Filtered category


In category theory, filtered categories generalize the notion of directed set understood as a category. There is a dual notion of cofiltered category which will be recalled below.

Filtered categories

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

Cofiltered categories

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

Ind-objects and pro-objects

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 . The category of pro-objects in is the opposite of the category of ind-objects in the opposite category.

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 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 is filtered if and only if there is a cocone over any finite diagram.
Extending this, given a regular cardinal κ, a category is defined to be κ-filtered if there is a cocone over every diagram in of cardinality smaller than κ.
A κ-filtered limit is a limit of a functor where is a κ-filtered category.