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).

A category J is filtered when

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

A diagram is said to be of cardinality κ if the morphism set of its domain is of cardinality κ. A category J is filtered if and only if there is a cone over any finite diagram $d: D\to J$; more generally, for a regular cardinal κ, a category J is said to be κ-filtered if for every diagram d in J of cardinality smaller than κ there is a cone over d.

A filtered colimit is a colimit of a functor $F:J\to C$ where J is a filtered category. This readily generalizes to κ-filtered limits. An ind-object in a category C is a presheaf of sets $C^{op}\to Set$ which is a small filtered colimit of representable presheaves. Ind-objects in a category C form a full subcategory Ind(C) in the category of functors $C^{op}\to Set$. The category Pro(C) = Ind(C^op)^op of pro-objects in C is the opposite of the category of ind-objects in the opposite category C^op.

Cofiltered categories

There is a dual notion of cofiltered category. A category J is cofiltered if the opposite category J^op is filtered. In detail, a category is cofiltered when

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

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