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{{redirect|Power object|power objects in topos theory|topos}}
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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.).
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==Definition==
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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]] &ndash;&times;''Y'' to ''Z''. (The functor &ndash;&times;''Y'' from ''C'' to ''C'' maps objects ''X'' to ''X''&times;''Y'' and [[morphism]]s &phi; to &phi;&times;id<sub>''Y''</sub>).
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Explicitly, the definition is as follows. An object ''Z''<sup>''Y''</sup>, together with a morphism
 
:<math>\mathrm{eval}\colon (Z^Y \times Y) \rightarrow Z</math>
 
is an exponential object if for any object ''X'' and morphism ''g'' : (''X''&times;''Y'') &rarr; ''Z'' there is a unique morphism
 
:<math>\lambda g\colon X\to Z^Y</math>
 
such that the following diagram [[commutative diagram|commutes]]:
 
[[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 &ndash;&times;''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>
 
(Note: In [[functional programming language]]s, the morphism ''eval'' is often called ''[[apply]]'', and the syntax <math>\lambda g</math> is often written ''[[currying|curry]]''(''g''). The morphism ''eval'' here must not to be confused with the [[eval]] function in some [[programming language]]s, which evaluates quoted expressions.)
 
The morphisms <math>g</math> and <math>\lambda g</math> are sometimes said to be ''exponential adjoints'' of one another.<ref name="Goldblatt">{{cite book | title = Topoi : the categorial analysis of logic | last1 = Goldblatt | first1 = Robert | authorlink = Robert Goldblatt | publisher = [[North-Holland Publishing Company|North-Holland]] | edition = Revised | year = 1984 | page = 72 | chapter = Chapter 3: Arrows instead of epsilon | isbn = 978-0-444-86711-7 | series = Studies in Logic and the Foundations of Mathematics #98}}</ref>
 
==Examples==
 
In the [[category of sets]], the exponential object <math>Z^Y</math> is the set of all functions from <math>Y</math> to <math>Z</math>. The map <math>\mathrm{eval}\colon (Z^Y \times Y) \to Z</math> is just the evaluation map which sends the pair (''f'', ''y'') to ''f''(''y''). For any map <math>g\colon (X \times Y) \rightarrow Z</math> the map <math>\lambda g\colon X\to Z^Y</math> is the [[currying|curried]] form of <math>g</math>:
:<math>\lambda g(x)(y) = g(x,y).\,</math>
 
In the [[category of topological spaces]], the exponential object ''Z''<sup>''Y''</sup> exists provided that ''Y'' is a [[locally compact Hausdorff space]]. In that case, the space ''Z''<sup>''Y''</sup> is the set of all [[continuous function (topology)|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''<sup>''Y''</sup> 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''<sup>''Y''</sup>  need not be locally compact for locally compact spaces ''Z'' and ''Y''.
 
== References ==
*{{cite book|last=Adámek|first=Jiří|coauthors=Horst Herrlich, George Strecker|title=Abstract and Concrete Categories (The Joy of Cats)|publisher=John Wiley & Sons|url=http://katmat.math.uni-bremen.de/acc/|origyear=1990|year=2006}}
{{reflist}}
 
== External links ==
*[http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page ] which generates examples of exponential objects and other categorical constructions. Written by [http://www.j-paine.org/ Jocelyn Paine].
 
[[Category:Objects (category theory)]]

Latest revision as of 23:21, 19 November 2014

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