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  "body": "\n## Definition\n\nLet $\\sigma$ be an algebraic signature and let $K$ be a class of $\\sigma$-algebras.\nLet $\\mathcal{K}$ be the [[category|A2DE66937D]] whose objects are the members of $K$ and whose morphisms are the [[homomorphisms|KBXSZC6BV1]] between them.\nThe *forgetful functor* $U \\colon \\mathcal{K} \\to \\mathbf{Set}$ is the [[functor|28NQ0KRHD3]] that sends each algebra $A$ to its underlying set.\nIt sends each homomorphism $f \\colon A \\to B$ to the same map $f$ between the underlying sets.\n\nMore generally, let $\\sigma' \\subseteq \\sigma$.\nThe functor that sends a $\\sigma$-algebra to its reduct to the operations of $\\sigma'$, and each homomorphism to itself, is also called forgetful.\n\n## Remarks\n\n$U$ is a functor because the identity homomorphism is the identity map and the composite of homomorphisms is their composite as maps.\nIt is faithful: two homomorphisms $A \\to B$ with the same underlying map are equal.\n\nExample: the forgetful functor $\\mathbf{Ring} \\to \\mathbf{Ab}$ keeps the addition of a ring and drops its multiplication and unit.\nA ring homomorphism is additive, so it is sent to a group homomorphism.\n\nNon-example: consider the functor $\\mathbf{Grp} \\to \\mathbf{Set}$ that sends a group $G$ to the set of its subgroups, and a homomorphism $f$ to $H \\mapsto f(H)$.\nIt is a functor, but not a forgetful functor.\nIt does not send $G$ to its underlying set.\n\nThe term is also used informally for functors such as $\\mathbf{Top} \\to \\mathbf{Set}$, which forget a topology rather than operations.\n",
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