# Who for the first time defined abelian categories?

1. Who for the first time defined additive categories?
2. Who for the first time defined abelian categories?

I am guessing it should be in an algebraic geometric paper, but who and when? Any reference will be appreciated.

The term "abelian category" and most of the crucial ideas behind the definition were introduced by Mac Lane in

MacLane, Saunders, Duality for groups, Bull. Am. Math. Soc. 56, 485-516 (1950). ZBL0041.36306.

What Mac Lane called "abelian categories" were actually much closer to what we now call additive categories, but he also defined a notion of "abelian bicategories" which was very similar to but not quite the same as the modern abelian categories.

What we now call abelian categories were first defined by Buchsbaum in

Buchsbaum, D. A., Exact categories and duality, Trans. Am. Math. Soc. 80, 1-34 (1955). ZBL0065.25502.

Buchsbaum called abelian categories "exact categories", and his axiomatization was a bit different from (but equivalent to) the one which is now standard. The name "abelian categories" for these categories and the standard axiomatization were introduced in Grothendieck's famous Tohoku paper

Grothendieck, A., Sur quelques points d’algèbre homologique, Tohoku Math. J., II. Ser. 9, 119-221 (1957). ZBL0118.26104.

The Tohoku paper also defined additive categories for the first time.

• To avoid confusion, maybe it should be said that there is a generalization of abelian category called an "exact category" (defined by Quillen?) as well; i.e. a category with a set of admissible epimorphisms and mono morphisms satisfying certain conditions – leibnewtz Oct 13 '18 at 18:55
• @leibnewtz As well as a dramatically different generalization of abelian category called an "exact category" by Barr. – Kevin Carlson Oct 13 '18 at 20:14