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 A filtered colimit is a colimit of a functor F:J\to C where J is a filtered category.

Cofiltered categories

A category J is cofiltered if the opposite category is filtered. In detail, a category is cofiltered when A cofiltered limit is a limit of a functor F:J \to C where J is a cofiltered category.

Ind-objects and pro-objects

Given a small category C, a presheaf of sets 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 Ind(C) in the category of functors (presheaves). The category of pro-objects in C is the opposite of the category of ind-objects in the opposite category C^{op}.

κ-filtered categories

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, , or. 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\to 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 colimit is a colimit of a functor F:J\to C where J is a κ-filtered category.