Exponential object

In mathematics, specifically in , an exponential object or map object is the generalization of a function space in set theory. with all and exponential objects are called . Categories (such as of ) without adjoined products may still have an exponential law. Let be a category, let and be of , and let have all with . An object together with a morphism is an exponential object if for any object and morphism there is a unique morphism (called the transpose of ) such that the following diagram commutes: This assignment of a unique to each establishes an isomorphism (bijection) of hom-sets, If exists for all objects in , then the functor defined on objects by and on arrows by , is a right adjoint to the product functor . For this reason, the morphisms and are sometimes called exponential adjoints of one another. Alternatively, the exponential object may be defined through equations: Existence of is guaranteed by existence of the operation . Commutativity of the diagrams above is guaranteed by the equality . Uniqueness of is guaranteed by the equality . The exponential is given by a universal morphism from the product functor to the object . This universal morphism consists of an object and a morphism . In the , an exponential object is the set of all functions . The map is just the evaluation map, which sends the pair to . For any map the map is the curried form of : A Heyting algebra is just a bounded lattice that has all exponential objects. Heyting implication, , is an alternative notation for . The above adjunction results translate to implication () being right adjoint to meet (). This adjunction can be written as , or more fully as: In the , the exponential object exists provided that is a locally compact Hausdorff space. In that case, the space is the set of all continuous functions from to 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.

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