Lemma 84.36.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $U$ be a simplicial algebraic space over $S$. Let $a : U \to X$ be an augmentation. If $a : U \to X$ is a proper hypercovering of $X$, then

\[ a^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \quad \text{and}\quad a^{-1} : \textit{Ab}(X_{\acute{e}tale}) \to \textit{Ab}(U_{\acute{e}tale}) \]

are fully faithful with essential image the cartesian sheaves and quasi-inverse given by $a_*$. Here $a : \mathop{\mathit{Sh}}\nolimits (U_{\acute{e}tale}) \to \mathop{\mathit{Sh}}\nolimits (X_{\acute{e}tale})$ is as in Section 84.32.

**Proof.**
We will prove the statement for sheaves of sets. It will be an almost formal consequence of results already established. Consider the diagram of Lemma 84.36.1. In the proof of this lemma we have seen that $h_{-1}$ is the morphism $a_ X$ of More on Cohomology of Spaces, Section 83.8. Thus it follows from More on Cohomology of Spaces, Lemma 83.8.1 that $(h_{-1})^{-1}$ is fully faithful with quasi-inverse $h_{-1, *}$. The same holds true for the components $h_ n$ of $h$. By the description of the functors $h^{-1}$ and $h_*$ of Lemma 84.5.2 we conclude that $h^{-1}$ is fully faithful with quasi-inverse $h_*$. Observe that $U$ is a hypercovering of $X$ in $(\textit{Spaces}/S)_{ph}$ as defined in Section 84.21 since a surjective proper morphism gives a ph covering by Topologies on Spaces, Lemma 72.8.3. By Lemma 84.21.1 we see that $a_{ph}^{-1}$ is fully faithful with quasi-inverse $a_{ph, *}$ and with essential image the cartesian sheaves on $(\textit{Spaces}/U)_{ph, total}$. A formal argument (chasing around the diagram) now shows that $a^{-1}$ is fully faithful.

Finally, suppose that $\mathcal{G}$ is a cartesian sheaf on $U_{\acute{e}tale}$. Then $h^{-1}\mathcal{G}$ is a cartesian sheaf on $(\textit{Spaces}/U)_{ph, total}$. Hence $h^{-1}\mathcal{G} = a_{ph}^{-1}\mathcal{H}$ for some sheaf $\mathcal{H}$ on $(\textit{Spaces}/X)_{ph}$. We compute using somewhat pedantic notation

\begin{align*} (h_{-1})^{-1}(a_*\mathcal{G}) & = (h_{-1})^{-1} \text{Eq}( \xymatrix{ a_{0, small, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, small, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ (h_{-1})^{-1}a_{0, small, *}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & (h_{-1})^{-1}a_{1, small, *}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{0, big, ph, *}h_0^{-1}\mathcal{G}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, big, ph, *}h_1^{-1}\mathcal{G}_1 } ) \\ & = \text{Eq}( \xymatrix{ a_{0, big, ph, *}(a_{0, big, ph})^{-1}\mathcal{H} \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, big, ph, *}(a_{1, big, ph})^{-1}\mathcal{H} } ) \\ & = a_{ph, *}a_{ph}^{-1}\mathcal{H} \\ & = \mathcal{H} \end{align*}

Here the first equality follows from Lemma 84.4.2, the second equality follows as $(h_{-1})^{-1}$ is an exact functor, the third equality follows from More on Cohomology of Spaces, Lemma 83.8.5 (here we use that $a_0 : U_0 \to X$ and $a_1: U_1 \to X$ are proper), the fourth follows from $a_{ph}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$, the fifth from Lemma 84.4.2, and the sixth we've seen above. Since $a_{ph}^{-1}\mathcal{H} = h^{-1}\mathcal{G}$ we deduce that $h^{-1}\mathcal{G} \cong h^{-1}a^{-1}a_*\mathcal{G}$ which ends the proof by fully faithfulness of $h^{-1}$.
$\square$

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