On general closure operators and quasi factorization structures

In this article the notions of quasi mono (epi) as a generalization of mono (epi), (quasi weakly hereditary) general closure operator C on a category X with respect to a class M of morphisms, and quasi factorization structures in a category X are introduced. It is shown that under certain conditions, if (ε,M) is a quasi factorization structure in X, then X has a quasi right M-factorization structure and a quasi left ε-factorization structure. It is also shown that for a quasi weakly hereditary and quasi idempotent QCD-closure operator with respect to a certain class M, every quasi factorization structure (ε,M) yields a quasi factorization structure relative to the given closure operator; and that for a closure operator with respect to a certain class M, if the pair of classes of quasi dense and quasi closed morphisms forms a quasi factorization structure, then the closure operator is both quasi weakly hereditary and quasi idempotent. Several illustrative examples are provided.