Factorizations, Localizations, and the Orthogonal Subcategory Problem

In this paper factorization structures of an abstract category are considered, depending on a class of morphisms which is not necessarily closed under composition; as soon as it is one obtains the usual factorization systems defined by the diagonal‐fill‐in property. General existence criteria for those factorization structures are proved, in particular for monotone‐light factorizations which are defined for abstract categories and which are considered in more detail. Finally, sufficient conditions for a positive solution of the Orthogonal Subcategory Problem are derived from the existence of certain factorization structures.