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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a category with [[zero morphisms|YF9F9EG6M3]], and let $X_1, X_2$ be objects of $\\mathcal{C}$.\nA *biproduct* of $X_1$ and $X_2$ is an object $X$ with morphisms $p_k \\colon X \\to X_k$ and $i_k \\colon X_k \\to X$ for $k = 1, 2$ such that:\n\n1. $p_k \\circ i_k = \\id_{X_k}$ for $k = 1, 2$, and $p_l \\circ i_k = 0$ for $k \\ne l$;\n2. $(X, p_1, p_2)$ is a [[product|S5744QZV4N]] of $X_1$ and $X_2$;\n3. $(X, i_1, i_2)$ is a [[coproduct|4FJ0S71P58]] of $X_1$ and $X_2$.\n\nThe biproduct is written $X_1 \\oplus X_2$; biproducts of finite families are defined in the same way.\n\n## Remarks\n\nIn a [[preadditive category|SC6FD5F4J0]], an object $X$ with maps $p_k$, $i_k$ satisfying clause 1 and $i_1 \\circ p_1 + i_2 \\circ p_2 = \\id_X$ is a biproduct, and every biproduct has this form.\nIn particular, in a preadditive category every binary product, and every binary coproduct, is a biproduct.\n\nExample: for [[abelian groups|082GHNPJ53]] $A$ and $B$, the direct sum $A \\oplus B$ with $p_1(a, b) = a$, $p_2(a, b) = b$, $i_1(a) = (a, 0)$, $i_2(b) = (0, b)$ is a biproduct.\nClause 1 is immediate, and $i_1 p_1 + i_2 p_2$ sends $(a, b)$ to $(a, 0) + (0, b) = (a, b)$.\n\nNon-example: in the category of groups, with the trivial group as zero object, the product $\\Z \\times \\Z$ with its inclusions satisfies clauses 1 and 2 but not clause 3.\nIf it were a coproduct, the maps $\\Z \\to F_2$ onto the two generators $a, b$ of the free group $F_2$ would extend to one homomorphism $\\Z \\times \\Z \\to F_2$.\nIts image would be abelian and contain $a$ and $b$, but $ab \\ne ba$ in $F_2$.\nThe coproduct in groups is the [[free product|TM7QS7P7X9]] instead.\n",
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