Let $\mathcal{N} = (N, <, \ldots)$ be an o-minimal structure and let $X \subset N^m$ be a definable set. Following the procedure stablished by Edmundo, Jones and Peatfield in "Sheaf cohomology in o-minimal structures." (Journal of Mathematical Logic 6.02 (2006): 163-179), we can define the spectrum of $X$, $\widetilde{X}$ as the set of ultrafilters in the boolean algebra of the definable subsets of $X$, or, equivalently, the set of all complete $m$-types which contains a formula defining $X$, endowed with the topology generated by the basis $\{\widetilde{O} | O \subset X $ is definable and open in $X\}$. Given a definable map $f\colon X \to Y$ we can define $\tilde{f} \colon \widetilde{X} \to \widetilde{Y}$ by $\tilde{f}(\alpha) = \{ B | f^{-1}(B) \in \alpha\}$, the o-minimal spectrum of $f$.
This tilde operator then defines a functor $\mathrm{DTOP} \to \widetilde{\mathrm{DTOP}}$, where $\mathrm{DTOP}$ the category of definable sets and continuous definable maps and $\widetilde{\mathrm{DTOP}}$ the category of o-minimal spectra of definable sets and o-minimal spectra of continous definable maps between definable sets. If for a definable set $X$ we define the o-minimal site consisting of definable open subsets of $X$, and admissable coverings those with a finite subcovering and denote by $\mathrm{Sh}_\mathrm{dtop}(X)$ the category of sheaves of abelian groups on $X$ with respect to the o-minimal site, the authors argue that $\mathrm{Sh}_\mathrm{dtop}(X)$ is isomorphic to $\mathrm{Sh}(\widetilde{X})$ (the category of sheaves of abelian groups on $\widetilde{X}$), but I don't quite understand how do we define it. Quoting the article:
"Thus, for a definable set $X$, we define the functors of the categories of sheaves of abelian groups $$\mathrm{Sh}_\mathrm{dtop}(X) \to \mathrm{Sh}(\widetilde{X})$$ which sends $\mathcal{F} \in \mathrm{Sh}_\mathrm{dtop}(X)$ into $\widetilde{\mathcal{F}}$ where, for $U$ an open definable subsets of $X$, we define $\widetilde{\mathcal{F}}(\widetilde{U}) = \{ \tilde{s}: s \in \mathcal{F}(U)\} \approx\mathcal{F}(U)$"
Since the topology on $\widetilde{X}$ is generated by $\tilde{U}$, it is clearly sufficient to define a sheaf on $\mathrm{Sh}(\widetilde{X})$ by its value on this basis elements. But I don't understand what does $\tilde{s}$ mean in this context.
