Exact category

Exact category
Name	Exact category
Field	Category theory, Homological algebra
Introduced	1980s (Quillen)
Introduced by	Daniel Quillen

Exact category

An exact category is an additive category equipped with a distinguished class of kernel–cokernel pairs that behave like short exact sequences in abelian categories and support a form of homological algebra. Originating in the work of Daniel Quillen on higher algebraic K-theory, exact categories provide a flexible framework linking constructions in Algebraic K-theory, Representation theory, Algebraic geometry, Functional analysis, and Topology by abstracting the notion of exactness found in module categories and categories of sheaves.

Definition

An exact category is an additive category C together with a class E of kernel–cokernel pairs (i: A → B, p: B → C) called conflations, where i is an inflation and p is a deflation, satisfying axioms modeled on short exact sequences in abelian categories. The axioms require that E is closed under isomorphism, contains all split exact sequences (those coming from direct sum decompositions related to split sequences), is stable under composition and base change for inflations and deflations, and satisfies axioms ensuring pushouts of inflations and pullbacks of deflations exist and remain in E. The standard reference axiomatizing these properties is due to Daniel Quillen and was later refined in works by Maxim Kontsevich, Henning Krause, and Tomasz Brzeziński among others.

Examples

Typical examples include full additive subcategories of abelian categories that are closed under extensions, such as the category of finitely generated projective modules over a ring R, the category of vector bundles on a scheme X (a subcategory of the category of quasi-coherent sheaves), and categories of coherent sheaves on a Noetherian scheme that are stable under extensions. Other instances arise in Representation theory: the category of finite-dimensional representations of a finite-dimensional algebra A that are filtered by a given set of standard modules, as studied in the context of highest weight categories and quasi-hereditary algebras. In Functional analysis, exact structures appear on categories of Banach spaces with short exact sequences split by bounded linear maps, and in Algebraic topology they appear when extracting exact subcategories from stable model categories or derivators.

Properties

Exact categories inherit many properties of abelian contexts while remaining more general. Every exact category is additive and has finite biproducts; conflations determine a notion of admissible monomorphisms and admissible epimorphisms that behave like kernels and cokernels in abelian categories. The Yoneda Ext groups Ext^n can be defined via derived functors of Hom or via resolutions built from conflations, linking exact categories to derived categories and triangulated categories. The Waldhausen S-construction and Quillen Q-construction apply to exact categories to produce Algebraic K-theory spectra; this underpins comparisons between K-theory of schemes, rings, and exact functors between categories. Stability under idempotent completion (Karoubi envelope) often plays a role: the idempotent completion of an exact category carries an induced exact structure, relevant to work of Max Karoubi and computations in higher K-theory.

Exact functors and subcategories

An exact functor between exact categories is an additive functor that preserves conflations, inflations, and deflations. Exact embeddings (fully faithful exact functors) identify exact subcategories; a full additive subcategory closed under extensions and under kernels of deflations and cokernels of inflations is an extension-closed exact subcategory. Important classes include resolving subcategories and coresolving subcategories that reflect projective and injective behavior, respectively; such notions are central in the study of cotorsion pairs, tilting theory, and the representation theory of algebras like Auslander–Reiten theory and cluster categories. Localizations and quotient exact categories occur by modding out by thick subcategories analogous to Serre subcategories in abelian categories; these operations are used in comparisons between model structures in derived categories and in applications to Noncommutative geometry.

Constructions and operations

Standard constructions include taking the idempotent completion, forming category of admissible subobjects, or passing to categories of complexes with the induced degreewise exact structure. The heart of a bounded t-structure in a derived category of an abelian category yields an exact (indeed abelian) category; conversely, one can produce triangulated categories from exact categories via the bounded homotopy category of complexes modulo acyclic complexes, leading to Verdier quotients and links to Brown representability in homotopy theory. The process of forming the Quillen Q-construction or Waldhausen S•-construction assigns algebraic K-theory spectra to exact categories, connecting to invariants like G-theory and K-theory of schemes exploited in comparisons by Thomason–Trobaugh and others.

Relation to abelian and triangulated categories

Exact categories sit between additive and abelian categories: every abelian category is exact with conflations the short exact sequences, while an exact category need not have all kernels and cokernels, so may fail to be abelian. Exact categories often serve as hearts of t-structures in triangulated categories, providing abelian or exact subcategories whose derived categories recover the ambient triangulated structure, a perspective developed in work involving Beilinson–Bernstein–Deligne and Verdier. The passage from an exact category to its bounded derived category produces a triangulated category; conversely, hearts in derived categories or in triangulated categories give exact or abelian structures that facilitate homological methods used in Algebraic geometry, Representation theory, and Algebraic K-theory.

Category:Exact categories