Filtered category

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 [math]\displaystyle{ J }[/math] is filtered when

• it is not empty,
• for every two objects [math]\displaystyle{ j }[/math] and [math]\displaystyle{ j' }[/math] in [math]\displaystyle{ J }[/math] there exists an object [math]\displaystyle{ k }[/math] and two arrows [math]\displaystyle{ f:j\to k }[/math] and [math]\displaystyle{ f':j'\to k }[/math] in [math]\displaystyle{ J }[/math],
• for every two parallel arrows [math]\displaystyle{ u,v:i\to j }[/math] in [math]\displaystyle{ J }[/math], there exists an object [math]\displaystyle{ k }[/math] and an arrow [math]\displaystyle{ w:j\to k }[/math] such that [math]\displaystyle{ wu=wv }[/math].

A filtered colimit is a colimit of a functor [math]\displaystyle{ F:J\to C }[/math] where [math]\displaystyle{ J }[/math] is a filtered category.

Cofiltered categories

A category [math]\displaystyle{ J }[/math] is cofiltered if the opposite category [math]\displaystyle{ J^{\mathrm{op}} }[/math] is filtered. In detail, a category is cofiltered when

• it is not empty,
• for every two objects [math]\displaystyle{ j }[/math] and [math]\displaystyle{ j' }[/math] in [math]\displaystyle{ J }[/math] there exists an object [math]\displaystyle{ k }[/math] and two arrows [math]\displaystyle{ f:k\to j }[/math] and [math]\displaystyle{ f':k \to j' }[/math] in [math]\displaystyle{ J }[/math],
• for every two parallel arrows [math]\displaystyle{ u,v:j\to i }[/math] in [math]\displaystyle{ J }[/math], there exists an object [math]\displaystyle{ k }[/math] and an arrow [math]\displaystyle{ w:k\to j }[/math] such that [math]\displaystyle{ uw=vw }[/math].

A cofiltered limit is a limit of a functor [math]\displaystyle{ F:J \to C }[/math] where [math]\displaystyle{ J }[/math] is a cofiltered category.

Ind-objects and pro-objects

Given a small category [math]\displaystyle{ C }[/math], a presheaf of sets [math]\displaystyle{ C^{op}\to Set }[/math] that is a small filtered colimit of representable presheaves, is called an ind-object of the category [math]\displaystyle{ C }[/math]. Ind-objects of a category [math]\displaystyle{ C }[/math] form a full subcategory [math]\displaystyle{ Ind(C) }[/math] in the category of functors (presheaves) [math]\displaystyle{ C^{op}\to Set }[/math]. The category [math]\displaystyle{ Pro(C)=Ind(C^{op})^{op} }[/math] of pro-objects in [math]\displaystyle{ C }[/math] is the opposite of the category of ind-objects in the opposite category [math]\displaystyle{ C^{op} }[/math].

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 [math]\displaystyle{ J }[/math] of the form [math]\displaystyle{ \{\ \ \}\rightarrow J }[/math], [math]\displaystyle{ \{j\ \ \ j'\}\rightarrow J }[/math], or [math]\displaystyle{ \{i\rightrightarrows j\}\rightarrow J }[/math]. 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 [math]\displaystyle{ J }[/math] is filtered (according to the above definition) if and only if there is a cocone over any finite diagram [math]\displaystyle{ d: D\to J }[/math].

Extending this, given a regular cardinal κ, a category [math]\displaystyle{ J }[/math] is defined to be κ-filtered if there is a cocone over every diagram [math]\displaystyle{ d }[/math] in [math]\displaystyle{ J }[/math] 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 [math]\displaystyle{ F:J\to C }[/math] where [math]\displaystyle{ J }[/math] 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.