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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 Category of elements ： ウィキペディア英語版 Category of elements In category theory, if C is a category and $F:\; C\; \backslash to\; \backslash mathbf$ is a set-valued functor, the category of elements of F $\backslash mathop(F)$ (also denoted by ∫CF) is the category defined as follows: * Objects are pairs $(A,a)$ where $A\; \backslash in\; \backslash mathop(C)$ and $a\; \backslash in\; FA$. * An arrow $(A,a)\; \backslash to\; (B,b)$ is an arrow $f:\; A\; \backslash to\; B$ in C such that $(Ff)a\; =\; b$. A more concise way to state this is that the category of elements of F is the comma category $\backslash ast\backslash downarrow\; F$, where $\backslash ast$ is a one-point set. The category of elements of F comes with a natural projection $\backslash mathop(F)\; \backslash to\; C$ that sends an object (A,a) to A, and an arrow $(A,a)\; \backslash to\; (B,b)$ to its underlying arrow in C. == The category of elements of a presheaf == Somewhat confusingly in some texts (e.g. Mac Lane, Moerdijk), the category of elements for a presheaf is defined differently. If $P\; \backslash in\backslash hat\; C\; :=\; \backslash mathbf^(P)$, or, to make the distinction to the above definition clear, ∫C P) is the category defined as follows: * Objects are pairs $(A,a)$ where $A\; \backslash in\; \backslash mathop(C)$ and $a\backslash in\; P(A)$. * An arrow $(A,a)\backslash to\; (B,b)$ is an arrow $f:A\backslash to\; B$ in C such that $(Pf)b\; =\; a$. As one sees, the direction of the arrows is reversed. One can, once again, state this definition in a more concise manner: the category just defined is nothing but $(\backslash ast\backslash downarrow\; P)^$. Consequentially, in the spirit of adding a "co" in front of the name for a construction to denote its opposite, one should rather call this category the category of coelements of P. For C small, this construction can be extended into a functor ∫C from $\backslash hat\; C$ to $\backslash mathbf$, the category of small categories. In fact, using the Yoneda lemma one can show that ∫CP $\backslash cong\; \backslash mathop\}:\; C\; \backslash to\; \backslash hat$ is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫C is naturally isomorphic to $\backslash mathop$. 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「Category of elements」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.