Exact category

In mathematics, an exact category is a concept of category theory due to Daniel Quillen which 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:

is exact in E. If D is a subcategory of E, it is an exact subcategory if the inclusion functor is fully faithful and exact.

Exact categories come from abelian categories in the following way. 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

is exact in A. Then E is an exact category in the above sense. 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.