Natural factorizations and related categorical structures

Abstract:
The basic notion of the homotopy theory is a Quillen's model category. A model structure assumes distinguished classes of morphisms which form two weak factorization systems. Recently discovered concept of a natural weak factorization [M. Grandis, W. Tholen, 2006], [R. Garner, 2007] open a promising possibility to algebraization of the whole subject. We consider some steps towards application of natural factorizations in the context of model categories and relations of these new ideas with the category theory itself.