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  "body": "\n## Definition\n\nLet $(\\mathcal{V}, \\otimes, I, \\alpha, \\lambda, \\rho)$ be a [[monoidal category|M3MSQ3NKAE]].\nA *$\\mathcal{V}$-enriched category* (or $\\mathcal{V}$-category) $\\mathcal{C}$ consists of:\n\n1. a class $\\mathrm{Ob}(\\mathcal{C})$ of objects;\n2. for each pair of objects $a, b$, an object $\\mathcal{C}(a, b)$ of $\\mathcal{V}$, the hom-object;\n3. for each triple $a, b, c$, a morphism $M_{a,b,c} \\colon \\mathcal{C}(b, c) \\otimes \\mathcal{C}(a, b) \\to \\mathcal{C}(a, c)$ of $\\mathcal{V}$, the composition;\n4. for each object $a$, a morphism $j_a \\colon I \\to \\mathcal{C}(a, a)$ of $\\mathcal{V}$, the identity.\n\nThese data satisfy, for all objects $a, b, c, d$:\n\n1. associativity: $M_{a,b,d} \\circ (M_{b,c,d} \\otimes \\id) = M_{a,c,d} \\circ (\\id \\otimes M_{a,b,c}) \\circ \\alpha$ as morphisms $(\\mathcal{C}(c, d) \\otimes \\mathcal{C}(b, c)) \\otimes \\mathcal{C}(a, b) \\to \\mathcal{C}(a, d)$;\n2. units: $M_{a,b,b} \\circ (j_b \\otimes \\id) = \\lambda$ and $M_{a,a,b} \\circ (\\id \\otimes j_a) = \\rho$ as morphisms $I \\otimes \\mathcal{C}(a, b) \\to \\mathcal{C}(a, b)$ and $\\mathcal{C}(a, b) \\otimes I \\to \\mathcal{C}(a, b)$.\n\n## Remarks\n\nExample: for $\\mathcal{V} = (\\mathbf{Set}, \\times, \\set{*})$, a $\\mathcal{V}$-category is a locally small [[category|A2DE66937D]]; for $\\mathcal{V} = (\\mathbf{Ab}, \\otimes, \\Z)$, it is a [[preadditive category|SC6FD5F4J0]].\nExample: let $\\mathcal{V} = [0, \\infty]$, with a morphism $x \\to y$ exactly when $x \\ge y$, tensor product $+$ and unit $0$.\nA $\\mathcal{V}$-category is a set $X$ with numbers $d(x, y)$ such that $d(y, z) + d(x, y) \\ge d(x, z)$ (composition) and $0 \\ge d(x, x)$ (identities), so every [[metric space|0VKD1TP7F3]] is one.\nNon-example: on $\\R$, the function $d(x, y) = (x - y)^2$ does not give a $[0, \\infty]$-category: composition fails, since $d(0, 1) + d(1, 2) = 2$ is less than $d(0, 2) = 4$.\nThe associativity and unit axioms are automatic in this example, because $[0, \\infty]$ is a preorder.\n",
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