In this paper we show that each factorization structure M on a small category X , satisfying certain conditions, yields a presheaf M on X and a morphism of presheaves m :. −→ M. We then give connections, and set up one to one correspondences, between subclasses of the following classes: (a) closure operators on X (b) subobjects of M (c) morphisms from M to (d) weak Lawvere-Tierney topologies (e) weak Grothendieck topologies (f) closure operators on Sets X op .
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.
In this paper the category, C = (C, D)
− −, of partial morphisms of a category C with respect to a certain class D of subobjects of C is formed and the universality of monomorphisms of C is investigated. The main result characterizes Cuniversality of monos, in terms of C-universality of monos and the existence of local C-implications.
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