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 formula_1 is filtered when A filtered colimit is a colimit of a functor formula_14 where formula_1 is a filtered category. Cofiltered categories. A category formula_1 is cofiltered if the opposite category formula_17 is filtered. In detail, a category is cofiltered when A cofiltered limit is a limit of a  functor formula_30 where formula_1 is a cofiltered category. Ind-objects and pro-objects. Given a small category formula_32, a presheaf of sets formula_33 that is a small filtered colimit of representable presheaves, is called an ind-object of the category formula_32. Ind-objects of a category formula_32 form a full subcategory formula_36 in the category of functors (presheaves) formula_33. The category formula_38 of pro-objects in formula_32 is the opposite of the category of ind-objects in the opposite category formula_40. 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 formula_1 of the form formula_42, formula_43, or formula_44. 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 formula_1 is filtered (according to the above definition) if and only if there is a cocone over any "finite" diagram formula_46. Extending this, given a regular cardinal κ, a category formula_1 is defined to be κ-filtered if there  is a cocone over every diagram formula_48 in formula_1 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 (co)limit is a (co)limit of a functor formula_14 where formula_1 is a κ-filtered category.