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  "body": "\n## Definition\n\nA *preadditive category* is a [[category|A2DE66937D]] $\\mathcal{C}$ with an [[abelian group|082GHNPJ53]] structure, written additively, on each set $\\Hom(X, Y)$.\nComposition must be bilinear:\n$$h \\circ (f + g) = h \\circ f + h \\circ g, \\qquad (f + g) \\circ k = f \\circ k + g \\circ k$$\nfor all $f, g \\colon X \\to Y$, $h \\colon Y \\to Z$ and $k \\colon W \\to X$.\n\n## Remarks\n\nEquivalently, a preadditive category is a category [[enriched|BPH9SJ30Y0]] over the monoidal category of abelian groups with the tensor product.\nThe zero element of $\\Hom(X, Y)$ is the zero morphism $0_{XY}$, and bilinearity gives $h \\circ 0 = 0$ and $0 \\circ k = 0$.\nSome authors call these categories $\\mathbf{Ab}$-categories.\n\nExample: for a [[ring|0B1261B5ED]] $R$, the category of left [[$R$-modules|0NZ8JA25WK]] with pointwise addition $(f + g)(x) = f(x) + g(x)$ is preadditive.\nThe sum of homomorphisms is a homomorphism, and $h(f(x) + g(x)) = h(f(x)) + h(g(x))$ because $h$ is additive, which gives the first identity; the second holds by definition of the sum.\n\nExample: a ring $R$ is a preadditive category with one object $\\ast$, with $\\Hom(\\ast, \\ast) = R$ and composition the multiplication of $R$.\n\nNon-example: the category of sets is not preadditive, since $\\Hom(\\set{0}, \\emptyset)$ is empty and an abelian group is never empty.\n",
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