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Many people are familiar with the notion of an additive category. This is a category with the following properties:

(1) It contains a zero object (an object which is both initial and terminal).

This implies that the category is enriched in pointed sets. Thus if a product $X \times Y$ and a coproduct $X \sqcup Y$ exist, then we have a canonical map from the coproduct to the product (given by "the identity matrix").

(2) Finite products and coproducts exist.

(3) The canonical map from the coproduct to the product is an equivalence.

A standard exercise shows this gives us a multiplication on each hom space making the category enriched in commutative monoids (with unit).

(4) An additive category further requires that these commutative monoids are abelian groups.

I want to know what standard terminology is for a category which satisfies the first three axioms but not necessarily the last.

I can't seem to find it using Google or Wikipedia. An obvious guess, "Pre-additive", seems to be standard terminology for a category enriched in abelian groups, which might not have products/coproducts.

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I'd just say category with biproducts, thereby avoiding adding yet another name! –  Mariano Suárez-Alvarez Jan 20 '10 at 13:51

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up vote 6 down vote accepted

One name that I have seen used is semiadditive category.

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"Category with biproducts" is probably the only standard name, but I'm not really fond of it because (at least in my experience) a more natural characterization of these categories satisfying (1)-(3) is as categories enriched in commutative monoids with finite coproducts. I would prefer to use "additive" for (1)-(3) (after all, "additive" doesn't say anything about being able to subtract!) and may have used that terminology in conversations with you, but I am unlikely to garner much support for this.

One sometimes encounters the term "R-additive category" for an additive category enriched in R-Mod. Given that, maybe "$\mathbb{N}$-additive category" is an alternative, pretending that the usual usage of "additive" is short for "$\mathbb{Z}$-additive"?

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But the enrichment in monoids is a consequence of the existence of biproducts---one does not say "finite group with Sylow subgroups for all primes". (It'd be so nice to be able to edit comments!) –  Mariano Suárez-Alvarez Jan 20 '10 at 15:04
    
I said "categories enriched in commutative monoids with finite coproducts", not "categories enriched in commutative monoids with finite biproducts". Of course the latter would be silly. (Or have I misunderstood you?) –  Reid Barton Jan 20 '10 at 15:06

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