Based on original work in [TV03], this paper gives a representation theorem for the Scott continuous maps between frames in terms of natural transformations between functors indexed by frames. The result specializes to frame homomorphisms, thereby giving a representation theorem for all locale maps (i.e. continuous maps as defined in locale theory).

Locales are Isabelle’s mechanism to deal with parametric theories. We present typical examples of locale specifications, along with interpretations between locales to change their hierarchic dependencies and interpretations to reuse locales in theory contexts and proofs. This tutorial is intended for locale novices; familiarity with Isabelle and Isar is presumed.

Semi-regular locales are extensions of the classical semiregular spaces. We investigate the conditions such that semi-regularization is a functor. We also investigate the conditions such that semi-regularization is a reflection or coreflection.

Department of Infectious Parasitic and ImmuneMediated Diseases, Istituto Superiore di Sanità, Rome, Italy (M L Ricci BiolD, S Fontana PhD, F Pinci BiolD, M Scaturro PhD); Azienda Unità Sanitaria Locale, Forlì, Italy (E Fiumana MD); Azienda Unità Sanitaria Locale, Cesena, Italy (M F Pedna BiolD); Agenzia Regionale Prevenzione e Ambiente, Bologna, Italy (M A Bucci Sabattini BiolD); and “G B Morga...

It it shown that geometric morphisms between elementary toposes can be represented as adjunctions between the corresponding categories of locales. These adjunctions are characterised as those that preserve the order enrichment, commute with the double power locale monad and whose right adjoints preserve finite coproduct. They are also characterised as those adjunctions that preserve the order e...

. On considère une classe de systèmes non-linéaires pour lesquels la technique de synthèse par backstepping n’est pas applicable. On présente un critère pour la synthèse d’une commande hybride faisant l’union d’une commande par backstepping et d’une commande locale. Cette commande hybride rend le système bouclé globalement asymptotiquement stable. Le critère de sélection repose sur le choix de ...

It is well known that the internal suplattices in the topos of sheaves on a locale are precisely the modules on that locale. Using enriched category theory and a lemma on KZ doctrines we prove (the generalization of) this fact in the case of ordered sheaves on a small quantaloid. Comparing module-equivalence with sheaf-equivalence for quantaloids and using the notion of centre of a quantaloid, ...