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

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

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

[[!include quasi-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{models}{Models}\dotfill \pageref*{models} \linebreak
\noindent\hyperlink{Limits}{Limits and colimits}\dotfill \pageref*{Limits} \linebreak
\noindent\hyperlink{equivalences}{Equivalences}\dotfill \pageref*{equivalences} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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 generalization of the notion of [[functor category]] from [[category theory]] to [[(∞,1)-category|(∞,1)]]-[[higher category theory]].

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

Let $C$ and $D$ be [[(∞,1)-categories]], taken in their incarnation as [[quasi-category|quasi-categories]]. Then

\begin{displaymath}
Func(C,D) := sSet(C,D)
\end{displaymath}
is the [[simplicial set]] of morphisms of simplicial sets between $C$ and $D$ (in the standard [[sSet]]-[[enriched category|enrichment]] of $SSet$):

\begin{displaymath}
sSet(C,D) := [C,D] := ([n] \mapsto Hom_{sSet}(\Delta[n]\times C,D))
  \,.
\end{displaymath}
The objects in $Fun(C,D)$ are the [[(∞,1)-functors]] from $C$ to $D$, the morphisms are the corresponding [[natural transformations]] or [[homotopy|homotopies]], etc.

\begin{prop}
\label{}\hypertarget{}{}
The simplicial set $Fun(C,D)$ is indeed a [[quasi-category]].

In fact, for $C$ and $D$ any simplicial sets, $Fun(C,D)$ is a [[quasi-category]] if $D$ is a [[quasi-category]].

\end{prop}
\begin{proof}
Using that [[sSet]] is a [[closed monoidal category]] the [[horn]] filling conditions

\begin{displaymath}
\itexarray{
    \Lambda[n]_i &\to& [C,D]
    \\
    \downarrow & \nearrow
    \\
    \Delta[n]
  }
\end{displaymath}
are equivalent to

\begin{displaymath}
\itexarray{
    C \times \Lambda[n]_i &\to& D
    \\
    \downarrow & \nearrow
    \\
    C \times \Delta[n]
  }
  \,.
\end{displaymath}
Here the vertical map is [[fibrations of quasi-categories|inner anodyne]] for inner horn inclusions $\Lambda[n]_i \hookrightarrow \Delta[n]$, and hence the lift exists whenever $D$ has all inner horn fillers, hence when $D$ is a [[quasi-category]].

\end{proof}
For the definition of $(\infty,1)$-functors in other models for $(\infty,1)$-categories see [[(∞,1)-functor]].

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

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

The projective and injective [[global model structure on functors]] as well as the [[Reedy model structure]] if $C$ is a [[Reedy category]] [[presentable (infinity,1)-category|presents]] $(\infty,1)$-categories of $(\infty,1)$-functors, at least when there exists a [[combinatorial simplicial model category]] model for the codomain.

Let

\begin{itemize}%
\item $C$ be a small [[sSet]]-[[enriched category]];


\item $A$ a [[combinatorial simplicial model category]] and $A^\circ$ its full [[sSet]]-subcategory of fibrant cofibrant objects;


\item $[C,A]$ the [[sSet]]-[[enriched functor category]] equipped with either the injective or projective [[global model structure on functors]] -- here: the injective or injective [[model structure on sSet-enriched presheaves]] -- and $[C,A]^\circ$ its full [[sSet]]-[[subcategory]] on fibrant-cofibrant objects.



\end{itemize}
Write $N : sSet Cat \to sSet$ for the [[homotopy coherent nerve]]. Since this is a [[right adjoint]] it preserves [[product]]s and hence we have a canonical morphism

\begin{displaymath}
N(C) \times N([C,A]) \simeq
  N(C \times [C,A])
  \stackrel{N(ev)}{\to}
  N(A)
\end{displaymath}
induced from the hom-[[adjunct]] of $Id : [C,A] \to [C,A]$.

The fibrant-cofibrant objects of $[C,A]$ are [[enriched functor]]s that in particular take values in fibrant cofibrant objects of $A$. Therefore this restricts to a morphism

\begin{displaymath}
N(C) \times N([C,A]^\circ) 
  \stackrel{N_{hc}(ev)}{\to}
  N(A^\circ)
  \,.
\end{displaymath}
By the [[internal hom]] [[adjunction]] this corresponds to a morphism

\begin{displaymath}
N([C,A]^\circ) 
  \stackrel{}{\to}
  sSet(N_{hc}(C),  N(A^\circ))
  \,.
\end{displaymath}
Here $A^\circ$ is [[Kan complex]] enriched by the axioms of an $sSet_{Quillen}$- [[enriched model category]], and so $N(A^\circ)$ is a [[quasi-category]], so that we may write this as

\begin{displaymath}
\cdots = Func(N(C), N(A^\circ))
  \,.
\end{displaymath}
\begin{prop}
\label{PresentationByModelStructuresOnFunctors}\hypertarget{PresentationByModelStructuresOnFunctors}{}
This canonical morphism

\begin{displaymath}
N([C,A]^\circ) 
  \stackrel{}{\to}
  Func(N(C),  N(A^\circ))
\end{displaymath}
is an $(\infty,1)$-equivalence in that it is a weak equivalence in the [[model structure for quasi-categories]].

