The concept is due to Daniel Quillen and is designed to encapsulate the properties of short exact sequences in abelian categories without requiring that morphisms actually possess kernels and cokernels, which is necessary for the usual definition of such a sequence.
An exact category E is an additive category possessing a class E of "short exact sequences": triples of objects connected by arrows satisfying the following axioms inspired by the properties of short exact sequences in an abelian category: Admissible monomorphisms are generally denoted
These axioms are not minimal; in fact, the last one has been shown by Bernhard Keller (1990) to be redundant.
Suppose A is abelian and let E be any strictly full additive subcategory which is closed under taking extensions in the sense that given an exact sequence in A, then if
We verify the axioms: Conversely, if E is any exact category, we can take A to be the category of left-exact functors from E into the category of abelian groups, which is itself abelian and in which E is a natural subcategory (via the Yoneda embedding, since Hom is left exact), stable under extensions, and in which a sequence is in E if and only if it is exact in A.