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  "body": "\n## Definition\n\nAn *additive category* is a [[preadditive category|SC6FD5F4J0]] $\\mathcal{C}$ such that:\n\n1. $\\mathcal{C}$ has a [[zero object|AQET9YHH6F]];\n2. every two objects $X_1, X_2$ of $\\mathcal{C}$ have a [[biproduct|BZEJKTQ76T]] $X_1 \\oplus X_2$.\n\n## Remarks\n\nThen every finite family of objects has a biproduct, and a morphism $\\bigoplus_{j} X_j \\to \\bigoplus_{i} Y_i$ is the same as a matrix $(f_{ij})$ with $f_{ij} \\colon X_j \\to Y_i$, composed by matrix multiplication.\nBeing additive is a property of the underlying category: the addition of morphisms is determined by the biproducts, through $f + g = \\nabla \\circ (f \\oplus g) \\circ \\Delta$, where $\\Delta$ is the diagonal and $\\nabla$ the codiagonal.\n\nExample: the category of left [[modules|0NZ8JA25WK]] over a ring $R$ is additive.\nThe zero module is a zero object, and the direct sum $M_1 \\oplus M_2$ is a biproduct.\n\nNon-example: the ring $\\Z$, as a preadditive category with one object $\\ast$, is not additive.\nIt has no zero object, since a zero object $Z$ has $\\Hom(Z, Z) = \\set{0}$, while $\\Hom(\\ast, \\ast) = \\Z$.\n",
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