\end{prop}
This is (\hyperlink{Lurie}{Lurie, prop. 4.2.4.4}).

\begin{proof}
The strategy is to show that the objects on both sides are [[exponential object]]s in the [[homotopy category]] of $sSet_{Joyal}$, hence isomorphic there.

That $Func(N(C), N(A^\circ)) \simeq (N(A^\circ))^{N(C)}$ is an exponential object in the homotopy category is pretty immediate.

That the left hand is an isomorphic exponential follows from (\hyperlink{Lurie}{Lurie, corollary A.3.4.12}), which asserts that for $C$ and $D$ $sSet$-[[enriched categories]] with $C$ cofibrant and $A$ as above, we have that composition with the evaluation map induces a bijection

\begin{displaymath}
Hom_{Ho(sSet Cat)}(D, [C,A]^\circ)
  \stackrel{\simeq}{\to}
  Hom_{Ho(sSet Cat)}(C \times D, A^\circ)
  \,.
\end{displaymath}
Since $Ho(sSet Cat_{Bergner}) \simeq Ho(sSet_{Joyal})$ this identifies also $N([C,A]^\circ)$ with the exponential object in question.

\end{proof}
\hypertarget{Limits}{}\subsubsection*{{Limits and colimits}}\label{Limits}

For $C$ an ordinary [[category]] that admits small [[limit]]s and [[colimit]]s, and for $K$ a [[small category]], the [[functor category]] $Func(D,C)$ has all small limits and colimits, and these are computed objectwise. See [[limits and colimits by example]]. The analogous statement is true for $(\infty,1)$-categories of $(\infty,1)$-functors

\begin{prop}
\label{}\hypertarget{}{}
Let $K$ and $C$ be [[quasi-categories]], such that $C$ has all [[limit in a quasi-category|colimits]] indexed by $K$.

Let $D$ be a small quasi-category. Then

\begin{itemize}%
\item The $(\infty,1)$-category $Func(D,C)$ has all $K$-indexed colimits;


\item A morphism $K^\triangleright \to Func(D,C)$ is a colimiting cocone precisely if for each object $d \in D$ the induced morphism $K^\triangleright \to C$ is a colimiting cocone.



\end{itemize}
\end{prop}
This is (\hyperlink{Lurie}{Lurie, corollary 5.1.2.3}).

\hypertarget{equivalences}{}\subsubsection*{{Equivalences}}\label{equivalences}

\begin{prop}
\label{}\hypertarget{}{}
A morphism $\alpha$ in $Func(D,C)$ (that is, a [[natural transformation]]) is an [[equivalence in an (infinity,1)-category|equivalence]] if and only if each component $\alpha_d$ is an equivalence in $C$.

\end{prop}
This is due to (\hyperlink{Joyal}{Joyal, Chapter 5, Theorem C}).

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

\begin{itemize}%
\item Between ordinary categories, it reproduces the ordinary [[category of functors]].


\item Since the standard [[model structure on simplicial sets]] presents [[? Grpd]]

\begin{displaymath}
(sSet_{Quillen})^\circ \simeq \infty Grpd
\end{displaymath}
the [[model structure on simplicial presheaves]] (more precisely and more generally the [[model structure on sSet-enriched presheaves]]) on the [[opposite (∞,1)-category]] $C^{op}$ [[presentable (infinity,1)-category|presents]] the [[(∞,1)-category of (∞,1)-presheaves]] on $C$:

\begin{displaymath}
N([C^{op},sSet_{Quillen}]^\circ) \simeq Func(C^{op},\infty Grpd)
  = PSh_{(\infty,1)}(C)
  \,.
\end{displaymath}


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

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


\item [[2-functor 2-category]]



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

The intrinsic definition is in section 1.2.7 of

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

\end{itemize}
The discussion of [[model category]] models is in A.3.4.

The theorem about equivalences is in

\begin{itemize}%
\item [[André Joyal]], \emph{The theory of quasicategories and its applications} lectures at \emph{\href{http://www.crm.es/HigherCategories/}{Simplicial Methods in Higher Categories}}, (\href{http://mat.uab.cat/~kock/crm/hocat/advanced-course/Quadern45-2.pdf}{pdf})

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

[[!redirects (∞,1)-categories of (∞,1)-functors]]

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

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



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