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 ${\displaystyle J}$  is filtered when

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

A filtered colimit is a colimit of a functor ${\displaystyle F:J\to C}$  where ${\displaystyle J}$  is a filtered category.

Cofiltered categories

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

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

A cofiltered limit is a limit of a functor ${\displaystyle F:J\to C}$  where ${\displaystyle J}$  is a cofiltered category.

Ind-objects and pro-objects

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

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

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