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 $\kappa$ if the morphism set of its domain is of cardinality $\kappa$ . 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 $\kappa$ , a category $J$ is said to be $\kappa$ -filtered if for every diagram $d$ in $J$ of cardinality smaller than $\kappa$ there is a 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 $\kappa$ -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

There is a dual notion of cofiltered category. 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.