Exact Categories
Theo Buehler
TL;DR
Exact categories provide a flexible framework for homological algebra beyond abelian categories, built from Quillen's axioms and diagram lemmas. The text develops a self-contained theory—from definitions, diagram lemmas, and functoriality to idempotent completion and weak idempotent completeness—then constructs derived categories and total derived functors without embedding into abelian categories, via Deligne–Keller methods. It covers projective/injective resolutions, classical derived functors, and extends to Frobenius categories, fully exact subcategories, and higher algebraic K-theory, with an appendix on the Gabriel–Quillen embedding. The resulting toolkit enables applications across functional-analytic settings and geometry, while clarifying foundational aspects of triangulated and derived categories in a non-abelian context.
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.