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.
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... Any abelian category is exact in the obvious way, according to the construction of #Motivation ... A less trivial example is the category Abtf of torsion-free abelian groups, which is a strictly full subcategory of the (abelian) category Ab of all abelian groups ... It is closed under extensions if is a short exact sequence of abelian groups in which are torsion-free, then is seen to be torsion-free by the following argument if is a torsion element, then its image in is zero ...
... A common generalization of these two concepts is given by the Grothendieck group of an exact category ... Simplified an exact category is an additive category together with a class of distinguished short sequences A → B → C ... The distinguished sequences are called "exact sequences", hence the name ...
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