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  "body": "\n## Summary\n\nAn exact category, in the sense of Quillen, is an [[additive category|B2VP96NJSK]] with a chosen class of kernel–cokernel pairs.\nThese pairs behave like [[short exact sequences|KJKNJMGKZZ]].\nIt allows homological algebra and algebraic K-theory in categories that are not abelian, such as the category of finitely generated projective modules.\nIt is a different notion from an [[exact category in the sense of Barr|G9ZNDNT3YY]].\n\n## Explanations\n\n### L1 human\n\nIn an abelian category every monomorphism has a cokernel and fits into a short exact sequence.\nMany useful categories, such as vector bundles or projective modules, are additive but not abelian.\nAn exact category keeps only a chosen family of sequences $A \\to B \\to C$, declared to be the short exact ones, and asks that this family behave well under composition, pullback and pushout.\nFor example, in the category of finitely generated projective modules one can choose the split exact sequences.\n",
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