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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 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 that 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. It should be noted that for in the category of sets, . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「exponential object」の詳細全文を読む スポンサード リンク
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