\begin{udef}
Let $A$ and $B$ be [[model categories]], $D$ a plain [[category]] and

\begin{displaymath}
\itexarray{
    D &\stackrel{r}{\to}& A
    \\
    & \searrow_\gamma 
    \\
    && B
  }
\end{displaymath}
two plain [[functor]]s. Say that a \textbf{model-category theoretic factorization} of $\gamma$ through $A$ is

\begin{enumerate}%
\item a [[Quillen adjunction]] $(L \dashv R) : A \stackrel{\overset{L}{\to}}{\underset{R}{\leftarrow}} B$


\item a [[natural transformation|natural]] weak equivalence $\eta  : L \circ r \to \gamma$

\begin{displaymath}
\itexarray{
    D &&\stackrel{r}{\to}&& A
    \\
    & \searrow_\gamma &{}^\eta\swArrow& \swarrow_L
    \\
    && B
  }
  \,.
\end{displaymath}


\end{enumerate}
Let the [[category]] of such factorizations have morphisms $((L \dashv R), \eta ) \to ((L' \dashv R'), \eta' )$ given by [[natural transformation]]s $L \to L'$ such that for all all objects $d \in D$ the diagrams

\begin{displaymath}
\itexarray{
     L\circ r(d) &&\to&& L'\circ r(d)
     \\
     & {}_{\eta_{d}}\searrow && \swarrow_{\eta'_{d}}
     \\
     &&
     \gamma()
  }
\end{displaymath}
commutes.

\end{udef}
Notice that the [[(∞,1)-category]] presented by a [[model category]] -- at least by a [[combinatorial model category]] -- has all [[limit in a quasi-category|(∞,1)-categorical colimits]], and that the Quillen left adjoint functor $L$ presents, via its [[derived functor]], a left [[adjoint (∞,1)-functor]] that preserves $(\infty,1)$-categorical colimits. So the notion of factorization as above is really about factorizations through colimit-preserving $(\infty,1)$-functors into $(\infty,1)$-categories that have all colimits.

\begin{utheorem}
\textbf{(model category presentation of free $(\infty,1)$-cocompletion)}

For $C$ a [[small category]], the projective global [[model structure on simplicial presheaves]] $[C^{op}, sSet]_{proj}$ on $C$ is universal with respect to such factorizations of functors out of $C$:

every functor $C \to B$ to any [[model category]] $B$ has a factorization through $[C^{op}, sSet]_{proj}$ as above, and the category of such factorizations is [[contractible]].

\end{utheorem}
\begin{proof}
This is theorem 1.1 in

\begin{itemize}%
\item [[Dan Dugger]], \emph{[[Universal homotopy theories]]} .

\end{itemize}
The proof is on page 30.

To produce the factorization $[C^{op},sSet] \to B$ given the functor $\gamma$, first notice that the ordinary [[Yoneda extension]] $[C^{op},Set] \to B$ would be given by the left [[Kan extension]] given by the [[coend]] formula

\begin{displaymath}
F \mapsto \int^{c \in C} \gamma(c) \cdot F(c)
  \,,
\end{displaymath}
where the dot in the integrand is the [[copower|tensoring]] of cocomplete category $B$ over [[Set]]. To refine this to a [[Quillen adjunction|left Quillen functor]] $L : [C^{op},sSet] \to B$, \emph{choose} a [[cosimplicial resolution]]

\begin{displaymath}
\Gamma : C \to [\Delta,B]
\end{displaymath}
of $\gamma$. Then set

\begin{displaymath}
L : F \mapsto 
  \int^{c \in C} 
  \int^{[n] \in \Delta}
  \Gamma^n(c) \cdot F_n(c)
  \,.
\end{displaymath}
The [[right adjoint]] $R : B \to [C^{op},sSet]$ of this functor is given by

\begin{displaymath}
R(X) : c \mapsto Hom_B(\Gamma^\bullet(c), X)
  \,.
\end{displaymath}
For $(L \dashv R) : [C^{op}, sSet]_{proj} \stackrel{\to}{\leftarrow} B$ to be a [[Quillen adjunction]], it is sufficient to check that $R$ preserves fibrations and acyclic fibrations. By definition of the projective model structure this means that for every (acyclic) fibration $b_1 \to b_2$ in $B$ we have for every object $c \in C$ that that

\begin{displaymath}
Hom_C(\Gamma^\bullet(c), b_1 \to b_2)
\end{displaymath}
is an (acyclic) fibration of simplicial sets. But this is one of the standard properties of [[cosimplicial resolution]]s.

