A functor is cocontinuous if it preserves small colimits. Usually one only considers cocontinuous functors whose domain and codomain are cocomplete categories (have all small colimits).
Note that is cocontinuous if and only if the functor between opposite categories is a continuous functor.
Not every functor is cocontinuous; an example of a dis-cocontinuous (or disco-continuous) functor is the forgetful functor from the category of pointed sets to the category of sets.
“Morally speaking,” a functor is cocontinuous if and only if it is a left adjoint (or equivalently has a right adjoint). Actually, only the ‘if’ part is true as stated; the ‘only if’ part has some conditions on it, given by the adjoint functor theorem.
Last revised on November 15, 2010 at 03:59:36. See the history of this page for a list of all contributions to it.