Theorem 19.6.3 (Barr–Beck monadicity, [Lurie (2017), Theorem 4.7.3.5]). Let \(G\colon D \to C\) be a conservative functor admitting a left adjoint. Assume that \(D\) admits geometric realizations of \(G\)-split simplicial objects and that \(G\) preserves them. Then \(G\) is monadic.
Proof. Let \(T\) be the monad associated to the adjunction and let \(K_G\colon D \to \LMod _T(C)\) be the comparison functor. Apply Theorem 21.7.3 to \(G\), the forgetful functor \(U_T\colon \LMod _T(C) \to C\), and \(K_G\). The identity \(U_TK_G \simeq G\) has mate \(F_T \to K_GF\), which is an isomorphism by the cited package. The same package supplies conservativity of \(U_T\) and the required geometric realizations on the Eilenberg–Moore side. All the hypotheses of the comparison theorem are therefore satisfied, so \(K_G\) is an equivalence. □
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