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  "body": "\n## Summary\n\nA category enriched over a [[monoidal category|M3MSQ3NKAE]] $\\mathcal{V}$ has, instead of a set of morphisms between two objects, an object of $\\mathcal{V}$.\nComposition and identities are morphisms of $\\mathcal{V}$ that satisfy associativity and unit laws.\nOrdinary locally small [[categories|A2DE66937D]] are the case $\\mathcal{V} = \\mathbf{Set}$; [[preadditive categories|SC6FD5F4J0]] are the case of abelian groups; strict 2-categories are the case of small categories.\nLawvere observed that [[metric spaces|0VKD1TP7F3]] are categories enriched over the non-negative reals.\n\n## Explanations\n\n### L1 human\n\nIn many categories the morphisms between two objects carry extra structure.\nLinear maps between vector spaces can be added, continuous maps between spaces form a space, and so on.\nAn enriched category builds this structure into the definition: the \"hom-set\" is replaced by a hom-object of some fixed kind.\nA surprising example: a metric space is a category whose objects are points and whose \"hom-object\" from $x$ to $y$ is the number $d(x, y)$.\nComposition becomes the triangle inequality.\n",
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