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.

Definition

be objects have all binary products together with a morphism is an exponential object (called the transpose) such that the following diagram commutes: establishes an isomorphism (bijection) of hom-sets

, then the functor, is a right adjoint are sometimes called exponential adjoints of one another.

Equational definition

Alternatively, the exponential object may be defined through equations:

• .

• .

• .

Universal property

is given by a universal morphism.

Examples

In the category of sets is the set of all functions. is just the evaluation map is the curried:

A Heyting algebra is just a bounded lattice) being right adjoint to meet

In the category of topological spaces is a locally compact Hausdorff space is the set of all continuous functions together with the compact-open topology. The evaluation map is the same as in the category of sets; it is continuous with the above topology. 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. 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 is often \operatorname{apply} is often \operatorname{curry}(g) should not be confused with the eval function in some programming languages, which evaluates quoted expressions.

See also

• Closed monoidal category

Notes

References

External links

• Interactive Web page which generates examples of exponential objects and other categorical constructions. Written by Jocelyn Paine.