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 J is filtered when

  • it is not empty,
  • for every two objects j and j' in J there exists an object k and two arrows f:j\to k and f':j'\to k in J,
  • for every two parallel arrows u,v:i\to j in J, there exists an object k and an arrow w:j\to k such that wu=wv.

A diagram is said to be of cardinality \kappa if the morphism set of its domain is of cardinality \kappa. A category J is filtered if and only if there is a cocone over any finite diagram d: D\to J; more generally, for a regular cardinal \kappa, a category J is said to be \kappa-filtered if for every diagram d in J of cardinality smaller than \kappa there is a cocone over d.

A filtered colimit is a colimit of a functor F:J\to C where J is a filtered category. This readily generalizes to \kappa-filtered limits. An ind-object in a category C is a presheaf of sets C^{op}\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^{op}\to Set. The category Pro(C)=Ind(C^{op})^{op} of pro-objects in C is the opposite of the category of ind-objects in the opposite category C^{op}.

Cofiltered categories[edit]

A category J is cofiltered if the opposite category J^{\mathrm{op}} is filtered. In detail, a category is cofiltered when

  • it is not empty
  • for every two objects j and j' in J there exists an object k and two arrows f:k\to j and f':k \to j' in J,
  • for every two parallel arrows u,v:j\to i in J, there exists an object k and an arrow w:k\to j such that uw=vw.

A cofiltered limit is a limit of a functor F:J \to C where J is a cofiltered category.

References[edit]