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.
is filtered when A filtered colimit is a colimit of a functor
is cofiltered if the opposite category
In detail, a category is cofiltered when A cofiltered limit is a limit of a functor
{\displaystyle C^{op}\to Set}
that is a small filtered colimit of representable presheaves, is called an ind-object of the category
form a full subcategory
in the category of functors (presheaves)
{\displaystyle C^{op}\to Set}
is the opposite of the category of ind-objects in the opposite category
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
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 (according to the above definition) 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
of cardinality smaller than κ.
(A small diagram is of cardinality κ if the morphism set of its domain is of cardinality κ.)