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

A category $J$ is cofiltered if the opposite category $J^{\mathrm{op}}$ 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.