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 $w u = w v$ .

A filtered colimit is a colimit of a functor $F : J \to C$ where $J$ is a filtered category.

Cofiltered categories

A category $J$ is cofiltered if the opposite category $J^{o p}$ 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 $u w = v w$ .

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

Ind-objects and pro-objects

Given a small category $C$ , a presheaf of sets $C^{o p} \to S e t$ that is a small filtered colimit of representable presheaves, is called an ind-object of the category $C$ . Ind-objects of a category $C$ form a full subcategory $I n d (C)$ in the category of functors (presheaves) $C^{o p} \to S e t$ . The category $P r o (C) = I n d (C^{o p})^{o p}$ of pro-objects in $C$ is the opposite of the category of ind-objects in the opposite category $C^{o p}$ .

κ-filtered categories

There is a variant of "filtered category" known as a "κ-filtered category", defined as follows. This begins with the following observation: the three conditions in the definition of filtered category above say respectively that there exists a cocone over any diagram in $J$ of the form ${} \to J$ , ${j j^{'}} \to J$ , or ${i ⇉ j} \to J$ . The existence of cocones for these three shapes of diagrams turns out to imply that cocones exist for any finite diagram; in other words, a category $J$ is filtered (according to the above definition) if and only if there is a cocone over any finite diagram $d : D \to J$ .

Extending this, given a regular cardinal κ, a category $J$ is defined to be κ-filtered if there is a cocone over every diagram $d$ in $J$ of cardinality smaller than κ. (A small diagram is of cardinality κ if the morphism set of its domain is of cardinality κ.)

A κ-filtered colimit is a colimit of a functor $F : J \to C$ where $J$ is a κ-filtered 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.