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  "body": "\n## Statement\n\nLet $\\sigma$ be an algebraic signature, and let $\\mathbf{Alg}(\\sigma)$ be the category of all $\\sigma$-algebras and their [[homomorphisms|KBXSZC6BV1]].\nLet $\\mathcal{V}$ be a [[variety|AJNZJD8442]] of $\\sigma$-algebras, viewed as a [[full subcategory|V3BFGKBJF7]].\nFor a $\\sigma$-algebra $A$, let $\\theta_A$ be the intersection of all [[congruences|DCSXXT4SER]] $\\theta$ of $A$ with $A / \\theta \\in \\mathcal{V}$, and let $q_A \\colon A \\to A / \\theta_A$ be the [[quotient map|N5KE7FH6CX]].\nThen $A / \\theta_A \\in \\mathcal{V}$, and for every $B \\in \\mathcal{V}$ and every homomorphism $f \\colon A \\to B$ there is a unique homomorphism $\\bar f \\colon A / \\theta_A \\to B$ with $\\bar f \\circ q_A = f$.\nHence $\\mathcal{V}$ is a [[reflective subcategory|FC7VB4WSG0]] of $\\mathbf{Alg}(\\sigma)$, with reflection $q_A$, and its reflector $A \\mapsto A / \\theta_A$ is left adjoint to the inclusion.\n\n## Remarks\n\nThe algebra $A / \\theta_A$ embeds into the product of the algebras $A / \\theta$ over all such $\\theta$, so it lies in $\\mathcal{V}$ by closure under subalgebras and products.\nFor $f \\colon A \\to B$ with $B \\in \\mathcal{V}$, the image $f(A)$ is a subalgebra of $B$, so $A / \\ker f \\cong f(A) \\in \\mathcal{V}$.\nThus $\\ker f \\supseteq \\theta_A$, and $f$ factors through $q_A$; the factorization is unique because $q_A$ is surjective.\n\nOnly closure under isomorphic copies, subalgebras and products is used, so the same holds for every quasivariety and every [[prevariety|EEAJ7AXBCT]].\nExample: for the variety of abelian groups inside groups, $\\theta_G$ is the congruence of the commutator subgroup $[G, G]$, and the reflection is the abelianization $G \\to G / [G, G]$.\nApplied to the absolutely free term algebra on a set $X$, the reflection gives the [[free algebra|GTS4DRRMZR]] of $\\mathcal{V}$ on $X$.\n\n## Explanations\n\n### L1 human\n\nAny algebra can be pushed into a given variety in a best possible way: impose just the equations of the variety, and nothing more.\nFor groups and the variety of abelian groups, this means forcing all elements to commute, which gives the abelianization.\nEvery homomorphism into an algebra of the variety factors through this best approximation in exactly one way.\n\n### L2 human\n\nThe reflection $A \\mapsto A / \\theta_A$ is a universal construction, and $\\theta_A$ is the congruence generated by the pairs $(s(\\bar a), t(\\bar a))$ for identities $s \\approx t$ of $\\mathcal{V}$ and tuples $\\bar a$ of $A$.\nSince the reflector is a left adjoint, it preserves colimits: the free product in $\\mathcal{V}$ of two members is the reflection of their free product in $\\mathbf{Alg}(\\sigma)$.\nComposing with the free functor of $\\mathbf{Alg}(\\sigma)$ recovers the [[free functor|N353JF8RBK]] of $\\mathcal{V}$.\nReflective subcategories are closed under limits, which matches the closure of varieties under products and under equalizers, which are subalgebras.\n",
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