Groupoid sheaves as Hilbert modules

We provide a new characterization of the notion of sheaf on an étale groupoid G, in terms of a particular kind of Hilbert module on the quantale O(G) of the groupoid. All the theory is developed in the context of the more general class of quantales known as stable quantal frames, of which examples are easy to construct because their category is algebraic. The homomorphisms of our Hilbert modules are necessarily adjointable and thus form a strongly self-dual category. By restriction we obtain, for any stable quantal frame, two isomorphic categories of sheaves whose morphisms are related by the duality.