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,
- 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.
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.
- Mac Lane, Saunders (1998), Categories for the Working Mathematician (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-98403-2, section IX.1.