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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a category with finite [[coproducts|4FJ0S71P58]].\nFor objects $A$ and $B$ consider the functor\n$$+ \\colon \\mathcal{C}/A \\times \\mathcal{C}/B \\to \\mathcal{C}/(A + B)$$\nthat sends a pair $(f \\colon X \\to A, \\, g \\colon Y \\to B)$ to $f + g \\colon X + Y \\to A + B$.\nHere $\\mathcal{C}/A$ is the [[slice category|1TC7W7BQT3]] over $A$ and the domain is a [[product category|AFNE9C4NH6]].\nThe category $\\mathcal{C}$ is *extensive*, or finitarily extensive, if this functor is an [[equivalence of categories|K7A04ZBD92]] for all objects $A$ and $B$.\n\n## Remarks\n\nAn equivalent form: $\\mathcal{C}$ has [[pullbacks|Q05K04QJ71]] along the coproduct injections, and the following holds.\nFor every commutative diagram whose bottom row is $A \\to A + B \\leftarrow B$ and whose top row is $X_1 \\to X \\leftarrow X_2$, the top row is a coproduct if and only if both squares are pullbacks.\nIn an extensive category coproducts are disjoint, and the initial object is strict.\n\nExample: the category of sets is extensive.\nA map $h \\colon X \\to A + B$ is, up to isomorphism, the sum of its restrictions $h^{-1}(A) \\to A$ and $h^{-1}(B) \\to B$, and this decomposition is unique.\n\nNon-example: the category of real vector spaces, where the coproduct is the direct sum.\nLet $\\Delta \\colon \\R \\to \\R \\oplus \\R$ be the diagonal $x \\mapsto (x, x)$.\nThe pullbacks of $\\Delta$ along both injections are the zero space, since $(x, x) = (a, 0)$ forces $x = 0$.\nBut $0 \\oplus 0 = 0$ is not $\\R$, so in the equivalent form the squares are pullbacks while the top row is not a coproduct.\n",
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