Exponential object
From Exampleproblems
In mathematics, specifically in category theory, an exponential object the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories
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 is an exponential object if there is an morphism
such that for any object X and morphism g : (X×Y) → Z there is a unique morphism
such that the following diagram commutes:
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 bijection between the hom-sets
Examples
In the category of sets, the exponential object ZY is the set of all functions from Y to Z. The map 
is just the evaluation map which sends the pair (f, y) to f(y). For any map 
the map 
is the curried form of g:
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. For this reason the category of topological spaces fails to be cartesian closed.
