Dear all, I built a category from another category by keeping "the same" objects and taking the morphisms to be diagrams from the old category that satisfy certain properties. The closest thing to what I'm doing that I have been able to find is factorization systems, but there are some major differences. To be more specific, I am starting with an Abelian category. If there are morphisms f: A --> B, g: X --> B, and h: A --> X, it makes sense to talk about the morphism f+gh: A --> B, which can be represented with a non-commutative triangle. I don't know how to draw that in plain text, so I will depict it as the ordered triple (f,g,h). The category I built takes this type of non-commutative triangle as a morphism (f,g,h): A --> B. That is, the new category is storing extra information in the morphisms by distinguishing between the part that goes directly from A to B and the part that takes a detour through an intermediate object, X. So while it may be possible for f+gh = f'+g'h' in the original category, (f,g,h) and (f',g',h') would be different morphisms in the new category unless f=f', g=g', and h=h'. One nice feature of this construction is the original category can be embedded in the new category by taking X to be the zero object. The people I have shown this to have told me they have never seen anything like my construction. I am at a loss for search terms -- everything I have tried either turns up nothing or thousands of unrelated articles. The closest I've found is factorization systems, which involve a commutative triangle, f=gh, for some g and h with certain properties. If any of you know a reference or keyword associated with expanding a category by replacing the morphisms with diagrams (with a specified property/shape), I would greatly appreciate the assistance. Sincerely, Jason Erbele [For admin and other information see: http://www.mta.ca/~cat-dist/ ]