The Stacks project

First proof. We will apply Lemma 92.4.1. Since the terms $P_ n$ are polynomial algebras we see the first assumption of that lemma is satisfied. The second assumption is proved as follows. By Simplicial, Lemma 14.34.3 the map $\epsilon$ is a homotopy equivalence of underlying simplicial sets. By Simplicial, Lemma 14.31.9 this implies $\epsilon$ induces a quasi-isomorphism of associated complexes of abelian groups. By Simplicial, Lemma 14.31.8 this implies that $\epsilon$ is a trivial Kan fibration of underlying simplicial sets. $\square$

Second proof. We will use the criterion of Cohomology on Sites, Lemma 21.39.7. Let $U = (Q, \beta )$ be an object of $\mathcal{C}$. We have to show that

is homotopy equivalent to a singleton. Write $Q = A[E]$ for some set $E$ (this is possible by our choice of the category $\mathcal{C}$). Using the notation of Remark 92.3.3 we see that

By Simplicial, Lemma 14.34.3 the map $i(P_\bullet \to B) \to i(B \to B)$ is a homotopy equivalence in $\mathcal{S}$. Hence $S_\bullet$ is homotopy equivalent to

as desired. $\square$