Dear All, Many thanks to Fred, and I apologize for my "blind spot" with infinite biproducts. I don't remember where from, but I also knew e.g. this fact about complete lattices (as we know of course, every sup-complete sup-semilattice is a complete lattice, and the only reason of mentioning "sup" is that we want morphisms to preserve arbitrary sups, but not necessarily infs), and, moreover, I believe it is a well known fact. I suddenly switched from monoids to ordinary additivity in my mind, and made that wrong statement about "indiscrete"; the correct statement would be that an additive category with infinite biproducts is indiscrete. Now, many thanks to everyone who supported the "category with biproducts" proposal. Can we still say that "biproducts" means "finite biproducts"? We certainly need the empty and binary biproducts, so we probably should say "category with finite biproducts", or, maybe, "category with (finite) biproducts". Or, if we still prefer "category with biproducts", we could go back to "Duality for Groups", and assume that biproducts (=free-and-direct products) are always binary and that they are defined only when our category category has zero object. Let me also comment to
"CATEGORY WITH BIPRODUCTS".
I want to second this. When I earlier wrote that I was using "category with direct sums" I did not mention that the alternative I use is "category with biproducts". The reason I mostly use direct sum rather than biproduct is that it is familiar to students where biproduct is not, but "category with biproducts" does seem a better choice for articles in category theory.
from Robert Knighten's message. I agree of course, but my additional reason of not saying "direct sums" would be the situation with the theory of Grothendieck categories, where traditionally (which was started by Grothendieck himself) infinite direct sums, defined as coproducts (and they are different from products), are seriously used. Best regards, George [For admin and other information see: http://www.mta.ca/~cat-dist/ ]