imported>JLLong: /* Examples */ Fix grammar

2024-10-09T18:49:12Z

Examples: Fix grammar

New page

{{Short description|Categorical generalization of a function space in set theory}}


In [[mathematics]], specifically in [[category theory]], an '''exponential object''' or '''map object''' is the [[category theory|categorical]] generalization of a [[function space]] in [[set theory]]. [[Category (mathematics)|Categories]] with all [[Product (category theory)|finite products]] and exponential objects are called [[Cartesian closed category|cartesian closed categories]]. Categories (such as [[Subcategory|subcategories]] of '''[[category of topological spaces|Top]]''') without adjoined products may still have an '''exponential law'''.<ref>{{nlab| id=exponential+law+for+spaces |title=Exponential law for spaces}}</ref><ref>{{nlab|id=convenient+category+of+topological+spaces |title=Convenient category of topological spaces}}</ref>

==Definition==

Let <math>\mathbf{C}</math> be a category, let <math>Z</math> and <math>Y</math> be [[object (category theory)|objects]] of <math>\mathbf{C}</math>, and let <math>\mathbf{C}</math> have all [[product (category theory)|binary products]] with <math>Y</math>. An object <math display="inline">Z^Y</math> together with a [[morphism]] <math display="inline">\mathrm{eval}\colon (Z^Y \times Y) \to Z</math> is an ''exponential object'' if for any object <math>X</math> and morphism <math display="inline">g \colon X\times Y \to Z</math> there is a unique morphism <math display="inline">\lambda g\colon X\to Z^Y</math> (called the ''transpose'' of <math>g</math>) such that the following diagram [[commutative diagram|commutes]]:
[[File:ExponentialObject-01.svg|center|Universal property of the exponential object]]
This assignment of a unique <math>\lambda g</math> to each <math>g</math> establishes an [[isomorphism]] ([[bijection]]) of [[hom-set]]s, <math display="inline">\mathrm{Hom}(X\times Y,Z) \cong \mathrm{Hom}(X,Z^Y).</math>

If <math display="inline">Z^Y</math>exists for all objects <math>Z, Y</math> in <math>\mathbf{C}</math>, then the [[functor]] <math>(-)^Y \colon \mathbf{C}\to \mathbf{C}</math> defined on objects by <math>Z \mapsto Z^Y</math> and on arrows by <math>(f\colon X\to Z) \mapsto (f^Y\colon X^Y \to Z^Y)</math>, is a [[right adjoint]] to the product functor <math>-\times Y</math>. For this reason, the morphisms <math>\lambda g</math> and <math>g</math> are sometimes called ''exponential adjoints'' of one another.<ref name="Goldblatt">{{cite book|title=Topoi : the categorial analysis of logic|publisher=[[North-Holland Publishing Company|North-Holland]]|year=1984|isbn=978-0-444-86711-7|edition=Revised|series=Studies in Logic and the Foundations of Mathematics #98|page=72|chapter=Chapter 3: Arrows instead of epsilon|last1=Goldblatt|first1=Robert|author-link=Robert Goldblatt}}</ref>

===Equational definition===

Alternatively, the exponential object may be defined through equations:
* Existence of <math>\lambda g</math> is guaranteed by existence of the operation <math>\lambda - </math>.
* Commutativity of the diagrams above is guaranteed by the equality <math>\forall g \colon X \times Y \to Z,\ \mathrm{eval} \circ (\lambda g \times \mathrm{id}_Y) = g</math>.
* Uniqueness of <math>\lambda g</math> is guaranteed by the equality <math>\forall h \colon X \to Z^Y, \ \lambda (\mathrm{eval} \circ (h \times \mathrm{id}_Y)) = h</math>.

===Universal property===

The exponential <math>Z^Y</math> is given by a [[universal morphism]] from the product functor <math>- \times Y</math> to the object <math>Z</math>. This universal morphism consists of an object <math>Z^Y</math> and a morphism <math display="inline">\mathrm{eval}\colon (Z^Y \times Y) \to Z</math>.

