Cover relations on categories (Q835748)

A cover relation on a category \(\mathcal C\) is a binary relation \(\sqsubset \) on the class of morphisms of \(\mathcal C\), which is defined only for those pairs of morphisms which have the same codomain, and which has the following two properties: (i) if \(f\sqsubset g\) and \(h\) is composable with \(f\), then \( hf\sqsubset hg,\) (ii) if \(f\sqsubset g\) and \(f\) is composable with \(e\) then \( fe\sqsubset g\). The aim of this article is to examine two different kinds of cover relations on a category \(\mathcal C\) -- those induced by a special type of factorizations systems on \(\mathcal C\), and those induce by a special type of monoidal structures on \(\mathcal C\). In both cases the base structure can be fully recovered from the induced cover relation.