Factorizations, Localizations, and the Orthogonal Subcategory Problem

In this paper factorization structures of an abstract category are considered, depending on a class 𝔈 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.

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