Tropological systems are points of quantales

Abstract

We address two areas in which quantales have been used. One is of a topological nature, whereby quantales or involutive quantales are seen as generalized noncommutative spaces, and its main purpose so far has been to investigate the spectrum of noncommutative C ∗ -algebras. The other sees quantales as algebras of abstract experiments on physical or computational systems, and has been applied to the study of the semantics of concurrent systems. We investigate connections between the two areas, in particular showing that concurrent systems, in the form of either set-theoretic or localic tropological systems, can be identified with points of quantales by means of a suitable adjunction, which indeed holds for a much larger class of so-called “tropological models”. We show that in the case of tropological models in factor quantales, which still generalize tropological systems, the identification of models and (generalized) points preserves all the information needed for describing the observable behaviour of systems. We also define a notion of morphism of models that generalizes previous definitions of morphism of systems, and show that morphisms, too, can be defined in terms of either side of the adjunction, in fact giving us isomorphisms of categories. The relation between completeness notions for tropological systems and spatiality for quantales is also addressed, and a preliminary partial preservation result is obtained.