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 filtered colimit is a colimit of a functor  where  is a filtered 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. Ind-objects and pro-objects

Given a small category , a presheaf of sets  that is a small filtered colimit of representable presheaves, is called an ind-object of the category . Ind-objects of a category  form a full subcategory  in the category of functors (presheaves) . The category  of pro-objects in  is the opposite of the category of ind-objects in the opposite category . 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  of the form , , or . 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  is filtered (according to the above definition) if and only if there is a cocone over any finite diagram . Extending this, given a regular cardinal κ, a category  is defined to be κ-filtered if there is a cocone over every diagram  in  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  where  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.
 , section IX.1. Category theory