Filtered Category

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).

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 diagram is said to be of cardinality if the morphism set of its domain is of cardinality . A category is filtered if and only if there is a cone over any finite diagram ; more generally, for a regular cardinal, a category is said to be -filtered if for every diagram in of cardinality smaller than there is a cone over .

A filtered colimit is a colimit of a functor where is a filtered category. This readily generalizes to -filtered limits. An ind-object in a category is a presheaf of sets which is a small filtered colimit of representable presheaves. Ind-objects in a category form a full subcategory in the category of functors . The category of pro-objects in is the opposite of the category of ind-objects in the opposite category .