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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 Filtered category ： ウィキペディア英語版 Filtered category In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered category which will be recalled below. ==Filtered categories== A category $J$ is filtered when * it is not empty, * for every two objects $j$ and $j\text{'}$ in $J$ there exists an object $k$ and two arrows $f:j\backslash to\; k$ and $f\text{'}:j\text{'}\backslash to\; k$ in $J$, * for every two parallel arrows $u,v:i\backslash to\; j$ in $J$, there exists an object $k$ and an arrow $w:j\backslash to\; k$ such that $wu=wv$. A diagram is said to be of cardinality $\backslash kappa$ if the morphism set of its domain is of cardinality $\backslash kappa$. A category $J$ is filtered if and only if there is a cocone over any finite diagram $d:\; D\backslash to\; J$; more generally, for a regular cardinal $\backslash kappa$, a category $J$ is said to be $\backslash kappa$-filtered if for every diagram $d$ in $J$ of cardinality smaller than $\backslash kappa$ there is a cocone over $d$. A filtered colimit is a colimit of a functor $F:J\backslash to\; C$ where $J$ is a filtered category. This readily generalizes to $\backslash kappa$-filtered limits. An ind-object in a category $C$ is a presheaf of sets $C^\backslash to\; Set$ which is a small filtered colimit of representable presheaves. Ind-objects in a category $C$ form a full subcategory $Ind(C)$ in the category of functors $C^\backslash to\; Set$. The category $Pro(C)=Ind(C^)^$ of pro-objects in $C$ is the opposite of the category of ind-objects in the opposite category $C^$. 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「Filtered category」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.