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 ZY 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×idY).

Explicitly, the definition is as follows. An object ZY, 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 ZY exists for all objects Z in C, then the functor which sends Z to ZY 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 ZY 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 ZY exists provided that Y is a locally compact Hausdorff space. In that case, the space ZY 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 ZY 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.

[edit] See also

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