Mehrfache Partialisierung von Kategorien. (Multiple partialization of categories) (Q1090753)

Let C be a locally small category with a given class M of monomorphisms in C containing all C-isomorphisms, closed under composition, and such that every pair \(f: A\to B\), \(g: C\to B\) of C-morphisms with \(f\in M\) has a pullback in C. For C-objects A,B define \(Part_ M(A,B)=\{(f,g)\); \(f: X\to A\in M\), \(g: X\to B\}/\sim_{A,B}\) where \((f,g)\sim_{A,B}(f',g')\) if and only if there exists an C-isomorphism h with \(f=f'\circ h\), \(g=g'\circ h\). Define a category \(Part_ M\) whose objects are C-objects and the morphisms from A to B are all representatives of \(\sim_{A,B}\). Basic properties of \(Part_ M\) are given.