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\begin{document}

%-------------------------------------------------------------------


\section*{cartesian closed functor}

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

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

[[!include category theory - contents]]

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - 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{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{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{cartesian closed functor} is a [[functor]] $F\colon \mathcal{C}\to \mathcal{D}$ between [[cartesian closed categories]] which preserves both [[products]] and [[exponential objects]]/[[internal homs]] (all the structure of cartesian closed categories).

More precisely, if $F\colon C\to D$ preserves products, then the canonical [[morphisms]] $F(A\times B) \to F A \times F B$ (for all [[objects]] $A,B \mathcal{C}$) are [[isomorphisms]], and we therefore have canonical induced morphism $F[A,B] \to [F A, F B]$ --- the [[adjuncts]] of the composite $F[A,B] \times F A \xrightarrow{\cong} F([A,B] \times A) \to F B$. $F$ is \textbf{cartesian closed} if these maps $F[A,B] \to [F A, F B]$ are also isomorphisms.

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
When cartesian closed categories are identified with [[cartesian monoidal categories]] that are also [[closed monoidal category|closed monoidal]], a cartesian closed functor can be identified with a [[strong monoidal functor]] which is also [[strong closed functor|strong closed]].

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

\begin{prop}
\label{FrobeniusReciprocity}\hypertarget{FrobeniusReciprocity}{}
\textbf{(Frobenius reciprocity)}

Let $R : \mathcal{C} \to \mathcal{D}$ be a [[functor]] between [[cartesian closed categories]] with a [[left adjoint]] $L$. Then $R$ is cartesian closed precisely if the [[natural transformation]]

\begin{displaymath}
(L \pi_1, \epsilon_A L \pi_2)
  : 
  L(B \times R(A))
  \to 
  L(B) \times A
\end{displaymath}
is an [[isomorphism]].

\end{prop}
\begin{proof}
The above natural transformation is the [[mate]] of the exponential comparison natural transformation $R[A,B] \to [R A, R B]$ under the composite adjunctions

\begin{displaymath}
\mathcal{C}
\underoverset{[R A, -]}{- \times R A}{\rightleftarrows}
\mathcal{C}
\underoverset{R}{L}{\rightleftarrows}
\mathcal{D}
\end{displaymath}
and

\begin{displaymath}
\mathcal{C} 
\underoverset{R}{L}{\rightleftarrows}
\mathcal{D}
\underoverset{[A,-]}{A\times -}{\rightleftarrows}
\mathcal{D}
\end{displaymath}
\end{proof}
This is called the \textbf{[[Frobenius reciprocity]]} law. It is discussed, for instance, as (\hyperlink{Johnstone}{Johnstone, lemma 1.5.8}). More generally for closed monoidal categories (not necessarily cartesian monoidal) it is discussed in ``[[Wirthmüller contexts]]'' in

Let still $R$ and $L$ be as above.

\begin{cor}
\label{}\hypertarget{}{}
If $R$ is [[full and faithful]] and $L$ preserves binary [[products]], then $R$ is cartesian closed.

\end{cor}
For instance (\hyperlink{Johnstone}{Johnstone, corollary A1.5.9}).

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

\begin{prop}
\label{}\hypertarget{}{}
For $\mathcal{C}$ a [[locally cartesian closed category]] and $f : X_1 \to X_2$ a [[morphism]], the [[base change]]/[[pullback]] functor between the [[slice categories]]

\begin{displaymath}
f^* : \mathcal{C}_{/X_2} \to \mathcal{C}_{/X_1}
\end{displaymath}
is cartesian closed.

In particular the [[inverse image]] functor of an [[étale geometric morphism]] between [[toposes]] is cartesian closed and hence constitutes a cartesian [[Wirthmüller context]].

\end{prop}
\begin{proof}
The functor $f^*$ has a [[left adjoint]]

\begin{displaymath}
\sum_f : \mathcal{C}_{/X_1} \to \mathcal{C}_{/X_2}
\end{displaymath}
given by postcomposition with $f$ (the [[dependent sum]] along $f$). Therefore by prop. \ref{FrobeniusReciprocity} it is sufficient to show that for all $(A \to X_2)$ in $\mathcal{C}_{/X_2}$ and $(B \stackrel{b}{\to} X_1) \in \mathcal{C}_{/X_1}$ that

\begin{displaymath}
B \times_{X_1} f^* A \simeq B \times_{X_2} A
\end{displaymath}
in $\mathcal{C}$. But this is the [[pasting law]] for pullbacks in $\mathcal{C}$, which says that the two consecutive pullbacks on the left of

\begin{displaymath}
\itexarray{
    B \times_{X_1} f^* A &\to& f^* A &\to& A 
    \\
    \downarrow && \downarrow && \downarrow
    \\
    B &\stackrel{b}{\to}& X_1 &\stackrel{f}{\to}& X_2  
  }  
  \;\;\;
  \simeq
  \;\;\;
  \itexarray{
    (b \circ f)^* A &\to&  &\to& A 
    \\
    \downarrow &&  && \downarrow
    \\
    B &\stackrel{b}{\to}& X_1 &\stackrel{f}{\to}& X_2  
  }
\end{displaymath}
are isomorphic to the direct pullback along the composite on the right.

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

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


\item [[cartesian closed category]], [[locally cartesian closed category]]


\item \textbf{cartesian closed functor}, [[locally cartesian closed functor]]


\item [[cartesian closed model category]], [[locally cartesian closed model category]]


\item [[cartesian closed (∞,1)-category]] [[locally cartesian closed (∞,1)-category]]



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

For instance section A1.5 of

\begin{itemize}%
\item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]}

\end{itemize}
Also

\begin{itemize}%
\item H. Fausk, P. Hu, [[Peter May]], \emph{Isomorphisms between left and right adjoints}, Theory and Applications of Categories , Vol. 11, 2003, No. 4, pp 107-131. (\href{http://www.tac.mta.ca/tac/volumes/11/4/11-04abs.html}{TAC}, \href{http://www.math.uiuc.edu/K-theory/0573/FormalFeb16.pdf}{pdf})

\end{itemize}
[[!redirects cartesian closed functors]]



\end{document}