Pedro Resende

Etale groupoids and their quantales

Abstract We establish close and previously unknown relations between quantales and groupoids. In particular, to each etale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic etale groupoids and their quantales, which are given a rather simple characterization and here are called inverse quantal frames. We show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids, and as a consequence we obtain a (non-functorial) correspondence between these and localic etale groupoids that generalizes more classical results concerning inverse semigroups and topological etale groupoids. This generalization is entirely algebraic and it is valid in an arbitrary topos. As a consequence of these results we see that a localic groupoid is etale if and only if its sublocale of units is open and its multiplication map is semiopen, and an analogue of this holds for topological groupoids. In practice we are provided with new tools for constructing localic and topological etale groupoids, as well as inverse semigroups, for instance via presentations of quantales by generators and relations. The characterization of inverse quantal frames is to a large extent based on a new quantale operation, here called a support, whose properties are thoroughly investigated, and which may be of independent interest.

Quantales, finite observations and strong bisimulation

Abstract It has often been claimed that bisimulation makes distinctions that cannot be observed in practice. Abramsky and Vickers proposed an algebraic framework based on quantales for describing observations on concurrent processes without hidden transitions and used it in order to provide an “observational explanation” of several process equivalences, ranging from Hoare trace equivalence to ready-simulation. We follow their approach and argue that (strong) bisimulation can be explained in the same way, at least in the case of “image-computable” labelled transition systems.

On quantales and spectra of C*-algebras

We study properties of the quantale spectrum Max A of an arbitrary unital C*-algebra A. In particular we show that the spatialization of Max A with respect to one of the notions of spatiality in the literature yields the locale of closed ideals of A when A is commutative. We study under general conditions functors with this property, in addition requiring that colimits be preserved, and we conclude in this case that the spectrum of A necessarily coincides with the locale of closed ideals of the commutative reflection of A. Finally, we address functorial properties of Max, namely studying (non-)preservation of limits and colimits. Although Max  is not an equivalence of categories, therefore not providing a direct generalization of Gelfand duality to the noncommutative case, it is a faithful complete invariant of unital C*-algebras.

Quantales and Observational Semantics

We illustrate the idea that quantales can be regarded as algebras of experimental observations on physical systems, and we give a survey of some research in computer science where this idea has been used. We extend the mathematical framework hitherto available so that it can be applied to more general systems than before, in particular to quantum systems and systems whose behaviour is partially unobservable.

Localic sup-lattices and tropological systems

The approach to process semantics using quantales and modules is topologized by considering tropological systems whose sets of states are replaced by locales and which satisfy a suitable stability axiom. A corresponding notion of localic sup-lattice (algebra for the lower powerlocale monad) is described, and it is shown that there are contravariant functors from sup-lattices to localic sup-lattices and, for each quantale Q, from left Q-modules to localic right Q-modules. A proof technique for third completeness due to Abramsky and Vickers is reset constructively, and an example of application to failures semantics is given.

Sheaves as Modules

We revisit sheaves on locales by placing them in the context of the theory of quantale modules. The local homeomorphisms p:X→B are identified with the Hilbert B-modules that are equipped with a natural notion of basis. The homomorphisms of these modules are necessarily adjointable, and the resulting self-dual category yields a description of the equivalence between local homeomorphisms and sheaves whereby morphisms of sheaves arise as the “operator adjoints” of the inverse images of the maps of local homeomorphisms.

A Note on Infinitely Distributive Inverse Semigroups

By an infinitely distributive inverse semigroup will be meant an inverse semigroup S such that for every subset X ⊆ S and every s ∈ S ,i f X exists then so does (sX), and furthermore (sX )= s X . One important aspect is that the infinite distributivity of E(S) implies that of S ; that is, if the multiplication of E(S) distributes over all the joins that exist in E(S) then S is infinitely distributive. This can be seen in Proposition 20, page 28, of Lawson’s book [1]. Although the statement of the proposition mentions only joins of nonempty sets, the proof applies equally to any subset. The aim of this note is to present a proof of an analogous property for binary meets instead of multiplication; that is, we show that for any infinitely distributive inverse semigroup the existing binary meets distribute over all the joins that exist. A useful consequence of this lies in the possibility of constructing, from infinitely distributive inverse semigroups, certain quantales that are also locales (due to the stability of the existing joins both with respect to the multiplication

Groupoid sheaves as quantale sheaves

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 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 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.

The Reification Dimension in Object-oriented Data Base Design

The reification of an (abstract) object base schema over another (ground) object base schema is discussed. The concepts of reification object base schema, incorporation (inheritance), derived attribute and transaction are identified as basic. The reification object base schema includes the object classes of the abstract and the ground. bases as well as a reification object class for each object class to be reified. The attributes of the abstract object classes under reification are defined as derived attributes and events are introduced as transactions. A brief outline of the semantics is also discussed.

An Algebra of Behavioural Types

We propose a process algebra, the Algebra of Behavioural Types, as a language for typing concurrent objects. A type is a higher-order labelled transition system that characterises all possible life cycles of a concurrent object. States represent interfaces of objects; state transitions model the dynamic change of object interfaces. Moreover, a type provides an internal view of the objects that inhabits it: a synchronous one, since transitions correspond to message reception. To capture this internal view of objects we define a notion of bisimulation, strong on labels and weak on silent actions. We study several algebraic laws that characterise this equivalence, and obtain completeness results for image-finite types.