Exponential object

In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory.

Categories with all finite products and exponential objects are called cartesian closed categories.

Categories (such as subcategories of Top) without adjoined products may still have an exponential law.

) such that the following diagram commutes: This assignment of a unique

establishes an isomorphism (bijection) of hom-sets,

exists for all objects

defined on objects by

, is a right adjoint to the product functor

are sometimes called exponential adjoints of one another.

[3] Alternatively, the exponential object may be defined through equations: The exponential

is given by a universal morphism from the product functor

This universal morphism consists of an object

In the category of sets, an exponential object

is the set of all functions

is just the evaluation map, which sends the pair

is just a bounded lattice that has all exponential objects.

The above adjunction results translate to implication (

In the category of topological spaces, the exponential object

exists provided that

is a locally compact Hausdorff space.

is the set of all continuous functions from

The evaluation map is the same as in the category of sets; it is continuous with the above topology.

is not locally compact Hausdorff, the exponential object may not exist (the space

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.

However, the category of locally compact topological spaces is not cartesian closed either, since

need not be locally compact for locally compact spaces

A cartesian closed category of spaces is, for example, given by the full subcategory spanned by the compactly generated Hausdorff spaces.

In functional programming languages, the morphism

should not be confused with the eval function in some programming languages, which evaluates quoted expressions.