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\begin{document}

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\section*{cartesian closed category}

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

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

[[!include monoidal categories - contents]]

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

[[!include category theory - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{some_basic_consequences}{Some basic consequences}\dotfill \pageref*{some_basic_consequences} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{inheritance_by_reflective_subcategories}{Inheritance by reflective subcategories}\dotfill \pageref*{inheritance_by_reflective_subcategories} \linebreak
\noindent\hyperlink{exponentials_of_cartesian_closed_categories}{Exponentials of cartesian closed categories}\dotfill \pageref*{exponentials_of_cartesian_closed_categories} \linebreak
\noindent\hyperlink{functional_completeness_theorem}{Functional completeness theorem}\dotfill \pageref*{functional_completeness_theorem} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{literature}{Literature}\dotfill \pageref*{literature} \linebreak


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

A \textbf{cartesian closed category} (sometimes: \textbf{ccc}) is a category with [[finite products]] which is [[closed monoidal category|closed]] with respect to its [[cartesian monoidal category|cartesian monoidal structure]].

The [[internal hom]] $[S,X]$ in a cartesian closed category is often called [[exponentiation]] and is denoted $X^S$.

A \emph{cartesian closed functor} between cartesian closed categories $C$, $D$ is a functor $F \colon C \to D$ that is [[product-preserving functor|product-preserving]] and that is also \emph{exponential-preserving}, meaning that the canonical map

\begin{displaymath}
F(a^b) \to F(a)^{F(b)},
\end{displaymath}
corresponding to the composite

\begin{displaymath}
F(a^b) \times F(b) \cong F(a^b \times b) \stackrel{F(eval_{a, b})}{\to} F(a),
\end{displaymath}
is an isomorphism.

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

\begin{itemize}%
\item Any [[topos]] or [[quasitopos]], such as [[Set]], is cartesian closed.

See also [[closed monoidal structure on presheaves]].


\item [[Cat]] is also cartesian closed. (Cf. e.g. p. 98 of [[Categories Work|Mac Lane]], 2nd ed., or the remark below Definition 4.3.9 in [[Category Theory in Context|Riehl]].)


\item Many [[nice category of spaces|nice categories of topological spaces]] are also cartesian closed, particularly the [[convenient category of topological spaces|convenient categories of spaces]].



\end{itemize}
\hypertarget{some_basic_consequences}{}\subsection*{{Some basic consequences}}\label{some_basic_consequences}

A category is cartesian closed if it has finite products and if for any two objects $X$, $Y$, there is an object $Y^X$ (thought of as a ``space of maps from $X$ to $Y$'') such that for any object $Z$, there is a bijection between the set of maps $Z \to Y^X$ and the set of maps $Z \times X \to Y$, and this bijection is natural in $Z$.

There is an evaluation map $ev_{X, Y}: Y^X \times X \to Y$, which by definition is the map corresponding to the identity map $Y^X \to Y^X$ under the bijection. Using the naturality, it may be shown that the bijection $\hom(Z, Y^X) \to \hom(Z \times X, Y)$ takes a map $\phi: Z \to Y^X$ to the composite

\begin{displaymath}
Z \times X \stackrel{\phi \times 1_X}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} Y
\end{displaymath}
and the universal property of $Y^X$ may be phrased thus: given any $\psi: Z \times X \to Y$, there exists a unique map $\phi: Z \to Y^X$ for which $\psi = ev_{X, Y} \circ (\phi \times 1_X)$.

Taking $Z = 1$ (the terminal object), maps $1 \to Y^X$ (or ``points'' of $Y^X$) are in bijection with maps $X \stackrel{\pi_{2}^{-1}}{\to} 1 \times X \to Y$. So the ``underlying set'' of $Y^X$, namely $\hom(1, Y^X)$, is identified with the set of maps from $X$ to $Y$. Let us denote the point corresponding to $f: X \to Y$ by $[f]: 1 \to Y^X$. Then, by definition,

\begin{displaymath}
(1 \times X \stackrel{[f] \times 1_X}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} Y) = (1 \times X \stackrel{\pi_2}{\to} X \stackrel{f}{\to} Y)
\end{displaymath}
The following lemma says that the internal evaluation map $ev_{X, Y}$ indeed behaves as an evaluation map at the level of underlying sets.

\begin{lemma}
\label{}\hypertarget{}{}
Given a map $f: X \to Y$ and a point $x: 1 \to X$, the composite

\begin{displaymath}
1 \stackrel{\langle [f], x \rangle}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} Y
\end{displaymath}
is the point $f x: 1 \to Y$.

