In mathematics, specifically in category theory, an exact category is a category equipped with short exact sequences. 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.

Definition

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 and admissible epimorphisms are denoted These axioms are not minimal; in fact, the last one has been shown by to be redundant. One can speak of an exact functor between exact categories exactly as in the case of exact functors of abelian categories: an exact functor F from an exact category D to another one E is an additive functor such that if is exact in D, then 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.

Motivation

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 in A, then if M', M are in E, so is M. We can take the class E'' to be simply the sequences in E which are exact in A; that is, is in E iff 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.

Examples