A filtered category is a categorification of the concept of directed set. In addition to having an upper bound (but not necessarily a coproduct) for every pair of objects, there must also be an upper bound (but not necessarily a coequaliser) for every pair of parallel morphisms.
A diagram where is a filtered category is called a filtered diagram. A colimit of a filtered diagram is called a filtered colimit.
A category whose opposite is filtered is called cofiltered.
A (finitely) filtered category (sometimes called a filtrant category, as for instance in Kashiwara–Schapira's book Categories and Sheaves) is a category in which any finite diagram has a cocone. That is, for any finite category and any functor , there exists an object and a natural transformation where is the constant diagram at .
Equivalently, filtered categories can be characterized as those categories where, for every finite diagram , the diagonal functor is final. This point of view can be generalized to other kinds of categories whose colimits are well-behaved with respect to a type of limit, such as sifted categories.
This can be rephrased in more elementary terms by saying that:
Just as all finite colimits can be constructed from initial objects, binary coproducts, and coequalizers, so a cocone on any finite diagram can be constructed from these three.
More generally, if is a cardinal number, then a -filtered category is one such that any diagram has a cocone where has fewer than arrows. Note that a preorder is -filtered as a category just when it is -directed as a preorder.
Every category with a terminal object is filtered.
Every category which has finite colimits is filtered.
A product of filtered categories is filtered.