# Difference between revisions of "Exponential object"

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{{redirect|Power object|power objects in topos theory|topos}} | {{redirect|Power object|power objects in topos theory|topos}} | ||

− | In [[mathematics]], specifically in [[category theory]], an '''exponential object''' is the categorical equivalent of a [[function space]] in [[set theory]]. Categories with all [[product (category theory)|finite products]] and exponential objects are called [[cartesian closed category|cartesian closed categories]]. An exponential object may also be called a '''power object''' or '''map object''' (but note that the term "power object" means something different in [[topos theory]], analogous to "power set"). | + | In [[mathematics]], specifically in [[category theory]], an '''exponential object''' is the categorical equivalent of a [[function space]] in [[set theory]]. Categories with all [[product (category theory)|finite products]] and exponential objects are called [[cartesian closed category|cartesian closed categories]]. An exponential object may also be called a '''power object''' or '''map object''' (but note that the term "power object" means something different in [[topos theory]], analogous to "power set"; see [[power set]] for a simplified explanation.). |

==Definition== | ==Definition== | ||

− | Let ''C'' be a category with [[product (category theory)|binary products]] and let ''Y'' and ''Z'' be objects of ''C''. The exponential object ''Z''<sup>''Y''</sup> 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 | + | Let ''C'' be a category with [[product (category theory)|binary products]] and let ''Y'' and ''Z'' be [[category theory|objects]] of ''C''. The exponential object ''Z''<sup>''Y''</sup> 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 [[morphism]]s φ to φ×id<sub>''Y''</sub>). |

Explicitly, the definition is as follows. An object ''Z''<sup>''Y''</sup>, together with a morphism | Explicitly, the definition is as follows. An object ''Z''<sup>''Y''</sup>, together with a morphism | ||

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[[File:ExponentialObject-01.png|center|Universal property of the exponential object]] | [[File:ExponentialObject-01.png|center|Universal property of the exponential object]] | ||

− | If the exponential object ''Z''<sup>''Y''</sup> exists for all objects ''Z'' in ''C'', then the functor which sends ''Z'' to ''Z''<sup>''Y''</sup> is a [[right adjoint]] to the functor –×''Y''. In this case we have a natural bijection between the [[hom-set]]s | + | If the exponential object ''Z''<sup>''Y''</sup> exists for all objects ''Z'' in ''C'', then the functor which sends ''Z'' to ''Z''<sup>''Y''</sup> is a [[right adjoint]] to the functor –×''Y''. In this case we have a natural [[bijection]] between the [[hom-set]]s |

:<math>\mathrm{Hom}(X\times Y,Z) \cong \mathrm{Hom}(X,Z^Y).</math> | :<math>\mathrm{Hom}(X\times Y,Z) \cong \mathrm{Hom}(X,Z^Y).</math> | ||

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[[Category:Objects (category theory)]] | [[Category:Objects (category theory)]] | ||

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## Revision as of 05:48, 16 June 2013

{{#invoke:Hatnote|hatnote}}Template:Main other

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. An exponential object may also be called a **power object** or **map object** (but note that the term "power object" means something different in topos theory, analogous to "power set"; see power set for a simplified explanation.).

## 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

is an exponential object if for any object *X* and morphism *g* : (*X*×*Y*) → *Z* there is a unique morphism

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

(Note: In functional programming languages, the morphism *eval* is often called *apply*, and the syntax is often written *curry*(*g*). The morphism *eval* here must not to be confused with the eval function in some programming languages, which evaluates quoted expressions.)

The morphisms and are sometimes said to be *exponential adjoints* of one another.^{[1]}

## Examples

In the category of sets, the exponential object is the set of all functions from to . 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 :

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} 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 *Z*^{Y} need not be locally compact for locally compact spaces *Z* and *Y*.

## References

- {{#invoke:citation/CS1|citation

|CitationClass=book }}

- ↑ {{#invoke:citation/CS1|citation |CitationClass=book }}

## External links

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