Concept
In , 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. A 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. A category is cofiltered if the 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 of a functor where is a cofiltered category. Given a , a 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 . 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 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 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.