Lemma 14.28.6. Let \mathcal{A} be an additive category. Let a, b : U \to V be morphisms of cosimplicial objects of \mathcal{A}. If a, b are homotopic, then s(a), s(b) : s(U) \to s(V) are homotopic maps of cochain complexes. If in addition \mathcal{A} is abelian, then Q(a), Q(b) : Q(U) \to Q(V) are homotopic maps of cochain complexes.
The (cosimplicial) Dold-Kan functor carries homotopic maps to homotopic maps.
Proof. Let (-)' : \mathcal{A} \to \mathcal{A}^{opp} be the contravariant functor A \mapsto A. By Lemma 14.28.5 the maps a' and b' are homotopic. By Lemma 14.27.1 we see that s(a') and s(b') are homotopic maps of chain complexes. Since s(a') = (s(a))' and s(b') = (s(b))' we conclude that also s(a) and s(b) are homotopic by applying the additive contravariant functor (-)'' : \mathcal{A}^{opp} \to \mathcal{A}. The result for the Q-complexes follows in the same manner using that Q(U)' = N(U'). \square
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Comment #852 by Bhargav Bhatt on