\end{lemma}
\begin{proof}
We have

\begin{displaymath}
\itexarray{
1 \stackrel{\langle [f], x \rangle}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} X & = & 1 \stackrel{\delta_1}{\to} 1 \times 1 \stackrel{[f] \times x}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} Y\\
 & = & 1 \stackrel{\delta_1}{\to} 1 \times 1 \stackrel{1 \times x}{\to} 1 \times X \stackrel{[f] \times 1_X}{\to} Y^X \times X \stackrel{ev_{X, Y}}{\to} Y\\
 & = & 1 \stackrel{\delta_1}{\to} 1 \times 1 \stackrel{1 \times x}{\to} 1 \times X \stackrel{\pi_2}{\to} X \stackrel{f}{\to} Y\\
 & = & 1 \stackrel{\delta_1}{\to} 1 \times 1 \stackrel{\pi_2}{\to} 1 \stackrel{x}{\to} X \stackrel{f}{\to} Y\\
 & = & 1 \stackrel{x}{\to} X \stackrel{f}{\to} Y
}
\end{displaymath}
where the penultimate equation uses naturality of the projection map $\pi_2$.

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
\textbf{Internal composition} $c_{X, Y, Z}: Z^Y \times Y^X \to Z^X$ is the unique map such that

\begin{displaymath}
(Z^Y \times Y^X \times X \stackrel{c \times 1_X}{\to} Z^X \times X \stackrel{ev_{X, Z}}{\to} Z) = (Z^Y \times Y^X \times X \stackrel{1 \times ev_{X, Y}}{\to} Z^Y \times Y \stackrel{ev_{Y, Z}}{\to} Z)
\end{displaymath}
\end{defn}
One may show that internal composition behaves as the usual composition at the underlying set level, in that given maps $g: Y \to Z$, $f: X \to Y$, we have

\begin{displaymath}
(1 \stackrel{\langle [g], [f] \rangle}{\to} Z^Y \times Y^X \stackrel{c_{X, Y, Z}}{\to} Z^X) = [g f]: 1 \to Z^X
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\begin{itemize}%
\item In a cartesian closed category, the product functors $A \times -$ have [[right adjoints]], so they preserve all [[colimits]]. In particular, a cartesian closed category that has finite [[coproducts]] is a [[distributive category]].


\item The [[internal logic]] of cartesian closed categories is the [[typed lambda-calculus]].



\end{itemize}
\hypertarget{inheritance_by_reflective_subcategories}{}\subsubsection*{{Inheritance by reflective subcategories}}\label{inheritance_by_reflective_subcategories}

In showing that a given category is cartesian closed, the following theorem is often useful (cf. A4.3.1 in the [[Elephant]]):

\begin{theorem}
\label{}\hypertarget{}{}
If $C$ is cartesian closed, and $D\subseteq C$ is a [[reflective subcategory]], then the reflector $L\colon C\to D$ preserves finite [[products]] if and only if $D$ is an [[exponential ideal]] (i.e. $Y\in D$ implies $Y^X\in D$ for any $X\in C$). In particular, if $L$ preserves finite products, then $D$ is cartesian closed.

\end{theorem}
\hypertarget{exponentials_of_cartesian_closed_categories}{}\subsubsection*{{Exponentials of cartesian closed categories}}\label{exponentials_of_cartesian_closed_categories}

The following observation was taken from a post of Mike Shulman at \href{http://mathoverflow.net/questions/104152/exponentials-in-functor-categories/104178#104178}{MathOverflow}.

If $\mathcal{C}$ is small and $\mathcal{D}$ is complete and cartesian closed, then $\mathcal{D}^{\mathcal{C}}$ is also complete and cartesian closed. Completeness is clear since limits in $D^C$ are computed pointwise. As for cartesian closure, we can compute exponentials in essentially the same way as for presheaves, motivated by $\mathcal{D}$-enriched category theory:

\begin{displaymath}
G^F(x) = \int_{y\in \mathcal{C}} \prod_{\mathcal{C}(x,y)} G(y)^{F(y)}.
\end{displaymath}
Then we can compute

\begin{displaymath}
\itexarray{
  \mathcal{D}^{\mathcal{C}}\left(H,G^F\right)
  &=& \int_{x\in \mathcal{C}} \mathcal{D}\left(H(x), G^F(x)\right)\\
  &=& \int_{x\in \mathcal{C}} \mathcal{D}\left(H(x), \int_{y\in \mathcal{C}} \prod_{\mathcal{C}(x,y)} G(y)^{F(y)}\right)\\
  &=& \int_{x\in \mathcal{C}} \int_{y\in \mathcal{C}} \prod_{\mathcal{C}(x,y)} \mathcal{D}\left(H(x), G(y)^{F(y)}\right)\\
  &=& \int_{y\in \mathcal{C}} \mathcal{D}\left( \int^{x\in \mathcal{C}} \sum_{\mathcal{C}(x,y)} H(x), G(y)^{F(y)}\right)\\
  &=&  \int_{y\in \mathcal{C}} \mathcal{D}\left(H(y), G(y)^{F(y)}\right)\\
  &=&  \int_{y\in \mathcal{C}} \mathcal{D}\left(H(y)\times F(y), G(y)\right)\\
  &=& \mathcal{D}^{\mathcal{C}}(H\times F, G).
}
\end{displaymath}
Here the antepenultimate step uses the [[co-Yoneda lemma]]. This appears to involve colimits in $\mathcal{D}$ as well, but the existence of these colimits is not actually an extra assumption: the co-Yoneda lemma tells us that $\int^{x\in \mathcal{C}} \sum_{\mathcal{C}(x,y)} H(x)$ \emph{exists and is isomorphic to} $H(y)$.

