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. 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: Admissible monomorphisms are generally denoted ↣ {displaystyle ightarrowtail } and admissible epimorphisms are denoted ↠ . {displaystyle woheadrightarrow .} These axioms are not minimal; in fact, the last one has been shown by Bernhard Keller (1990) 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 {displaystyle 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. 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 ″ {displaystyle M',M''} are in E, so is M {displaystyle M} . We can take the class E to be simply the sequences in E which are exact in A; that is,