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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 Generator (category theory) ： ウィキペディア英語版 Generator (category theory) In category theory in mathematics a family of generators (or family of separators) of a category $\backslash mathcal\; C$ is a collection $\backslash$ of objects, indexed by some set ''I'', such that for any two morphisms $f,\; g:\; X\; \backslash rightarrow\; Y$ in $\backslash mathcal\; C$, if $f\backslash neq\; g$ then there is some ''i∈I'' and morphism $h\; :\; G\_i\; \backslash rightarrow\; X$, such that the compositions $f\; \backslash circ\; h\; \backslash neq\; g\; \backslash circ\; h$. If the family consists of a single object ''G'', we say it is a generator (or separator). Generators are central to the definition of Grothendieck categories. The dual concept is called a cogenerator or coseparator. ==Examples== * In the category of abelian groups, the group of integers $\backslash mathbf\; Z$ is a generator: If ''f'' and ''g'' are different, then there is an element $x\; \backslash in\; X$, such that $f(x)\; \backslash neq\; g(x)$. Hence the map $\backslash mathbf\; Z\; \backslash rightarrow\; X,$ $n\; \backslash mapsto\; n\; \backslash cdot\; x$ suffices. * Similarly, the one-point set is a generator for the category of sets. In fact, any nonempty set is a generator. * In the category of sets, any set with at least two objects is a cogenerator. 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「Generator (category theory)」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.