EXPONENTIAL OBJECT

In mathematics, specifically in category theory, an 'exponential object' or 'power 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.

Contents

Definition

Examples

See also

References

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'').
Explicitly, the definition is as follows. An object ''Z''^''Y'', together with a morphism
:$mathrm\{eval\}colon\; (Z^Y\; imes\; Y)$
ightarrow Z,
is an exponential object if for any object ''X'' and morphism ''g'' : (''X''×''Y'') → ''Z'' there is a unique morphism
:$lambda\; gcolon\; X\; o\; Z^Y,$
such that the following diagram commutes:

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 natural bijection between the hom-sets
:$mathrm\{Hom\}(X\; imes\; Y,Z)\; cong\; mathrm\{Hom\}(X,Z^Y).$

Examples

In the category of sets, the exponential object $Z^Y$ is the set of all functions from $Y$ to $Z$. The map $mathrm\{eval\}colon\; (Z^Y\; imes\; Y)\; o\; Z$ is just the evaluation map which sends the pair (''f'', ''y'') to ''f''(''y''). For any map $gcolon\; (X\; imes\; Y)$
ightarrow Z the map $lambda\; gcolon\; X\; o\; Z^Y$ is the curried form of $g$:
:$lambda\; g(x)(y)\; =\; g(x,y).,$
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 ''Y'' is locally compact Hausdorff). For this reason the category of topological spaces fails to be cartesian closed.

See also

★ Cartesian closed category

References

★ Abstract and Concrete Categories (The Joy of Cats), , Jiří, Adámek, , 2006,