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.
for every two objects and in there exists an object and two arrows and in ,
for every two parallel arrows in , there exists an object and an arrow such that .
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
it is not empty,
for every two objects and in there exists an object and two arrows and in ,
for every two parallel arrows in , there exists an object and an arrow such that .
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 (presheaves) . 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 (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 in 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 where is a κ-filtered category.