The Stacks project

Lemma 85.2.5. Let $X$ be a simplicial space. The functor $X_{n, Zar} \to X_{Zar}$, $U \mapsto U$ is continuous and cocontinuous. The associated morphism of topoi $g_ n : \mathop{\mathit{Sh}}\nolimits (X_ n) \to \mathop{\mathit{Sh}}\nolimits (X_{Zar})$ satisfies

  1. $g_ n^{-1}$ associates to the sheaf $\mathcal{F}$ on $X$ the sheaf $\mathcal{F}_ n$ on $X_ n$,

  2. $g_ n^{-1} : \mathop{\mathit{Sh}}\nolimits (X_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X_ n)$ has a left adjoint $g^{Sh}_{n!}$,

  3. $g^{Sh}_{n!}$ commutes with finite connected limits,

  4. $g_ n^{-1} : \textit{Ab}(X_{Zar}) \to \textit{Ab}(X_ n)$ has a left adjoint $g_{n!}$, and

  5. $g_{n!}$ is exact.

Proof. Besides the properties of our functor mentioned in the statement, the category $X_{n, Zar}$ has fibre products and equalizers and the functor commutes with them (beware that $X_{Zar}$ does not have all fibre products). Hence the lemma follows from the discussion in Sites, Sections 7.20 and 7.21 and Modules on Sites, Section 18.16. More precisely, Sites, Lemmas 7.21.1, 7.21.5, and 7.21.6 and Modules on Sites, Lemmas 18.16.2 and 18.16.3. $\square$

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