Groupoid sheaves as quantale sheaves. (English) Zbl 1231.06020
Summary: 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 étale groupoid \(G\) (the classifying topos of \(G\)) is equivalent to the category of sheaves on its involutive quantale \(\mathcal O(G)\). As a means toward 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 \(\mathcal O(G)\) by placing it in the wider context of stably supported quantales, and in the wider context of a module-theoretic description of arbitrary actions of étale groupoids, both of which may be interesting in their own right.
MSC:
| 06F07 | Quantales |
| 06D22 | Frames, locales |
| 18B25 | Topoi |
| 18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
| 22A22 | Topological groupoids (including differentiable and Lie groupoids) |
| 54B40 | Presheaves and sheaves in general topology |
Keywords:
equivariant sheaves; localic étale groupoids; classifying topos; category of sheaves; involutive quantales; stably supported quantales; actions of étale groupoidsReferences:
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