(AB1) pre-abelian category
(AB2) abelian category
(AB5) Grothendieck category
A full additive subcategory of an abelian category is called Quillen exact category if it is closed under extensions (if in extension , and are in then is in ). It is viewed as a pair where is a class of all short exact sequences in which are exact in .
All which appear as in an exact sequence as above are called inflations or admissible monomorphisms. All which appear in an exact sequence as above are called deflations or admissible epimorphisms.
A Quillen exact category is a pair of an additive category and a class of sequences called ‘exact’. The following axioms are required for :
(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.
(QE3) If a morphism having a kernel can factor a deflation as then it is a deflation. If a morphism having a cokernel can factor an inflation as then it is also an inflation.
For every small exact category in the sense of a pair , there is an embedding into an abelian category such that is a class of all sequences which are (short) exact in .
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.
Quillen introduced exact categories in above sense in the article
Alexander Rosenberg introduced 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 distinguisheed 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.