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\begin{document}

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\section*{exponential object}

\begin{quote}%
This entry is about the concept in [[category theory]]. For exponential functions see at \emph{[[exponential map]]}.


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\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{mapping_space}{}\paragraph*{{Mapping space}}\label{mapping_space}

[[!include mapping space - 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{related_notions}{Related notions}\dotfill \pageref*{related_notions} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{exponentiation_of_sets_and_of_numbers}{Exponentiation of sets and of numbers}\dotfill \pageref*{exponentiation_of_sets_and_of_numbers} \linebreak
\noindent\hyperlink{more_examples}{More examples}\dotfill \pageref*{more_examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

An \textbf{exponential object} $X^Y$ is an [[internal hom]] $[Y,X]$ in a [[cartesian closed category]]. It generalises the notion of [[function set]], which is an exponential object in [[Set]].

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

The above is actually a complete definition, but here we spell it out.

Let $X$ and $Y$ be objects of a [[category]] $C$ such that all binary [[product]]s with $Y$ exist. (Usually, $C$ actually has all binary products.) Then an \textbf{exponential object} is an object $X^Y$ equipped with an \textbf{[[evaluation map]]} $ev: X^Y \times Y \to X$ which is universal in the sense that, given any object $Z$ and map $e: Z \times Y \to X$, there exists a unique map $u: Z \to X^Y$ such that

\begin{displaymath}
Z \times Y \stackrel{u \times id_Y}\to X^Y \times Y \stackrel{ev}\to X
\end{displaymath}
equals $e$.

The map $u$ is known by various names, such as the \emph{exponential transpose} or \emph{[[currying]]} of $e$. It is sometimes denoted $\lambda(e)$ in a hat tip to the [[lambda-calculus]], since in the [[internal logic]] of a cartesian closed category this is the operation corresponding to $\lambda$-abstraction. It is also sometimes denoted $e^\flat$ (as in music notation), being an instance of the more general notion of [[adjunct]] or [[mate]].

As with other [[universal construction]]s, an exponential object, if any exists, is [[generalized the|unique up to unique isomorphism]]. It can also be characterized as a [[distributivity pullback]].

\hypertarget{related_notions}{}\subsection*{{Related notions}}\label{related_notions}

As before, let $C$ be a category and $X,Y\in C$.

\begin{itemize}%
\item If $X^Y$ exists, then we say that $X$ \textbf{exponentiates} $Y$.


\item If $Y$ is such that $X^Y$ exists for all $X$, we say that $Y$ is \textbf{exponentiable} (or \emph{powerful}, cf. Street-Verity \href{http://www.emis.de/journals/TAC/volumes/23/3/23-03.pdf}{pdf}). Then $C$ is \textbf{[[cartesian closed category|cartesian closed]]} if it has a [[terminal object]] and every object is exponentiable.


\item More generally, a morphism $f\colon Y \to A$ is \textbf{exponentiable} (or \emph{powerful}) when it is exponentiable in the [[over category]] $C/A$. This is equivalent to saying that the [[base change]] functor $f^*$ has a [[right adjoint]], usually denoted $\Pi_f$ and called a [[dependent product]]. In particular, $C$ is \textbf{[[locally cartesian closed category|locally cartesian closed]]} iff every morphism is exponentiable, iff all pullback functors have right adjoints.


\item Conversely, if $X$ is such that $X^Y$ exists for all $Y$, we say that $X$ is \textbf{exponentiating}. Again, $C$ is \textbf{cartesian closed} if it has a terminal object and every object is exponentiating. (The reader should beware that some authors say ``exponentiable'' for what is here called ``exponentiating.'')



\end{itemize}
Dually, a \textbf{coexponential object} in $C$ is an exponential object in the [[opposite category]] $C^{op}$. A \textbf{[[cocartesian coclosed category]]} has all of these (and an [[initial object]]). Some coexponential objects occur naturally in algebraic categories (such as [[rings]] or [[frames]]) whose opposites are viewed as categories of spaces (such as [[schemes]] or [[locales]]). Cf. also [[cocartesian closed category]].

When $C$ is not cartesian but merely monoidal, then the analogous notion is that of a [[residual|left/right residual]].

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

Of course, in any cartesian closed category every object is exponentiable and exponentiating. In general, exponentiable objects are more common and important than exponentiating ones, since the existence of $X^Y$ is usually more related to properties of $Y$ than properties of $X$.

\hypertarget{exponentiation_of_sets_and_of_numbers}{}\subsubsection*{{Exponentiation of sets and of numbers}}\label{exponentiation_of_sets_and_of_numbers}

In the cartesian closed category [[Set]] of [[set]]s, for $X,S \in Set$ to sets, their exponentiation $X^S$ is the set of [[function]]s $S\to X$.

Restricted to [[finite set]]s and under the [[cardinality]] operation $|-| : FinSet \to \mathbb{N}$ this induces an exponentiation operation on [[natural number]]s

\begin{displaymath}
|X^S| = |X|^{|S|}
  \,.
\end{displaymath}
This exponentiation operation on numbers $(-)^{(-)} : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is therefore the [[decategorification]] of the canonically defined [[internal hom]] of sets. It sends numbers $a,b \in \mathbb{N}$ to the product

\begin{displaymath}
a^b = a \times a \times \cdots \times a \;\; (b \; factors)
  \,.
\end{displaymath}
If $b = 0$ is [[zero]], the expression on the right is 1, reflecting the fact that $0$ is the [[cardinality]] of the [[empty set]], which is the [[initial object]] in [[Set]].

