Exact Categories and Quillen’s Q-Construction

Abstract

For our purposes, an exact category ζ is an additive category ζ embedded as a full (additive) subcategory of an abelian category a, such that if 0→M′→M→M″→0 is an exact sequence in a with M′, M″ε ζ, then M is isomorphic to an objecj of ζ. An exact sequence in ζ is then defined to be an exact sequence in a whose terms lie in ζ. Let ξ be the class of exact sequences in ζ. One can give an intrinsic definition of an exact category ζ in terms of a class ξ of diagrams in the additive category ζ, satisfying suitable axioms (see Quillen’s paper for details). In all cases relevant to us, the category embeds naturally in some abelian category a, such that ζ, is closed under extensions in a.