Lemma 14.28.7. Let \mathcal{A} be an additive category. Let a : U \to V be a morphism of cosimplicial objects of \mathcal{A}. If a is a homotopy equivalence, then s(a) : s(U) \to s(V) is a homotopy equivalence of chain complexes. If in addition \mathcal{A} is abelian, then also Q(a) : Q(U) \to Q(V) is a homotopy equivalence of chain complexes.
Proof. Omitted. See Lemma 14.28.6 above. \square
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