It's well-known that sheaves over a topological space are equivalent to etale spaces over the same space.
Now if we replace "topological space" by "locale", we can still define sheaves over a locale, and we can define etale locales over the same locale (at least it seems to me : the notion of local homeomorphism of locales seems pretty easy to guess). Are they still the same (i.e. equivalent) ?
I could (easily) prove that given an etale locale over a given locale $X$ we may form a sheaf over $X$ in the obvious way (the "sheaf of sections") but the reverse direction is giving me more imaginative trouble : the construction I know for topological spaces of the etale space of a sheaf is not that easily transportable to locales (the construction I know uses stalks at points of the space, but the point of locales is to avoid points); at least not at first sight.
But I believe (hope ?) that there is such a construction and that if I was more comfortable with sheaves on a space I would probably see it; but it turns out I'm wondering about this before being comfortable with them, so I wanted to know: thus there are two questions :
Is it true that for a locale $X$, $\mathbf{Sh}(X)\simeq \mathrm{Etale}(X)$ (with obvious notations ) ? If it is, how may we construct the etale locale of a given sheaf over $X$ ?