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 category whose opposite is filtered is called cofiltered.
That is, for any finite category and any functor , there exists an object and a natural transformation where is the constant diagram at . If is the result of freely adjoining a terminal object to a category , then the condition is the same as that any functor with finite domain admits an extension .
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.
In constructive mathematics, the elementary rephrasing above is equivalent to every Bishop-finite diagram admitting a cocone.
More generally, if is an infinite regular cardinal (or an arity class), then a -filtered category is one such that any diagram has a cocone where has arrows, or equivalently that any functor whose domain has fewer than morphisms admits an extension . The usual filtered categories are then the case , i.e., where the have fewer than morphisms (in other words are finite). (We could also say in this case ”-filtered”, but -filtered is more usual in the literature.)
Even more generally, if is a class of small categories, a category is called -filtered if -colimits commute with -limits in Set. When is the class of all -small categories for an infinite regular cardinal , then -filteredness is the same as -filteredness as defined above. See ABLR.
If is the class consisting of the terminal category and the empty category — which is to say, the class of -small categories when is the finite regular cardinal — then being -filtered in this sense is equivalent to being connected. Note that this is not what the explicit definition given above for infinite regular cardinals would specialize to by simply setting (that would be simply inhabitation).
Every category with a terminal object is filtered.
Every category which has finite colimits is filtered.
A product of filtered categories is filtered.