Groupoid sheaves as quantale sheaves

TL;DR

This paper establishes an equivalence between equivariant sheaves on localic etale groupoids and sheaves on their associated involutive quantales, expanding the understanding of sheaf theory in the context of quantale and groupoid structures.

Contribution

It introduces a new concrete example of sheaves on involutive quantales by linking them to equivariant sheaves on localic etale groupoids, and develops a module-theoretic framework for these sheaves.

Findings

Equivariant sheaves on localic etale groupoids are equivalent to sheaves on their involutive quantales.

Replaces matrix sheaves with complete Hilbert Q-modules for better categorical understanding.

Places the example within the broader context of stably supported quantales and groupoid actions.

Abstract

Several notions of sheaf on various types of quantale have been proposed and studied in the last twenty five years. It is fairly standard that for an involutive quantale Q satisfying mild algebraic properties the sheaves on Q can be defined to be the idempotent self-adjoint Q-valued matrices. These can be thought of as Q-valued equivalence relations, and, accordingly, the morphisms of sheaves are the Q-valued functional relations. Few concrete examples of such sheaves are known, however, and in this paper we provide a new one by showing that the category of equivariant sheaves on a localic etale groupoid G (the classifying topos of G) is equivalent to the category of sheaves on its involutive quantale O(G). As a means towards this end we begin by replacing the category of matrix sheaves on Q by an equivalent category of complete Hilbert Q-modules, and we approach the envisaged example where Q is an inverse quantal frame O(G) by placing it in the wider context of stably supported quantales, on one hand, and in the wider context of a module theoretic description of arbitrary actions of \'etale groupoids, both of which may be interesting in their own right.