The concept of a filtered category is a categorification of the concept of a 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. The dual notion of filtered category is that of cofiltered category: a category whose opposite is filtered. More in details, this requirement is that: 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. All this may be rephrased in more elementary terms by saying that: There exists an object of (the case when ) For any two objects , there exists an object and morphisms and . For any two parallel morphisms in , there exists a morphism such 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. However, the reduction is a little more sophisticated; see Theorem below. 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 when has arrows, or equivalently that any functor whose domain...