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Abelian category An additive category C, which satisfies the following conditions, for any morphism f ∈ HomC(X, Y ):

(i.) f has a kernel (a morphism i ∈ HomC (X′, X) such that f i = 0) and a co-kernel (a morphism p ∈ HomC(Y, Y ′) such that pf = 0);

(ii.) f may be factored as the composition of an epic (onto morphism) followed by a monic (one-to-one morphism) and this factorization is unique up to equivalent choices for these morphisms;

(iii.) if f is a monic, then it is a kernel; if f is an epic, then it is a co-kernel.