In mathematics, specifically in category theory, an exponential object is the categorical equivalent of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in most applications, it is a topological space or/and a vector space. ... Set theory is the mathematical theory of sets, which represent collections of abstract objects. ... In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. ... In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. ...

Definition

Let C be a category with binary products and let Y and Z be objects of C. The exponential object Z^Y 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). In category theory, one defines products to generalize constructions such as the cartesian product of sets, the product of groups, the product of rings and the product of topological spaces. ... In category theory, abstract algebra and other fields of mathematics, frequently constructions are defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ...

Explicitly, the definition is as follows. An object Z^Y 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: In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

If the exponential object Z^Y exists for all objects Z in C, then the functor which sends Z to Z^Y is a right adjoint to the functor –×Y. In this case we have a bijection between the hom-sets Image File history File links ExponentialObject-01. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... In mathematics, a morphism is an abstraction of a function or mapping between two spaces. ...

Examples

In the category of sets, the exponential object Z^Y 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 mathematics, the category of sets is the category whose objects are all sets and whose morphisms are all functions. ... In computer science, currying is the technique of transforming a function taking multiple arguments into a function that takes a single argument (the first of the arguments to the original function) and returns a new function that takes the remainder of the arguments and returns the result. ...

In the category of topological spaces, the exponential object Z^Y exists provided that Y is a locally compact Hausdorff space. In that case, the space Z^Y 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 (the space Z^Y exists, but fails to be an exponential object because the adjunction with the product only holds when Z is locally compact Hausdorff). For this reason the category of topological spaces fails to be cartesian closed. The category Top has topological spaces as objects and continuous maps as morphisms. ... In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure. ... In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. ...

See also

• cartesian closed category