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.
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 → k and f ′ : j ′ → k in J ,for every two parallel arrows u , v : i → j in J , there exists an object k and an arrow w : j → k such that w u = w v .A filtered colimit is a colimit of a functor F : J → C where J is a filtered category.
A category J is cofiltered if the opposite category J o p is filtered. In detail, a category is cofiltered when
it is not emptyfor every two objects j and j ′ in J there exists an object k and two arrows f : k → j and f ′ : k → j ′ in J ,for every two parallel arrows u , v : j → i in J , there exists an object k and an arrow w : k → j such that u w = v w .A cofiltered limit is a limit of a functor F : J → C where J is a cofiltered category.
Ind-objects and pro-objects
Given a small category C , a presheaf of sets C o p → S e t that is a small filtered colimit of representable presheaves, is called an ind-object of the category C . Ind-objects of a category C form a full subcategory I n d ( C ) in the category of functors (presheaves) C o p → S e t . The category P r o ( C ) = I n d ( C o p ) o p of pro-objects in C is the opposite of the category of ind-objects in the opposite category C o p .
There is a variant of "filtered category" known as a "κ-filtered category", defined as follows. This begins with the following observation: the three conditions in the definition of filtered category above say respectively that there exists a cocone over any diagram in J of the form { } → J , { j j ′ } → J , or { i ⇉ j } → J . The existence of cocones for these three shapes of diagrams turns out to imply that cocones exist for any finite diagram; in other words, a category J is filtered (according to the above definition) if and only if there is a cocone over any finite diagram d : D → J .
Extending this, given a regular cardinal κ, a category J is defined to be κ-filtered if there is a cocone over every diagram d in J of cardinality smaller than κ. (A small diagram is of cardinality κ if the morphism set of its domain is of cardinality κ.)
A κ-filtered (co)limit is a (co)limit of a functor F : J → C where J is a κ-filtered category.
