Let C be an additive category. For every pair X,Y of objects of C, let A(X,Y) be a subgroup of the additive group Hom(X,Y). Assume that for every quadruple W,X,Y,Z of objects of C, we have Hom(W,X)xA(X,Y)xHom(Y,Z) --> Hom(W,Z) (by composition ) maps into A(W,Z). That seems like having an ideal and it seems like I should then be able to form a category in which the set of morphisms from an object X to an object Y is Hom(X,Y)/A(X,Y).

Is there a name for such a system A and is there a name for the construction of modding out by A? Where is this formalized?

Allan Adler

(I assume by "additive category" you just mean Abelian-group-enriched category. These are sometimes called "ringoids" on the analogy of the generalization from groups to groupoids.) This is just the start of a well-known theory of ringoids whose basic message is that they behave very much like non-commutative rings. I'd call the systems A you refer to just ideals. The main reference is Barry Mitchell's "Rings with several objects", pp. 1-161 in Advances in Mathematics 8 (1972), and I'm sure you'd find the construction there. I haven't checked, but it's conceivable it's also in N. Popescu's "Abelian categories with applications to rings and modules" (LMS Monographs 3, Academic Press, London, 1973). There is also a discussion and some results in my thesis. Incidentally, one way of viewing the theory is that it says that modules and homomorphisms are "really" just functors and natural transformations (and I guess that's what you see in Kelly's book on enriched categories). However, I've also found it helpful to see it the other way round: even in the non-additive context, it lets you see some functors (specifically functors to Sets, e.g. presheaves) and natural transformations as modules and homomorphisms, and that's helpful in understanding tensor products of functors, flatness, etc. In "Quantales, observational logic and process semantics", with Samson Abramsky (shortly to appear in Mathematical Structures in Computer Science), we used "quantaloids", or sup-lattice enriched categories, and there are similar results there. A good book about how to treat categories as algebras (though it doesn't cover the additive case) is Philip Higgins' "Notes on categories and groupoids" (Van Nostrand Reinhold, London, 1971). Steve Vickers. ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++