Quasi Left Factorization Structures as Presheaves

In this article the notion of quasi left factorization structure in a category X is given. It is proved that quasi left factorization structures correspond to subobjects of a predefined object in the category of laxed preordered valued presheaves, Lax(PrOrdX op ). The main result proves this correspondence is one to one when quasi left factorization structures are restricted to the so called QLF-codomains.