Filtered category

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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[edit]

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 small 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.

Given a small category , a presheaf of sets that is a small filtered colimit of representable presheaves, is called an ind-object of the category . Ind-objects of a category form a full subcategory in the category of functors (presheaves) . The category of pro-objects in is the opposite of the category of ind-objects in the opposite category .

Cofiltered categories[edit]

A category is cofiltered if the opposite category is filtered. In detail, a category is cofiltered 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 cofiltered limit is a limit of a functor where is a cofiltered category.

References[edit]