Finally, to find the natural weak equivalence $\eta : L \circ j \simeq \gamma$, write $j : C \to [C^{op},sSet]$ for the [[Yoneda embedding]] and notice that by [[Yoneda reduction]] it follows that for $x \in C$ we have

\begin{displaymath}
L(j(x))
  =
  \int^{c \in C} \int^{[n] \in \Delta}
  \Gamma^n(c) \cdot C(c,x)
  =
  \Gamma^0(x)
\end{displaymath}
(where equality signs denote [[isomorphism]]s).

By the very definition of cosimplicial resolutions, there is a natural weak equivalence $\Gamma(x) \stackrel{\simeq}{\to}$. We can take this to be the component of $\eta$.

\end{proof}
\begin{ucor}
The [[(∞,1)-Yoneda embedding]] $j : C \to PSh(C)$ generates $PSh(C)$ under small colimits:

a full [[sub-quasi-category|(∞,1)-subcategory]] of $PSh(C)$ that contains all representables and is closed under forming $(\infty,1)$-colimits is already equivalent to $PSh(C)$.

\end{ucor}
\begin{proof}
This is [[Higher Topos Theory|HTT, corollary 5.1.5.8]].

\end{proof}
\hypertarget{WithOvercategories}{}\subsubsection*{{Relation to slicing}}\label{WithOvercategories}

The following analog of the corresponding result for 1-[[categories of presheaves]] holds for $(\infty,1)$-presheaves. See [[functors and comma categories]].

\begin{prop}
\label{SlicingCommutesWithFormingPresheaves}\hypertarget{SlicingCommutesWithFormingPresheaves}{}
\textbf{(slicing commutes with passing to presheaves)}

Let $\mathcal{C}$ be a [[small (∞,1)-category]] and $p \colon \mathcal{K} \to \mathcal{C}$ a [[diagram]].

Write $\mathcal{C}_{/p}$ and $PSh_\infty(\mathcal{C})/_{y p}$ for the corresponding [[over quasi-category|over categories]], where $y \colon \mathcal{C} \to PSh_\infty(\mathcal{C})$ is the [[(∞,1)-Yoneda embedding]].

Then we have an [[equivalence of quasi-categories|equivalence of (∞,1)-categories]]

\begin{displaymath}
PSh_\infty(\mathcal{C}_{/p}) 
    \stackrel{\simeq}{\to} 
  PSh_\infty(\mathcal{C})_{/y p}
  \,.
\end{displaymath}
\end{prop}
This appears as [[Higher Topos Theory|HTT, 5.1.6.12]].

\hypertarget{subcategories_of_presheaf_categories}{}\subsection*{{$(\infty,1)$-subcategories of $(\infty)$-presheaf categories}}\label{subcategories_of_presheaf_categories}

\hypertarget{locally_presentable_categories}{}\subsubsection*{{Locally presentable $(\infty,1)$-categories}}\label{locally_presentable_categories}

A [[reflective (∞,1)-subcategory]] of an $(\infty,1)$-category of $(\infty,1)$-presheaves is called a [[presentable (∞,1)-category]].

\hypertarget{sheaf_categories}{}\subsubsection*{{$(\infty,1)$-Sheaf $(\infty,1)$-categories}}\label{sheaf_categories}

If that [[left adjoint|left]] [[adjoint (∞,1)-functor]] to the embedding of the [[reflective (∞,1)-subcategory]] furthermore preserves finite [[limit]]s, then the subcategory is an [[(∞,1)-category of (∞,1)-sheaves]]: an [[(∞,1)-topos]]

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

[[!include locally presentable categories - table]]

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

This is the topic of section 5.1 of

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

\end{itemize}
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[[!redirects free (∞,1)-cocompletion]] [[!redirects free (infinity,1)-cocompletion]]

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

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

[[!redirects infinity-category of infinity-presheaves]] [[!redirects infinity-categories of infinity-presheaves]]



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