Factorization of Functors Having Left Adjoints

Abstract

The composition of functors with left adjoints with epimorphic front adjunctions may yield a functor whose left adjoint does not have epimorphic front adjunctions as in the case of the inclusion functors from Comp to Haus and from Haus to Top. In fact it is shown that any functor T : A → X with a left adjoint can be written as T = R ◯ S where R has a left adjoint having isomorphic front adjunctions and S has a left adjoint with epimorphic front adjunctions. However if T = R ◯ S, where the properties of R and S are interchanged from the above, then the left adjoint of T must have epimorphic front adjunctions.

In an (E,M)-category X , certain lifting properties of the functor T : A → X are shown to characterize the situation where T has a left adjoint with front adjunctions in E , These results can be further specialized to characterize possibly non-full E-reflective subcategories of (E,M)-categories.