Groupoid sheaves as quantale sheaves

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.Comment: 62 pages. Structure of preprint has changed. It now contains the contents of former arXiv:0807.3859 (withdrawn), and the definition of Q-sheaf applies only to inverse quantal frames (Hilbert Q-modules with enough sections are given no special name for more general quantales

Introducing sheaves over commutative semicartesian quantales

We extend the classic definition of sheaves on locales introducing an original notion of sheaves on semicartesian quantales. We show that the resulting category and the category of sheaves on locales share similar categorical properties, and discuss the difficulties in concluding whether our sheaves on quantales form a Grothendieck topos. We also prove a base change theorem, which may be useful not only to study the relation between sheaves on locales and sheaves on quantales, but also may be applied in the presence of an isomorphism of commutative and unital rings.Comment: 28 page

The Convergence of Filters on Quantales and Its Hausdorffness

In this paper, we introduce the definition of conergence of filters on quantale. Some characterizations of finit completeness and compactness of quantales are studied. At last, the Hausdorff property in quantale using the converence structure is presented.key words: Quantale; Point; ideal; Congerencen of filter; Hausdorff Propert

Groupoid Quantales: a non \'etale setting

It is well known that if G is an \'etale topological groupoid then its topology can be recovered as the sup-lattice generated by G-sets, i.e. by the images of local bisections. This topology has a natural structure of unital involutive quantale. We present the analogous construction for any non \'etale groupoid with sober unit space G_0. We associate a canonical unital involutive quantale with any inverse semigroup of G-sets which is also a sheaf over G_0. We introduce axiomatically the class of quantales so obtained, and revert the construction mentioned above by proving a representability theorem for this class of quantales, under a natural spatiality condition

Algebraic Properties of the Category of Q-P Quantale Modules

In this paper, the definition of a Q-P quantale  module and some relative concepts were introduced. Based on which, some properties of the Q-P quantale module, and the structure of the free Q-P quantale  modules generated by a set were obtained. It was proved that the category of Q-P quantale modules is algebraic