Exponential object

In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an exponential law.[1][2]

DefinitionEdit

Let   be a category, let   and   be objects of  , and let   have all binary products with  . An object   together with a morphism   is an exponential object if for any object   and morphism   there is a unique morphism   (called the transpose of  ) such that the following diagram commutes:

This assignment of a unique   to each   establishes an isomorphism of hom-sets,  

If  exists for all objects   in  , then the functor   defined on objects by   and on arrows by  , is a right adjoint to the product functor  . For this reason, the morphisms   and   are sometimes called exponential adjoints of one another.[3]

Equational definitionEdit

Alternatively, the exponential object may be defined through equations:

  • Existence of   is guaranteed by existence of the operation  .
  • Commutativity of the diagrams above is guaranteed by the equality  .
  • Uniqueness of   is guaranteed by the equality  .

Universal propertyEdit

The exponential   is given by a universal morphism from the product functor   to the object  . This universal morphism consists of an object   and a morphism  .

ExamplesEdit

In the category of sets, an exponential object   is the set of all functions  .[4] The map   is just the evaluation map, which sends the pair   to  . For any map   the map   is the curried form of  :

 

A Heyting algebra   is just a bounded lattice that has all exponential objects. Heyting implication,  , is an alternative notation for  . The above adjunction results translate to implication ( ) being right adjoint to meet ( ). This adjunction can be written as  , or more fully as:

 

In the category of topological spaces, the exponential object   exists provided that   is a locally compact Hausdorff space. In that case, the space   is the set of all continuous functions from   to   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.[5] If   is not locally compact Hausdorff, the exponential object may not exist (the space   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   need not be locally compact for locally compact spaces   and  . A cartesian closed category of spaces is, for example, given by the full subcategory spanned by the compactly generated Hausdorff spaces.

In functional programming languages, the morphism   is often called  , and the syntax   is often written  . The morphism   here must not to be confused with the eval function in some programming languages, which evaluates quoted expressions.

See alsoEdit

NotesEdit

  1. ^ Exponential law for spaces in nLab
  2. ^ Convenient category of topological spaces in nLab
  3. ^ Goldblatt, Robert (1984). "Chapter 3: Arrows instead of epsilon". Topoi : the categorial analysis of logic. Studies in Logic and the Foundations of Mathematics #98 (Revised ed.). North-Holland. p. 72. ISBN 978-0-444-86711-7.
  4. ^ Mac Lane, Saunders (1978). "Chapter 4: Adjoints". categories for the working mathematician. graduate texts in mathematics. 5 (2nd ed.). Springer-Verlag. p. 98. doi:10.1007/978-1-4757-4721-8_5. ISBN 978-0387984032.
  5. ^ Joseph J. Rotman, An Introduction to Algebraic Topology (1988) Springer-Verlag ISBN 0-387-96678-1 (See Chapter 11 for proof.)

ReferencesEdit

External linksEdit