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:

Admissible monomorphisms are generally denoted ${\displaystyle \rightarrowtail }$ and admissible epimorphisms are denoted ${\displaystyle \twoheadrightarrow .}$ These axioms are not minimal; in fact, the last one has been shown by Bernhard Keller (1990) to be redundant.

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