Exact Categories

TL;DR

This paper provides a comprehensive survey of homological algebra in exact categories, including diagram lemmas, derived categories, and functor constructions, extending classical results beyond abelian categories.

Contribution

It introduces methods to construct derived categories and functors directly in exact categories without embedding into abelian categories, and discusses key theorems like Gabriel-Quillen embedding.

Findings

Direct proofs of diagram lemmas from axioms

Construction of derived categories without abelian embedding

Extension of classical derived functors to exact categories

Abstract

We survey the basics of homological algebra in exact categories in the sense of Quillen. All diagram lemmas are proved directly from the axioms, notably the five lemma, the 3 x 3-lemma and the snake lemma. We briefly discuss exact functors, idempotent completion and weak idempotent completeness. We then show that it is possible to construct the derived category of an exact category without any embedding into abelian categories and we sketch Deligne's approach to derived functors. The construction of classical derived functors with values in an abelian category painlessly translates to exact categories, i.e., we give proofs of the comparison theorem for projective resolutions and the horseshoe lemma. After discussing some examples we elaborate on Thomason's proof of the Gabriel-Quillen embedding theorem in an appendix.