cocontinuous

Cocontinuous is a term used in category theory to describe a functor that preserves colimits. If F: C → D is cocontinuous, then for every small diagram D: J → C that has a colimit colim D in C, the image diagram F ∘ D has a colimit in D and F(colim D) is canonically isomorphic to colim(F ∘ D). In short, F preserves all small colimits.

This notion is dual to continuity, which describes functors that preserve limits. Consequently, a cocontinuous functor preserves coproducts, coequalizers, pushouts, and other colimits that exist in its domain. It is common to speak of a category being cocomplete when all small colimits exist, and of a functor being cocontinuous when it preserves those colimits.

A fundamental source of cocontinuous functors is left adjoints. If F is left adjoint to a functor G, then F preserves all colimits that exist, hence F is cocontinuous. Classic examples include the free group functor from sets to groups, which assigns to each set X the free group on X; this functor preserves colimits and thus is cocontinuous. More broadly, many algebraic constructions that are left adjoints—such as free object constructions and tensor-hom adjunctions in suitable settings—are cocontinuous.

Notes and scope: The property depends on the existence of colimits in the source category. A functor may preserve limits without preserving colimits, and vice versa. In practical contexts, cocontinuity is an important condition when transferring colimit computations along functors, such as in constructions within algebra, topology, and sheaf theory.