The paper develops the previously proposed approach to constructing factorization systems in general categories. This approach is applied to the problem of finding conditions under which a functor (not necessarily admitting a right adjoint) `reflects' factorization systems. In particular, a generalization of the well-known Cassidy-Héebert-Kelly factorization theorem is given. The problem of relating a factorization system to a pointed endofunctor is considered. Some relevant examples in concrete categories are given.
Keywords: (local) factorization system, family of adjunctions between slice categories, semi-left-exact reflection, fibration, (co)pointed endofunctor
2000 MSC: 18A20, 18A32, 18A25
Theory and Applications of Categories,
Vol. 12, 2004,
No. 11, pp 326-354.