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

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

## 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

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