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 \mathbf{C} on a category \mathcal{X} with respect to a class \mathcal{M} of morphisms, and quasi factorization structures in a category \mathcal{X} are introduced. It is shown that under certain conditions, if (\mathcal{E}, \mathcal{M}) is a quasi factorization structure in \mathcal{X}, then \mathcal{X} has a quasi right \mathcal{M}-factorization structure and a quasi left \mathcal{E}-factorization structure. It is also shown that for a quasi weakly hereditary and quasi idempotent QCD-closure operator with respect to a certain class \mathcal{M}, every quasi factorization structure (\mathcal{E}, \mathcal{M}) yields a quasi factorization structure relative to the given closure operator; and that for a closure operator with respect to a certain class \mathcal{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.

Keywords:

Quasi mono (epi) , quasi (right , left) factorization structure , (quasi weakly hereditary , quasi idempotent) general closure operator