Duality for noncommutative frames (Q2215637)

In this interesting investigation, the authors give a noncommutative facet of the well-known equivalence between spatial frames and sober spaces. This new equivalence is given by the adjoint situation for the categories of left-handed noncommutative frames and sheaves for the front topology (this theorem can be seen as a reminisces of the well know fact: every sheaf over a spatial frame is isomorphic to the sheaf of global section of a suitable étale space over the space associated to the spatial frame in play). Among other interesting subjects as noncommutative Priestly duality and sheaves over the dissolution of a locale ( here the dissolution of a locale is the assembly of the locale), with these elements the authors describes an equivalence between \(\mathrm{Shv}(PS)\) the category of pairs \((S,\mathcal{F})\) where \(S\) is a priestley space and \(\mathcal{F}\) is a sheaf on \(S\) with global support and the category \(SDL\) of left-hamded strongly distributive skew lattices with \(0\). As a consequence of this duality the authors obtain the following: These exists a duality between left-handed noncommutative frames and the category of \(\mathrm{Shv}(\mathrm{Loc}_{d})\) the category of sheaves on dissolution locales.