# The many groupoids of a stably Gelfand quantale

## TL;DR

This paper explores the structure of stably Gelfand quantales, showing how projections lead to associated pseudogroups and groupoids, and simplifies the axiomatization of inverse quantal frames.

## Contribution

It introduces a new perspective on projections in stably Gelfand quantales and provides a simplified axiomatization of inverse quantal frames.

## Key findings

- Projections in stably Gelfand quantales determine pseudogroups and localic étale groupoids.
- A new simplified axiomatization of inverse quantal frames is established.
- Inverse quantal frames correspond to unital stably Gelfand quantal frames with covering partial units.

## 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 $Q$.