Timeline for Name for abelian category in which every short exact sequence splits

7 events

when toggle format	what		by	license	comment
Apr 13, 2019 at 12:56	history	edited	Mare	CC BY-SA 4.0	deleted 1 character in body
Apr 13, 2019 at 12:55	comment	added	Donu Arapura		I almost regret making my original suggestion, but if I were given a choice between "semisimple" and "abelian category with global dimension 0", I'd pick the first. At least it's semisimpler.
Apr 13, 2019 at 12:15	comment	added	Jeremy Rickard		I think that a good reason for avoiding the term “semisimple” for abelian categories, at least without explanation, is that there are at least three different uses that I have seen: (i) every short exact sequence splits, (ii) every object is a coproduct of simples, or (iii) every object is a finite coproduct of simples. Sometimes we can have clearly correct opinions on what the terminology should be ... but we’re too late.
Apr 13, 2019 at 12:08	history	edited	Mare	CC BY-SA 4.0	deleted 74 characters in body
Apr 13, 2019 at 12:07	comment	added	Leonid Positselski		Yes, "abelian categories of global/homological dimension 0" is a good terminology for abelian categories in which all short exact sequences split. Existence of projectives or injectives is not needed for that (as one can always define the Ext functor in an abelian category using Yoneda's construction).
Apr 13, 2019 at 12:04	comment	added	Leonid Positselski		It seems to me that the proper references are sections III.2.3 and III.5.6, and Exercise IV.1.1 in the book of Gelfand and Manin.
Apr 13, 2019 at 11:43	history	answered	Mare	CC BY-SA 4.0