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,

• ,

• .

A filtered colimit is a colimit of a functor is a filtered category.

Cofiltered categories

is cofiltered if the opposite category is filtered. In detail, a category is cofiltered when

• it is not empty,

• ,

• .

A cofiltered limit is a limit of a functor is a cofiltered category.

Ind-objects and pro-objects

Given a small category, a presheaf that is a small filtered colimit of representable presheaves, is called an ind-object.

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. The existence of cocones for these three shapes of diagrams turns out to imply that cocones exist for any is filtered (according to the above definition) if and only if there is a cocone over any finite.

Extending this, given a regular cardinal 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 is a κ-filtered category.

References

• Artin, M., Grothendieck, A. and Verdier, J.-L. Séminaire de Géométrie Algébrique du Bois Marie (SGA 4). Lecture Notes in Mathematics 269, Springer Verlag, 1972. Exposé I, 2.7.

• , section IX.1.