Exponential object

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In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories.

[edit] Definition

Let C be a category with binary products and let Y and Z be objects of C. The exponential object Z^Y can be defined as a universal morphism from the functor –×Y to Z. (The functor –×Y from C to C maps objects X to X×Y and morphisms φ to φ×id_Y).

Explicitly, the definition is as follows. An object Z^Y, together with a morphism

$\mathrm{eval}\colon (Z^Y \times Y) \rightarrow Z\,$

is an exponential object if for any object X and morphism g : (X×Y) → Z there is a unique morphism

$\lambda g\colon X\to Z^Y\,$

such that the following diagram commutes:

Universal property of the exponential object

If the exponential object Z^Y exists for all objects Z in C, then the functor which sends Z to Z^Y is a right adjoint to the functor –×Y. In this case we have a natural bijection between the hom-sets

$\mathrm{Hom}(X\times Y,Z) \cong \mathrm{Hom}(X,Z^Y).$

[edit] Examples

In the category of sets, the exponential object $Z Y$ is the set of all functions from $Y$ to $Z$ . The map $\mathrm{eval}\colon (Z^Y \times Y) \to Z$ is just the evaluation map which sends the pair (f, y) to f(y). For any map $g\colon (X \times Y) \rightarrow Z$ the map $\lambda g\colon X\to Z^Y$ is the curried form of $g$ :

$\lambda g(x)(y) = g(x,y).\,$

In the category of topological spaces, the exponential object Z^Y exists provided that Y is a locally compact Hausdorff space. In that case, the space Z^Y is the set of all continuous functions from Y to Z together with the compact-open topology. The evaluation map is the same as in the category of sets. If Y is not locally compact Hausdorff, the exponential object may not exist (the space Z^Y exists, but fails to be an exponential object because the adjunction with the product only holds when Z is locally compact Hausdorff). For this reason the category of topological spaces fails to be cartesian closed.