Quasi Right Factorization Structures as Presheaves

Abstract

In this article the notion of quasi right factorization structure in a category X is given. The main result is a one to one correspondence between certain classes of quasi right factorization structures and 2-reflective subobjects of a predefined object in Lax(PrOrd X op ). Also a characterization of quasi right factorization structures in terms of images is given. As an application, the closure operators are discussed and it is shown that quasi closed members of certain collections are quasi right factorization structures. Finally several examples are furnished.

On semi weak factorization structures

“…In [9] a one to one correspondence between certain classes of quasi right factorization structures and 2-reflective subobjects of a predefined object in the category of laxed preordered valued presheaves, Lax(P rOrd X op ), has been studied. In [7] it has been shown that quasi left factorization structures correspond to subobjects of predefined objects in Lax(P rOrd X op ).…”

“…The notions of (right, left) factorization structure appeared in [2], while weak factorization structures introduced in [1]. In [9] and [7] the notions of quasi right, respectively, quasi left, factorization structure and some related results has been given. Since in various categories, there are important classes of morphisms that are not factorization structures nor even weak factorization structures, a weaker notion of factorization structure is deemed necessary; so the notion of semi weak factorization structure is introduced.…”

“…With g E = {ge|ge is defined and e ∈ E} (for E the class of all morphisms, g E is just a principal sieve, see [7]), we have Definition 1.4. See [9]. Suppose that M is a class of morphisms in X .…”

On General Closure Operators and Quasi Factorization Structures

“…Let C be the category of torsion free modules, [7], and M be the class of retractions. Then M is a quasi right factorization structure in C, [13]. A morphism f : X / / Y in C can be factored as follows:…”

“…Weakly hereditary and idempotent closure operators play an important role, as they arise from factorization structures. In [13], quasi right factorization structures are introduced and their connection with closure operators are investigated, while quasi left factorization structures appear in [10].…”