Exponential object

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"Power object" redirects here. For power objects in topos theory, see topos.

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. An exponential object may also be called a power object or map object.

[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).

(Note: In functional programming languages, the morphism eval is often called apply, and the syntax λg is often written curry(g). The morphism eval here must not to be confused with the eval function in some programming languages, which evaluates quoted expressions.)

[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 still exists, but it may fail to be an exponential object since the evaluation function need not be continuous). For this reason the category of topological spaces fails to be cartesian closed.


[edit] References