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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Exponential object ： ウィキペディア英語版 Exponential object 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 :$\backslash mathrm\backslash colon\; (Z^Y\; \backslash times\; Y)\; \backslash rightarrow\; Z$ is an exponential object if for any object ''X'' and morphism ''g'' : (''X''×''Y'') → ''Z'' there is a unique morphism :$\backslash lambda\; g\backslash colon\; X\backslash to\; Z^Y$ 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 :$\backslash mathrm(X\backslash times\; Y,Z)\; \backslash cong\; \backslash mathrm(X,Z^Y).$ (Note: In functional programming languages, the morphism ''eval'' is often called ''apply'', and the syntax $\backslash lambda\; g$ 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 $g$ and $\backslash lambda\; g$ are sometimes said to be ''exponential adjoints'' of one another. It should be noted that for $A,B$ in the category of sets, $A^B\; =\; \backslash mathrm(B,A)$. 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 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.