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  "body": "\n## Definition\n\nA [[concrete category|FYBRP16KAN]] is a pair $(\\mathcal{C}, U)$ of a category $\\mathcal{C}$ and a faithful functor $U \\colon \\mathcal{C} \\to \\mathbf{Set}$.\nFaithful means: for all morphisms $f, g \\colon A \\to B$, $U(f) = U(g)$ implies $f = g$.\nLet $\\sigma$ be an algebraic signature and $K$ a class of $\\sigma$-algebras closed under isomorphism.\nLet $\\mathcal{K}$ be the category with the members of $K$ as objects and the [[homomorphisms|KBXSZC6BV1]] between them as morphisms.\nThe *concrete category of algebras* of $K$ is the pair $(\\mathcal{K}, U)$, where $U$ is the [[forgetful functor|WV3BPYCHVG]].\n\n## Remarks\n\nThe functor $U$ is faithful because a homomorphism is determined by its underlying map.\n\nExample: the category $\\mathbf{Grp}$ of groups with its forgetful functor to $\\mathbf{Set}$ is a concrete category of algebras.\n\nNon-example: the category $\\mathbf{Grp}$ with the functor that sends $G$ to the underlying set of its abelianization $G / [G, G]$ is not a concrete category, since this functor is not faithful.\nThe two homomorphisms $\\Z \\to S_3$ that send $1$ to the transpositions $(1\\,2)$ and $(1\\,3)$ are different, but both induce the map $\\Z \\to \\Z / 2\\Z$ that sends $1$ to the nontrivial element.\n\nNot every category admits a faithful functor to $\\mathbf{Set}$: by a theorem of Freyd, the homotopy category of topological spaces does not.\n",
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