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 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^{\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. ==Ind-objects and pro-objects==

Given a small category C, a presheaf of sets C^{op}\to Set 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 Ind(C) in the category of functors (presheaves) 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}. ==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 \{\ \ \}\rightarrow J, \{j\ \ \ j'\}\rightarrow J, or \{i\rightrightarrows j\}\rightarrow 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==