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The many groupoids of a stably Gelfand quantale (1706.06545v2)
Published 20 Jun 2017 in math.RA and math.CT
Abstract: We study the projections of an arbitrary stably Gelfand quantale $Q$ and show that each projection determines a pseudogroup $S\subset Q$ (and a corresponding localic \'etale groupoid $G$) together with a map of involutive quantales $p:Q\to\mathcal L{\bigvee}(S)\ [=\mathcal O(G)]$. As an application we obtain a simplified axiomatization of inverse quantal frames (= quantales of \'etale groupoids) whereby such a quantale is shown to be the same as a unital stably Gelfand quantal frame whose partial units cover $Q$.