==Examples==

In the [[category of sets]], an exponential object <math>Z^Y</math> is the set of all [[function (mathematics)|functions]] <math>Y \to Z</math>.<ref name="Lane">{{cite book|title=Categories for the working mathematician|volume = 5|last=Mac Lane|first=Saunders|publisher=Springer-Verlag|year=1978|isbn=978-0387984032|edition=2nd|series=graduate texts in mathematics|page=98|chapter=Chapter 4: Adjoints|author-link=Saunders Mac Lane|doi = 10.1007/978-1-4757-4721-8_5}}</ref> The map <math>\mathrm{eval}\colon (Z^Y \times Y) \to Z</math> is just the [[apply|evaluation map]], which sends the pair <math>(f, y)</math> to <math>f(y)</math>. For any map <math>g\colon X \times Y \to 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>

A [[Heyting algebra]] <math>H</math> is just a bounded [[lattice (order)|lattice]] that has all exponential objects. Heyting implication, <math>Y \Rightarrow Z</math>, is an alternative notation for <math>Z^Y</math>. The above adjunction results translate to implication (<math>\Rightarrow : H \times H \to H</math>) being [[right adjoint]] to [[Join and meet|meet]] (<math>\wedge : H \times H \to H</math>). This adjunction can be written as <math>(- \wedge Y) \dashv (Y \Rightarrow -)</math>, or more fully as:
<math display="block">(- \wedge Y): H \stackrel {\longrightarrow} {\underset {\longleftarrow}{\top}} H: (Y \Rightarrow -)</math>

In the [[category of topological spaces]], the exponential object <math>Z^Y</math> exists provided that <math>Y</math> is a [[Locally compact space|locally compact]] [[Hausdorff space]]. In that case, the space <math>Z^Y</math> is the set of all [[continuous function (topology)|continuous functions]] from <math>Y</math> to <math>Z</math> 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.<ref>[[Joseph J. Rotman]], ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag {{ISBN|0-387-96678-1}} ''(See Chapter 11 for proof.)''</ref> If <math>Y</math> is not locally compact Hausdorff, the exponential object may not exist (the space <math>Z^Y</math> 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 <math>Z^Y</math> need not be locally compact for locally compact spaces <math>Z</math> and <math>Y</math>. A cartesian closed category of spaces is, for example, given by the [[Subcategory#Formal_definition|full subcategory]] spanned by the [[compactly generated space|compactly generated]] Hausdorff spaces.

In [[functional programming language]]s, the morphism <math>\operatorname{eval}</math> is often [[Apply| called <math>\operatorname{apply}</math>]], and the syntax <math>\lambda g</math> is often [[function application#Representation|written <math>\operatorname{curry}(g)</math>]]. The morphism <math>\operatorname{eval}</math> should not be confused with the [[eval|<code>eval</code>]] function in some [[programming language]]s, which evaluates quoted expressions.

==See also==
*[[Closed monoidal category]]

== Notes ==
{{reflist}}

== References ==
* {{cite book|last=Adámek|first=Jiří|author2=[[Horst Herrlich]]|author3=George Strecker|title=Abstract and Concrete Categories (The Joy of Cats)|publisher=John Wiley & Sons|url=http://katmat.math.uni-bremen.de/acc/|orig-year=1990|year=2006}}
* {{cite book |last=Awodey|first=Steve|authorlink = Steve Awodey|title=Category theory|publisher=Oxford University Press|location=Oxford New York|year=2010|isbn=978-0199237180|chapter=Chapter 6: Exponentials}}
* {{cite book |last=Mac Lane|first=Saunders|title=[[Categories for the working mathematician]]|publisher=Springer|location=New York|year=1998|isbn=978-0387984032|chapter=Chapter 4: Adjoints}}

== External links ==
*[https://web.archive.org/web/20080916162345/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 [https://web.archive.org/web/20081223001815/http://www.j-paine.org/ Jocelyn Paine].

{{Category theory}}

[[Category:Objects (category theory)]]

Exponential object - Revision history

imported>JLLong: /* Examples */ Fix grammar