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  "body": "\n## Definition\n\nA [[category|A2DE66937D]] $\\mathcal{C}$ is *Cartesian closed* if:\n\n1. $\\mathcal{C}$ has a [[terminal object|EK85VSSRMR]].\n2. Any two objects $X$ and $Y$ of $\\mathcal{C}$ have a [[product|S5744QZV4N]] $X \\times Y$.\n3. Any two objects $Y$ and $Z$ of $\\mathcal{C}$ have an [[exponential object|RHWD8DY41R]] $Z^Y$.\n\n## Remarks\n\nClauses 1 and 2 say that $\\mathcal{C}$ has finite products.\nClause 3 then says that each functor $- \\times Y$ has a right adjoint $(-)^Y$.\nSo there are bijections $\\Hom(X \\times Y, Z) \\cong \\Hom(X, Z^Y)$, natural in $X$ and $Z$.\n\nExample: the category of sets is Cartesian closed.\nA one-point set is terminal, $X \\times Y$ is the Cartesian product, and $Z^Y$ is the set of maps $Y \\to Z$ with evaluation.\n\nNon-example: the category of abelian groups is not Cartesian closed, although clauses 1 and 2 hold.\nIn a Cartesian closed category, $- \\times Y$ is a left adjoint, so it preserves the initial object $0$, and $0 \\times Y \\cong 0$.\nIn abelian groups $0 \\times \\Z \\cong \\Z$, which is not $0$, so clause 3 fails.\n",
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