The display locale construction defines a fully faithful functor from cosheaves of sets on a locale to the slice category of locales over .
The construction in this article describes a functor in the opposite direction that yields the inverse of the above functor once restricted to the appropriate subcategory.
Thus, the functor described here plays the same role in the equivalence between cosheaves of sets on and complete spreads over as the sheaf of sections construction plays in the equivalence between sheaves of sets on and etale locales? over .
Suppose is a map of locales, where is a locally connected locale. Define a precosheaf on as follows. Send an open in to the set of connected components of , which is a locally connected locale because so is . Send an inclusion of opens to the induced map on the sets of connected components.
(Funk, Proposition 2.1.) This precosheaf is a cosheaf.
Created on April 18, 2020 at 06:00:43. See the history of this page for a list of all contributions to it.