nLab Quillen exact category

Definition

Via embedding

A full additive subcategory $A$ of an abelian category $B$ is called Quillen exact category if it is closed under extensions (if in extension $0\to X\stackrel{j}\to Y\stackrel{p}\to Z\to 0$, $X$ and $Z$ are in $A$ then $Y$ is in $A$). It is viewed as a pair $(A,E)$ where $E$ is the class of all short exact sequences in $A$ which are exact in $B$.

All $j$ which appear as $j$ in an exact sequence as above are called inflations or admissible monomorphisms. All $p$ which appear in an exact sequence as above are called deflations or admissible epimorphisms.

Via exact structure

A Quillen exact category is a pair $(A,E)$ of an additive category $A$ and a class of sequences $E$ called ‘exact’. The following axioms are required for $(A,E)$:

(QE1) The class of ‘exact’ sequences is closed under isomorphisms and it contains all split extensions. For any ‘exact’ sequence the deflation is the cokernel of inflation and the inflation is the kernel of the deflation.

(QE2) The class of deflations is closed under composition and base change by arbitrary maps. The class of inflations is closed under compositions and cobase change by arbitrary maps.

(QE3) If a morphism $M\to M'$ having a kernel can factor a deflation $N\to M'$ as $N\to M\to M'$ then it is a deflation. If a morphism $I\to I'$ having a cokernel can factor an inflation $I\to J$ as $I\to I'\to J$ then it is also an inflation.

Properties

Quillen-Gabriel embedding theorem

For every small exact category in the sense of a pair $(A,E)$, there is an embedding $A\hookrightarrow B$ into an abelian category such that $E$ is a class of all sequences which are (short) exact in $B$.

Relation to Waldhausen categories and algebraic K-theory

Every Quillen exact category can be made into a Waldhausen category. However some information is lost in the process. Moreover, not every Waldhausen category comes from a Quillen exact category. Both Quillen exact categories and Waldhausen categories are devised in order to do algebraic K-theory. The K-theory spectrum based on Quillen's Q-construction and an exact category agrees with the K-theory spectrum based on the Waldhausen S-construction of the K-theory spectrum from its associated Waldhausen category.

References

Quillen introduced exact categories in above sense in the article

• Daniel Quillen, “Higher algebraic K-theory”, in Higher K-theories, pp. 85–147, Proc. Seattle 1972, Lec. Notes Math. 341, Springer 1973.

A nonadditive generalization of exact categories has been introduced by Dyckerhoff and Kapranov and named a proto-exact category.

Alexander Rosenbergintroduced one sided generalizations of Quillen exact categories: right ‘exact’ categories involving deflations, and left ‘exact’ categories involving inflations. One of the motivations an alternative definition of higher K-theory of (right exact) categories not involving spectra. In this setup the K-theory is an example of a derived functor in nonabelian homological algebra utilizing roughly the left ‘exact’ structure on the category of essentially small right ‘exact’ categories. It is not known if this K-theory when restricted to the category of essentially small Quillen exact categories agrees with Quillen K-theory. But it has the standard properties of Quillen K-theory (devissage, exactness and so on).

The one-sided generalization inspired by ideas introduced by Keller and Vossieck in the build up of the theory of suspended categories.

A right ‘exact’ category is a category with an initial object and a Grothendieck pretopology consisting of single maps which are strict epimorphisms. The distinguished class of strict epimorphisms is called a right ‘exact’ structure, or the class of deflations. The construction of derived functors in this generality involves a version of satellites.

• Dmitry Kaledin, Wendy Lowen, Cohomology of exact categories and (non-)additive sheaves, Adv. Math. 272 (2015) 652–698 arXiv:1102.5756