nLab
exponential object

Idea

An 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.

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 products with $Y$ exist. (Usually, $C$ actually has all binary products.) Then an exponential object is an object $X^{Y}$ equipped with an 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

Z \times Y \overset{u \times {id}_{Y}}{\to} X^{Y} \times Y \overset{ev}{\to} X

equals $e$ .

As with other universal constructions, an exponential object, if any exists, is unique up to unique isomorphism.

Related notions

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

If $X^{Y}$ exists, then we say that $X$ exponentiates $Y$ .
If $Y$ is such that $X^{Y}$ exists for all $X$ , we say that $Y$ is exponentiable. Then $C$ is cartesian closed if it has a terminal object and every object is exponentiable.
More generally, a morphism $f : Y \to A$ is exponentiable when it is exponentiable in the over category $C / A$ . This is equivalent to saying that the pullback functor $f^{*}$ has a right adjoint, usually denoted $Π_{f}$ and called a dependent product. In particular, $C$ is locally cartesian closed iff every morphism is exponentiable.
Conversely, if $X$ is such that $X^{Y}$ exists for all $Y$ , we say that $X$ is exponentiating. Again, $C$ is 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.”)

Dually, a coexponential object in $C$ is an exponential object in the opposite category $C^{op}$ . A cocartesian coclosed category? has all of these (and an initial object).

David: How should entries involving the cocartesian property be organised? How many of the eight possibilities (co)cartesian (co)monoidal (co)closed are worth mentioning? Sixteen with (co)category?

Toby: Potentially all of them, but in practice only the ones that come up. This one only came up since I wanted to say what a coexponential object was and (in context) it was natural to ask what is a category that has all of these. But that doesn't mean that anybody actually has to create the page, much less the others. On the other hand, if there's something interesting to say about them, then we should have them!

Mike: One place where coexponential objects occur naturally is in algebraic categories whose opposites are viewed as categories of spaces. So for instance rings, or frames, or the categories of loci? used in synthetic differential geometry, have some interesting coexponential objects (although none of them is actually cocartesian coclosed).

There is no difference between monoidal and comonoidal (there is a bijection between monoidal structures on $C$ and on $C^{op}$ ), so your eight possibilities are really only four. And there aren’t many cocartesian closed (or cartesian coclosed) categories; that would mean you have an object $[Y, X]$ and an isomorphism

C (Z, [Y, X]) ≅ C (Z ⊔ Y, X) ≅ C (Z, X) \times C (Y, X)

Taking $Z$ to be the initial object, we see that $C (Y, X) ≅ *$ for any objects $X, Y$ . So the only cocartesian closed categories are (-1)-categories.

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 $X$ .

In Top (the category of all topological spaces), locally compact Hausdorff spaces are exponentiable. Note, though, that most nice categories of spaces are cartesian closed.
There are similar characterizations of exponentiable locales and toposes.
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.
In a type theory with dependent products, every display morphism is exponentiable in the category of contexts —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.

However, exponentiating objects do matter sometimes.

In Abstract Stone Duality, Sierpinski space is exponentiating.
Toby Bartels has argued that predicative mathematics can have a set of truth values as long as this set is not exponentiating (or even exponentiates only finite sets).

Properties

As with other internal homs, the currying isomorphism

{hom}_{C} (Z, X^{Y}) ≅ {hom}_{C} (Z \times Y, X)

is a natural isomorphism of sets. By the usual Yoneda arguments, this isomorphism can be internalized to an isomorphism in $C$ :

(X^{Y})^{Z} ≅ X^{Y \times Z} .

Similarly, $X ≅ X^{1}$ , where $1$ is a terminal object. Thus, a product of exponentiable objects is exponentiable.

Other natural isomorphisms that match equations from ordinary algebra include:

$(X \times Y)^{Z} = X^{Z} \times Y^{Z}$ ;
$1^{Z} ≅ 1$ .

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:

$X^{Y + Z} ≅ X^{Y} \times X^{Z}$ ;
$X^{0} ≅ 1$ .

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).