When the natural numbers are embedded into larger [[rig]]s or [[ring]]s, the operation of exponentiation may extend to these larger context. It yields for instance an exponentiation operation on the [[positive real numbers]].

\hypertarget{more_examples}{}\subsubsection*{{More examples}}\label{more_examples}

\begin{itemize}%
\item In [[Top]] (the category of \emph{all} topological spaces), the exponentiable spaces are precisely the [[core-compact spaces]]. In particular, this includes [[locally compact Hausdorff space]]s. However, most [[nice category of spaces|nice categories of spaces]] are cartesian closed, so that all objects are exponentiable; note that usually the cartesian product in such categories has a slightly different topology than it does in $Top$.


\item There are similar characterizations of exponentiable [[locales]] (see [[locally compact locale]] and [[continuous poset]] and (in the appropriate higher-categorical sense) [[toposes]] and [[(∞,1)-toposes]] (see [[metastably locally compact locale]] and [[continuous category]] as well as [[exponentiable topos]]).


\item In [[algebraic set theory]] one often assumes that only small objects (and morphisms) are exponentiable. This is analogous to how in material [[set theory]] one can talk about the class of functions $Y\to X$ when $Y$ is a set and $X$ a class, but not the other way round.


\item In a [[type theory]] with [[dependent product]]s, every display morphism is exponentiable in the category of [[context]]s ---even in a type theory without identity types, so that not every morphism is display and the relevant slice category need not have all products.


\item In a [[functor category]] $D^C$, a [[natural transformation]] $\alpha:F\to G$ is exponentiable if (though probably not ``only if'') it is [[cartesian natural transformation|cartesian]] and each component $\alpha_c:F c \to G c$ is exponentiable in $D$. Given $H\to F$, we define $\Pi_\alpha(H)(c) = \Pi_{\alpha_c}(H c)$; then for $u:c\to c'$ to obtain a map $\Pi_{\alpha_c}(H c) \to \Pi_{\alpha_{c'}}(H c')$ we need a map $\alpha_{c'}^*(\Pi_{\alpha_c}(H c)) \to H c'$. But since $\alpha$ is cartesian, $\alpha_{c'}^*(\Pi_{\alpha_c}(H c)) \cong \alpha_c^* (\Pi_{\alpha_c}(H c))$, so we have the counit $\alpha_c^* (\Pi_{\alpha_c}(H c)) \to H c$ that we can compose with $H u$.



\end{itemize}
However, exponentiating objects do matter sometimes.

\begin{itemize}%
\item In \href{http://www.paultaylor.eu/ASD/}{Abstract Stone Duality}, [[Sierpinski space]] is exponentiating.


\item [[Toby Bartels]] has \href{http://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c023187}{argued} that [[predicative mathematics]] can have a set of [[truth value]]s as long as this set is not exponentiating (or even exponentiates only [[finite set]]s).


\item A [[dialogue category]] is a [[symmetric monoidal category]] equipped with an exponentiating object.



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

As with other internal homs, the \textbf{[[currying]]} isomorphism

\begin{displaymath}
hom_C(Z,X^Y) \cong hom_C(Z \times Y,X)
\end{displaymath}
is a [[natural isomorphism]] of sets. By the usual [[Yoneda lemma|Yoneda]] arguments, this isomorphism can be internalized to an isomorphism in $C$:

\begin{displaymath}
(X^Y)^Z \cong X^{Y\times Z}.
\end{displaymath}
Similarly, $X \cong X^1$, where $1$ is a [[terminal object]]. Thus, a product of exponentiable objects is exponentiable.

Other natural isomorphisms that match equations from ordinary algebra include:

\begin{itemize}%
\item $(X \times Y)^Z = X^Z \times Y^Z$;
\item $1^Z \cong 1$.

\end{itemize}
These show that, in a cartesian monoidal category, a product of exponentiating objects is also exponentiating.

Now suppose that $C$ is a [[distributive category]]. Then we have these isomorphisms:

\begin{itemize}%
\item $X^{Y + Z} \cong X^Y \times X^Z$;
\item $X^0 \cong 1$.

\end{itemize}
Here $Y + Z$ is a [[coproduct]] of $Y$ and $Z$, while $0$ is an [[initial object]]. Thus in a distributive category, the exponentiable objects are closed under coproducts.

Note that any cartesian closed category with finite coproducts must be distributive, so all of the isomorphisms above hold in any closed [[2-rig]] (such as [[Set]], of course).

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

\begin{itemize}%
\item [[exponential ideal]]


\item [[exponentiable topos]]


\item [[reflexive object]]



\end{itemize}
[[!redirects exponential object]] [[!redirects exponential objects]] [[!redirects exponentiable object]] [[!redirects exponentiable objects]] [[!redirects exponentiable morphism]] [[!redirects exponentiable morphisms]] [[!redirects powerful morphism]] [[!redirects powerful morphisms]] [[!redirects powerful object]] [[!redirects powerful objects]]

[[!redirects exponentiating object]] [[!redirects exponentiating objects]]



\end{document}