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\begin{document}

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\section*{functor category}

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

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{functor_categories}{}\section*{{Functor categories}}\label{functor_categories}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{usage}{Usage}\dotfill \pageref*{usage} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{limits_and_colimits_and_closure}{Limits and colimits and closure}\dotfill \pageref*{limits_and_colimits_and_closure} \linebreak
\noindent\hyperlink{LocalPresentability}{Accessibility and local presentability}\dotfill \pageref*{LocalPresentability} \linebreak
\noindent\hyperlink{size_issues}{Size issues}\dotfill \pageref*{size_issues} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

Given [[category|categories]] $C$ and $D$, the \textbf{functor category} -- written $D^C$ or $[C,D]$ -- is the category whose

\begin{itemize}%
\item [[object]]s are [[functor|functors]] $F: C \to D$


\item [[morphism]]s are [[natural transformation|natural transformations]] between these functors.



\end{itemize}
\hypertarget{usage}{}\subsection*{{Usage}}\label{usage}

Functor categories serve as the [[hom-category|hom-categories]] in the [[strict 2-category]] [[Cat]].

In the context of [[enriched category theory]] the functor category is generalized to the [[enriched functor category]].

In the absence of the [[axiom of choice]] (including many [[internal category|internal]] situations), the appropriate notion to use is often instead the [[anafunctor category]].

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

\hypertarget{limits_and_colimits_and_closure}{}\subsubsection*{{Limits and colimits and closure}}\label{limits_and_colimits_and_closure}

If $D$ has [[limits]] or [[colimits]] of a certain shape, then so does $[C,D]$ and they are computed pointwise. (However, if $D$ is not complete, then other limits in $[C,D]$ can exist ``by accident'' without being pointwise.)

If $C$ is small and $D$ is [[cartesian closed category|cartesian closed]] and [[complete category|complete]], then $[C,D]$ is cartesian closed. See at \emph{[[cartesian closed category]]} for a proof.

\hypertarget{LocalPresentability}{}\subsubsection*{{Accessibility and local presentability}}\label{LocalPresentability}

Functor categories enjoy the following accessibility and local presentability properties, as explained by [[Zhen Lin Low]] at \href{http://nforum.mathforge.org/discussion/6152}{nForum}.

\begin{itemize}%
\item $\kappa$-[[accessible functors]] from a $\kappa$-[[accessible category]] to any accessible category form an accessible category. (It is not so easy to say what the accessibility rank is here.)


\item $\kappa$-accessible functors from a $\kappa$-accessible category to any locally $\lambda$-[[locally presentable category|presentable category]] form a locally $\lambda$-presentable category.


\item Cocontinuous functors between [[locally presentable categories]] form a locally presentable category. More precisely, if $C$~and~$D$ are locally $\kappa$-presentable, then so is~$[C,D]$.


\item Continuous accessible functors between locally presentable categories form the opposite of a locally presentable category. More precisely, if $C$~and~$D$ are locally $\kappa$-presentable, then so is $[C,D]^{\rm op}$.



\end{itemize}
Indeed, the point is this: given a $\kappa$-[[accessible category]] $\mathcal{C} \simeq Ind^\kappa (\mathcal{A})$ ($\mathcal{A}$ essentially small), the category of $\kappa$-accessible functors $\mathcal{C} \to \mathcal{D}$ (for arbitrary $\mathcal{D}$; here by ``$\kappa$-accessible'' we mean simply ``preserves $\kappa$-filtered colimits'') is naturally equivalent to the category of all $\mathcal{A} \to \mathcal{D}$. It should be well known that:

\begin{enumerate}%
\item If $\mathcal{D}$ is accessible, then so is $[\mathcal{A}, \mathcal{D}]$.


\item If $\mathcal{D}$ is locally $\lambda$-presentable, then so is $[\mathcal{A}, \mathcal{D}]$.


\item Colimit-preserving functors out of a locally $\kappa$-presentable category are $\kappa$-accessible.


\item A right adjoint between locally $\kappa$-presentable categories is $\kappa$-accessible if and only if its left adjoint is strongly $\kappa$-accessible (i.e. preserves $\kappa$-presentable objects as well as $\kappa$-filtered colimits); and every limit-preserving accessible functor between locally presentable categories is a right adjoint.



\end{enumerate}
Statements 1 and 2 are proved in Adamek and Rosick\'y{}, \emph{Locally presentable and accessible categories}, statement 3 is obvious, and statement 4 is a straightforward exercise. Thus the claims follow.

In general, accessible functors between accessible categories do not form an accessible category due to size issues. The best one can hope for is a [[class-accessible category]]. Let $\mathcal{C}$ be an accessible category that is \emph{not} essentially small. Consider the category $\mathcal{A}$ of all accessible functors $\mathcal{C} \to \mathbf{Set}$. This is the same as the smallest full replete subcategory of $[\mathcal{C}, \mathbf{Set}]$ containing all representable functors and closed under small colimits. In particular, $\mathcal{A}$ is accessible if and only if $\mathcal{A}$ locally presentable. We claim $\mathcal{A}$ is not accessible.

Indeed, suppose $\mathcal{A}$ has a small generating family, say $\mathcal{G}$. Then for some regular cardinal $\kappa$, every member of $\mathcal{G}$ is $\kappa$-accessible. So consider $\mathcal{C} (X, -)$ for some object $X$ that is \emph{not} $\kappa$-presentable. (Such an $X$ exists because $\mathcal{C}$ is \emph{not} essentially small.) Since $\mathcal{G}$ generates, there is a small diagram of $\kappa$-accessible functors whose colimit is $\mathcal{C} (X, -)$. But then $\mathcal{C} (X, -)$ is a retract of a $\kappa$-accessible functor and hence $\kappa$-accessible: a contradiction. That said, $\mathcal{A}$ is a [[class-locally presentable category]].

\hypertarget{size_issues}{}\subsection*{{Size issues}}\label{size_issues}

If $C$ and $D$ are [[small category|small]], then $[C,D]$ is also small.

If $C$ is small and $D$ is [[locally small category|locally small]], then $[C,D]$ is still locally small.

Even if $C$ and $D$ are locally small, if $C$ is not small, then $[C,D]$ will usually not be locally small.

As a partial converse to the above, if $C$ and $[C,Set]$ are locally small, then $C$ must be [[essentially small category|essentially small]]; see \href{http://tac.mta.ca/tac/volumes/1995/n9/1-09abs.html}{Freyd \& Street (1995)}.

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

\begin{itemize}%
\item [[2-functor 2-category]]


\item [[(infinity,1)-category of (infinity,1)-functors]]



\end{itemize}
[[!redirects functor category]] [[!redirects functor categories]] [[!redirects diagram category]] [[!redirects diagram categories]]

[[!redirects category of functors]] [[!redirects categories of functors]]



\end{document}