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Preadditive Category

A preadditive category or \(\mathbf{Ab}-\mathbf{Cat}\) is a category \(\mathcal{A}\) which is enriched over the category of abelian groups, i.e. for any pair of objects \(X,Y\in\mathcal{A}\) their hom set \(\mathrm{Hom}_\mathcal{A}(X,Y)\) carries the structure of an abelian group, such that composition is bilinear; \((f+g)\circ(h+k)=f\circ h+f\circ k+ g\circ h+g\circ k\) whenever \(X,Y,Z\in\mathcal{A},\;h,k:X\to Y,\; f,g:Y\to Z\).