Spahn sheaf on a sheaf (Rev #2, changes)

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Motivation

Let $X\in \mathrm{<span><del\; class="diffmod">\; S</del><ins\; class="diffmod">\; H</ins></span>}$ X\in S H be a space (an object of the a category of spaces), let$\mathrm{<span><del\; class="diffmod">\; Sh</del><ins\; class="diffmod">\; H</ins></span>}(X)$ Sh(X) H be the category of sheaves spaces), on let the frame of opens on$\mathrm{Sh}(X)$ X Sh(X) , let be the category of sheaves on the frame of opens on$(S/X{)}^{\mathrm{et}}$ (S/X)^{et} X , denote let the wide subcategory of$S(H/X{)}^{\mathrm{et}}$ S/X (H/X)^{et} denote the wide subcategory of $H/X$ with only étale morphisms. Then there is an adjoint equivalence

where

• $\Gamma$ sends an étale morphism $f:U\to X$ to the sheaf of local sections of $f$.

• $L$ sends a sheaf on $X$ to its espace étale.