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  "body": "\n## Statement\n\nLet $U \\colon \\mathcal{D} \\to \\mathcal{C}$ be a functor.\nA pair of morphisms $f, g \\colon d \\to d'$ in $\\mathcal{D}$ is *$U$-split* if $U(f), U(g)$ have a split coequalizer in $\\mathcal{C}$.\nA split coequalizer of $p, q \\colon x \\to y$ is a morphism $e \\colon y \\to z$ with $s \\colon z \\to y$ and $t \\colon y \\to x$ such that $e p = e q$, $e s = \\id_z$, $p t = \\id_y$ and $q t = s e$.\nThen $U$ is [[monadic|DNTDCKZBKK]] if and only if these three conditions hold.\n\n1. $U$ has a [[left adjoint|TZHF6ZM92S]].\n2. $U$ reflects isomorphisms: if $U(h)$ is an isomorphism, then $h$ is an isomorphism.\n3. Every $U$-split pair in $\\mathcal{D}$ has a [[coequalizer|DVSTWGVCSZ]] in $\\mathcal{D}$, and $U$ preserves these coequalizers.\n\n## Remarks\n\nA split coequalizer is a coequalizer, and every functor preserves it, because the equations that define it are preserved.\nFor the proof, let $T = U F$ and let $(A, a)$ be an algebra in the [[Eilenberg–Moore category|FEPRJR3MND]].\nThe pair $F(a), \\varepsilon_{F(A)} \\colon F U F (A) \\to F(A)$ is $U$-split, and its coequalizer defines a left adjoint of the comparison functor.\nConditions 2 and 3 then make the unit and counit of this adjunction isomorphisms.\n\nExample: the forgetful functor from groups to sets satisfies all three conditions, so it is monadic.\nThe same holds for rings, modules over a ring, and every variety of universal algebras.\n\nWithout condition 2 the conclusion fails.\nThe forgetful functor from topological spaces to sets has a left adjoint, but a continuous bijection need not be a homeomorphism, and this functor is not monadic.\n\n## Explanations\n\n### L1 human\n\nThe theorem gives a practical test for deciding whether a category is algebraic over another one.\nOne checks that free objects exist, that the forgetful functor detects isomorphisms, and that certain quotients exist and are computed underneath.\nWhen the test passes, the objects are exactly the algebras of a monad.\n\n### L2 human\n\nBeck's theorem is the standard tool for proving monadicity, for example of compact Hausdorff spaces over sets or of modules over a ring.\nThere is also a variant that uses reflexive coequalizers: a right adjoint that reflects isomorphisms and in whose domain reflexive coequalizers exist and are preserved is monadic.\nThe dual statement characterizes comonadic functors, and in algebraic geometry it is the basis of descent theory.\nIn the strict form, $U$ creates coequalizers of $U$-split pairs, and the comparison functor is then an isomorphism.\n",
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