Similarly, the above argument can be interpreted to say that even if $\mathcal{D}$ is not complete, then the exponential $G^F$ in $\mathcal{D}^{\mathcal{C}}$ exists if and only if the particular limits above exist, and in that case they are isomorphic.

A more abstract argument using [[comonadic functor|comonadicity]] and the [[adjoint triangle theorem]], which also applies to [[locally cartesian closed categories]], can be found in Theorem 2.12 of

\begin{itemize}%
\item Street and Verity, \emph{The comprehensive factorization and torsors}, \href{http://www.tac.mta.ca/tac/volumes/23/3/23-03abs.html}{TAC}

\end{itemize}
and is reproduced in the setting of closed monoidal categories at [[closed monoidal category]].

\hypertarget{functional_completeness_theorem}{}\subsubsection*{{Functional completeness theorem}}\label{functional_completeness_theorem}

\begin{theorem}
\label{coKleisliOfComonadIsCartesianClosed}\hypertarget{coKleisliOfComonadIsCartesianClosed}{}
Let $C$ be a [[cartesian closed category]], and $c$ an object of $C$. Then the [[Kleisli category]] of the [[comonad]] $c \times - \colon C \to C$, denoted $C[c]$, is also a cartesian closed category.

\end{theorem}
\begin{remark}
\label{}\hypertarget{}{}
The [[Kleisli category of a comonad|Kleisli category of the comonad]] $c \times -$ on $C$ is canonically [[equivalence of categories|equivalent]] to the [[Kleisli category]] of the [[monad]] $(-)^c$ on $C$.

\end{remark}
\begin{proof}
of Theorem \ref{coKleisliOfComonadIsCartesianClosed}

Let $a$ and $b$ be objects of $C[c]$. Claim: the product $a \times b$ in $C$, considered as an object of $C[c]$, is the product of $a$ and $b$ in $C[c]$, according to the following series of natural bijections:

\begin{displaymath}
\itexarray{
\underline{z \to a \qquad z \to b} & C[c] \\
\underline{c \times z \to a, c \times z \to b} & C \\
\underline{c \times z \to a \times b} & C \\
z \to a \times b & C[c] 
}
\end{displaymath}
Claim: the exponential $a^b$ in $C$, considered as an object of $C[c]$, is the exponential of $a$ and $b$ in $C[c]$, according to the following series of natural bijections:

\begin{displaymath}
\itexarray{
\underline{z \to a^b} & C[c] \\
\underline{c \times z \to a^b} & C \\
\underline{c \times z \times b \to a} & C \\
z \times b \to a & C[c]
}
\end{displaymath}
where in the last step, we used the prior claim that the product of objects in $C$, when viewed as an object of $C[c]$, gives the product in $C[c]$.

\end{proof}
Observe that the object $c$ in $C[c]$ has an element $e \colon 1 \to c$, corresponding to the canonical isomorphism $c \times 1 \cong c$ in $C$. We call this element the ``generic element'' of $c$ in $C[c]$, according to the following theorem. This theorem can be viewed as saying that in the [[doctrine]] of cartesian closed categories, $C[c]$ is the result of freely adjoining an arrow $1\to c$ to $C$.

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Functional completeness)}

Let $C$ and $D$ be cartesian closed categories, and let $F \colon C \to D$ be a cartesian closed functor. Let $c$ be an object of $C$, and let $t: F(1) \to F(c)$ be an element in $D$. Then there exists an extension of $F$ to a cartesian closed functor $\tilde{F} \colon C[c] \to D$ that takes the generic element $e \colon 1 \to c$ in $C[c]$ to the element $t$, and this extension is unique up to isomorphism.

\end{theorem}
\begin{proof}
On objects, $\tilde{F}(a) = F(a)$. Let $f \colon a \to b$ be a map of $C[c]$, i.e., let $g \colon c \times a \to b$ be the corresponding map in $C$, and define $\tilde{F}(f)$ to be the composite

\begin{displaymath}
F(a) \cong 1 \times F(a) \stackrel{t \times 1}{\to} F(c) \times F(a) \cong F(c \times a) \stackrel{F(g)}{\to} F(b)
\end{displaymath}
It is straightforward to check that $\tilde{F}$ is a cartesian closed functor and is, up to isomorphism, the unique cartesian closed functor taking $e$ to $t$.

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

\begin{itemize}%
\item \textbf{cartesian closed category}, [[locally cartesian closed category]]


\item [[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]]


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



\end{itemize}
\hypertarget{literature}{}\subsection*{{Literature}}\label{literature}

Some of the major examples are treated in

\begin{itemize}%
\item S. Niefield, \emph{Cartesianness: topological spaces, uniform spaces and affine schemes}, J. Pure Appl. Algebra \textbf{23} (1982) 147-167 

\end{itemize}
[[!redirects cartesian closed category]] [[!redirects cartesian closed categories]] [[!redirects cartesian closed]] [[!redirects closed cartesian category]] [[!redirects closed cartesian categories]]

[[!redirects Cartesian closed category]] [[!redirects Cartesian closed categories]]



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