Lemma 85.2.12. Let $X$ be a topological space. Let $X_\bullet $ be the constant simplicial topological space with value $X$. The functor

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

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

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

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

## Comments (0)

There are also: