Terminology: Is there a name for a category with biproducts? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T13:17:43Z http://mathoverflow.net/feeds/question/12410 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12410/terminology-is-there-a-name-for-a-category-with-biproducts Terminology: Is there a name for a category with biproducts? Chris Schommer-Pries 2010-01-20T13:23:52Z 2010-01-20T19:13:23Z <p>Many people are familiar with the notion of an additive category. This is a category with the following properties:</p> <p>(1) It contains a zero object (an object which is both initial and terminal).</p> <p>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"). </p> <p>(2) Finite products and coproducts exist.</p> <p>(3) The canonical map from the coproduct to the product is an equivalence.</p> <p>A standard exercise shows this gives us a multiplication on each hom space making the category enriched in commutative monoids (with unit). </p> <p>(4) An additive category further requires that these commutative monoids are abelian groups.</p> <blockquote> <p>I want to know what standard terminology is for a category which satisfies the first three axioms but not necessarily the last. </p> </blockquote> <p>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. </p> http://mathoverflow.net/questions/12410/terminology-is-there-a-name-for-a-category-with-biproducts/12414#12414 Answer by Reid Barton for Terminology: Is there a name for a category with biproducts? Reid Barton 2010-01-20T14:49:25Z 2010-01-20T14:49:25Z <p>"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 <em>categories enriched in commutative monoids with finite coproducts</em>. 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.</p> <p>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"?</p> http://mathoverflow.net/questions/12410/terminology-is-there-a-name-for-a-category-with-biproducts/12419#12419 Answer by Mike Shulman for Terminology: Is there a name for a category with biproducts? Mike Shulman 2010-01-20T15:29:25Z 2010-01-20T15:29:25Z <p>One name that I have seen used is <a href="http://ncatlab.org/nlab/show/biproduct#semiadditive%5Fcategories%5F3" rel="nofollow">semiadditive category</a>.</p>