The Stacks project

Lemma 115.8.7. In Simplicial Spaces, Situation 85.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Simplicial Spaces, Remark 85.4.1. Suppose given strictly full weak Serre subcategories

Then

1. the collection of abelian sheaves $\mathcal{F}$ on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_ n$ is in $\mathcal{A}_ n$ for all $n$ is a strictly full weak Serre subcategory $\mathcal{A}_{total} \subset \textit{Ab}(\mathcal{C}_{total})$.

If $a_ n^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_ n$ for all $n$, then

1. $a^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_{total}$ and

2. $a^{-1}$ sends $D_\mathcal {A}(\mathcal{D})$ into $D_{\mathcal{A}_{total}}(\mathcal{C}_{total})$.

If $R^ qa_{n, *}$ sends $\mathcal{A}_ n$ into $\mathcal{A}$ for all $n, q$, then

1. $R^ qa_*$ sends $\mathcal{A}_{total}$ into $\mathcal{A}$ for all $q$, and

2. $Ra_*$ sends $D_{\mathcal{A}_{total}}^+(\mathcal{C}_{total})$ into $D_\mathcal {A}^+(\mathcal{D})$.