Exponential Object - Definition

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×id_Y).

Explicitly, the definition is as follows. An object ZY, together with a morphism

is an exponential object if 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 natural bijection between the hom-sets

(Note: In functional programming languages, the morphism eval is often called apply, and the syntax 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.)

The morphisms and are sometimes said to be exponential adjoints of one another.