In a paper of 1974, Brian Day employed a notion of factorization system in the context of enriched category theory, replacing the usual diagonal lifting property with a corresponding criterion phrased in terms of hom-objects. We set forth the basic theory of such enriched factorization systems. In particular, we establish stability properties for enriched prefactorization systems, we examine the relation of enriched to ordinary factorization systems, and we provide general results for obtaining enriched factorizations by means of wide (co)intersections. As a special case, we prove results on the existence of enriched factorization systems involving enriched strong monomorphisms or strong epimorphisms.
Keywords: factorization systems; factorisation systems; enriched categories; strong monomorphisms; strong epimorphisms; monoidal categories; closed categories
2010 MSC: 18A20, 18A30, 18A32, 18D20
Theory and Applications of Categories, Vol. 29, 2014, No. 18, pp 475-495.