# 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 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 diagram is said to be of cardinality if the morphism set of its domain is of cardinality . A category is filtered if and only if there is a cocone over any finite diagram ; more generally, for a regular cardinal , a category is said to be -filtered if for every diagram in of cardinality smaller than there is a cocone over .

A **filtered colimit** is a colimit of a functor where is a filtered category. This readily generalizes to -filtered limits. An **ind-object** in a category is a presheaf of sets which is a small filtered colimit of representable presheaves. Ind-objects in a category form a full subcategory in the category of functors . The category of pro-objects in is the opposite of the category of ind-objects in the opposite 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.

## 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. - {{#invoke:citation/CS1|citation

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