The Stacks project

Lemma 14.25.1. Let $\mathcal{A}$ be an abelian category.

1. The functor $s : \text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$ is exact.

2. The maps $s(U)^ n \to Q(U)^ n$ define a morphism of cochain complexes.

3. There exists a functorial direct sum decomposition $s(U) = D(U) \oplus Q(U)$ in $\text{CoCh}_{\geq 0}(\mathcal{A})$.

4. The functor $Q$ is exact.

5. The morphism of complexes $s(U) \to Q(U)$ is a quasi-isomorphism.

6. The functor $U \mapsto Q(U)^\bullet $ defines an equivalence of categories $\text{CoSimp}(\mathcal{A}) \to \text{CoCh}_{\geq 0}(\mathcal{A})$.