The Stacks project

Lemma 84.2.12. Let $X$ be a topological space. Let $X_\bullet $ be the constant simplicial topological space with value $X$. The functor

\[ X_{\bullet , Zar} \longrightarrow X_{Zar},\quad U \longmapsto U \]

is continuous and cocontinuous and defines a morphism of topoi $g : \mathop{\mathit{Sh}}\nolimits (X_{\bullet , Zar}) \to \mathop{\mathit{Sh}}\nolimits (X)$ as well as a left adjoint $g_!$ to $g^{-1}$. We have

  1. $g^{-1}$ associates to a sheaf on $X$ the constant cosimplicial sheaf on $X$,

  2. $g_!$ associates to a sheaf $\mathcal{F}$ on $X_{\bullet , Zar}$ the sheaf $\mathcal{F}_0$, and

  3. $g_*$ associates to a sheaf $\mathcal{F}$ on $X_{\bullet , Zar}$ the equalizer of the two maps $\mathcal{F}_0 \to \mathcal{F}_1$.

Proof. The statements about the functor are straightforward to verify. The existence of $g$ and $g_!$ follow from Sites, Lemmas 7.21.1 and 7.21.5. The description of $g^{-1}$ is immediate from Sites, Lemma 7.21.5. The description of $g_*$ and $g_!$ follows as the functors given are right and left adjoint to $g^{-1}$. $\square$

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