Exact category

In mathematics, an exact category is a concept of due to Daniel Quillen which is designed to encapsulate the properties of short exact sequences in without requiring that morphisms actually possess , which is necessary for the usual definition of such a sequence. An exact category E is an 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 : E is closed under isomorphisms and contains the canonical ("split exact") sequences: Suppose occurs as the second arrow of a sequence in E (it is an admissible epimorphism) and is any arrow in E. Then their exists and its projection to is also an admissible epimorphism. , if occurs as the first arrow of a sequence in E (it is an admissible monomorphism) and is any arrow, then their exists and its coprojection from is also an admissible monomorphism. (We say that the admissible epimorphisms are "stable under pullback", resp. the admissible monomorphisms are "stable under pushout".); Admissible monomorphisms are s of their corresponding admissible epimorphisms, and dually. The composition of two admissible monomorphisms is admissible (likewise admissible epimorphisms); Suppose is a map in E which admits a kernel in E, and suppose is any map such that the composition is an admissible epimorphism. Then so is Dually, if admits a cokernel and is such that is an admissible monomorphism, then so is 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 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. Exact categories come from abelian categories in the following way.