An exact category, in Quillen's sense, is an additive category 
equipped with a chosen class of composable pairs of morphisms 
in which 
is a kernel of 
and 
is a cokernel of 
. This class, called the exact structure, is closed under isomorphism of such pairs and satisfies the axioms below
(Bühler 2010, §2). The first and second morphisms
in a chosen pair are called admissible monomorphisms
and admissible epimorphisms, respectively.
1. Every identity morphism is both an admissible monomorphism and an admissible epimorphism.
2. Composites of admissible monomorphisms are admissible monomorphisms, and composites of admissible epimorphisms are admissible epimorphisms.
3. The pushout of an admissible monomorphism along any morphism exists and is an admissible monomorphism. Dually, the pullback of an admissible epimorphism along any morphism exists and is an admissible epimorphism.
A chosen pair is written as a short exact sequence 
. Every Abelian
category has an exact structure consisting of all its short
exact sequences. Every additive category
has the split exact structure consisting of pairs isomorphic
to the canonical inclusion and projection 
, where 
is a biproduct.
Different exact structures on the same category can
give different Grothendieck groups 
(Weibel 2013, pp. 129-130).
