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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 is filtered when * it is not empty, * for every two objects and in there exists an object and two arrows and in , * for every two parallel arrows in , there exists an object and an arrow such that . A diagram is said to be of cardinality if the morphism set of its domain is of cardinality . A category is filtered if and only if there is a cocone over any finite diagram ; more generally, for a regular cardinal , a category is said to be -filtered if for every diagram in of cardinality smaller than there is a cocone over . A filtered colimit is a colimit of a functor where is a filtered category. This readily generalizes to -filtered limits. An ind-object in a category is a presheaf of sets which is a small filtered colimit of representable presheaves. Ind-objects in a category form a full subcategory in the category of functors . The category of pro-objects in is the opposite of the category of ind-objects in the opposite category . 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
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