The Stacks project

Proof. For the underlying diagram of topoi we refer to the discussion in the proof of Lemma 85.33.1. The sheaf $\mathcal{O}_ U$ is the structure sheaf of the simplicial algebraic space $U$ as defined in Section 85.32. The sheaf $\mathcal{O}_ X$ is the usual structure sheaf of the algebraic space $X$. The sheaves of rings $\mathcal{O}_{big, total}$ and $\mathcal{O}_{big}$ come from the structure sheaf on $(\textit{Spaces}/S)_{fppf}$ in the manner explained in Section 85.22 which also constructs $a_{fppf}$ as a morphism of ringed topoi. The component morphisms $h_ n = a_{U_ n}$ and $h_{-1} = a_ X$ are morphisms of ringed topoi by More on Cohomology of Spaces, Section 84.7. Finally, since the continuous functor $u : U_{spaces, {\acute{e}tale}} \to (\textit{Spaces}/U)_{fppf, total}$ used to define $h$¹ is given by $V/U_ n \mapsto V/U_ n$ we see that $h_*\mathcal{O}_{big, total} = \mathcal{O}_ U$ which is how we endow $h$ with the structure of a morphism of ringed simplicial sites as in Remark 85.7.1. Then we obtain $h$ as a morphism of ringed topoi by Lemma 85.7.2. Please observe that the morphisms $h_ n$ indeed agree with the morphisms $a_{U_ n}$ described above. We omit the verification that the diagram is commutative (as a diagram of ringed topoi – we already know it is commutative as a diagram of topoi). $\square$