Taking advantage of the quantale-theoretic description \'etale groupoids we study principal bundles, Hilsum-Skandalis maps, and Morita equivalence in terms modules on inverse quantal frames. The Hilbert module quantale sheaves leads naturally to a formulation bimodules that resemble imprimitivity C*-algebras.

(Q,≦,∨) is a complete join semilattice (Q,・,e) is a monoid (∨S)・a=∨{b・a | b∈S} for a∈Q, S⊆Q a・(∨S)=∨{a・b | b∈S} for a∈Q, S⊆Q 2 Background: An Example of Quantale (Rel(A),≦,∨,・,e) is a quantale where Rel(A) is the set of all binary relations on A ≦ is the inclusion ∨ is the union ・ is the composition Q・R={ (a,c) | ∃b. (a,b)∈Q, (b,c)∈R } e is the identity relation 3 Our Goal: Relational Represent...

We associate a canonical unital involutive quantale to a topological groupoid. When the groupoid is also étale, this association is compatible with but independent from the theory of localic étale groupoids and their quantales [19] of P. Resende. As a motivating example, we describe the connection between the quantale and the C∗-algebra that both classify Penrose tilings, which was left as an o...

In this paper we prove that any quantale Q is (isomorphic to) a quantale of suitable relations on Q. As a consequence two isomorphism theorems are also shown with suitable sets of functions of Q into Q. These theorems are the mathematical background one needs in order to give natural and complete semantics for (non-commutative) Linear Logic using relations. 1991 Mathematics Subject Classificati...

Our work is a foundational study of the notion of approximation inQ-categories and in (U,Q)-categories, for a quantale Q and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of Qand (U,Q)-categories. We fully characterize continuous Q-categories (resp. (U,Q)-categories) among all cocomplete Q-categories (...

For an arbitrary localic étale groupoid G we provide simple descriptions, in terms of modules over the quantale O(G) of the groupoid, of the continuous actions of G, including actions on open maps and sheaves. The category of G-actions is isomorphic to a corresponding category of O(G)-modules, and as a corollary we obtain a new quantale based representation of étendues.

In this communication we generalize some recent results of Rump to categories enriched in a commutative quantale V. Using these results, show that every quantale-enriched multicategory admits an injective hull. Finally, expose connection between the Isbell adjunction and construction hulls for topological spaces made by Banaschewski 1973.

We pursue the idea that predicate logic is a “fibred algebra” while propositional logic is a single algebra; in the context of intuitionism, this algebraic understanding of predicate logic goes back to Lawvere, in particular his concept of hyperdoctrine. Here, we aim at demonstrating that the notion of monad-relativised hyperdoctrines, which are what we call fibred algebras, yields algebraisati...