arXiv subjects

Francesco Ciraulo

Publications and source records attributed to Francesco Ciraulo.

7 recordsLinked to original sources

arXiv2023

Overlap Algebras as Almost Discrete Locales

Boolean locales are "almost discrete", in the sense that a spatial Boolean locale is just a discrete locale (that is, it corresponds to the frame of open subsets of a discrete space, namely the powerset of a set). This basic fact, however, cannot be proven constructively, that is, over intuitionistic logic, as it requires the full law of excluded middle (LEM). In fact, discrete locales are never Boolean constructively, except for the trivial locale. So, what is an almost discrete locale constructively? Our claim is that Sambin's overlap algebras have good enough features to deserve to be called that. Namely, they include the class of discrete locales, they arise as smallest strongly dense sublocales (of overt locales), and hence they coincide with the Boolean locales if LEM holds.

math.LO↗

arXiv2022

$σ$-locales in Formal Topology

A $σ$-frame is a poset with countable joins and finite meets in which binary meets distribute over countable joins. The aim of this paper is to show that $σ$-frames, actually $σ$-locales, can be seen as a branch of Formal Topology, that is, intuitionistic and predicative point-free topology. Every $σ$-frame $L$ is the lattice of Lindelöf elements (those for which each of their covers admits a countable subcover) of a formal topology of a specific kind which, in its turn, is a presentation of the free frame over $L$. We then give a constructive characterization of the smallest (strongly) dense $σ$-sublocale of a given $σ$-locale, thus providing a "$σ$-version" of a Boolean locale. Our development depends on the axiom of countable choice.

math.LO↗

arXiv2020

Overlap Algebras: a Constructive Look at Complete Boolean Algebras

The notion of a complete Boolean algebra, although completely legitimate in constructive mathematics, fails to capture some natural structures such as the lattice of subsets of a given set. Sambin's notion of an overlap algebra, although classically equivalent to that of a complete Boolean algebra, has powersets and other natural structures as instances. In this paper we study the category of overlap algebras as an extension of the category of sets and relations, and we establish some basic facts about mono-epi-isomorphisms and (co)limits; here a morphism is a symmetrizable function (with classical logic this is just a function which preserves joins). Then we specialize to the case of morphisms which preserve also finite meets: classically, this is the usual category of complete Boolean algebras. Finally, we connect overlap algebras with locales, and their morphisms with open maps between locales, thus obtaining constructive versions of some results about Boolean locales.

cs.LO↗

arXiv2018

Factorizing the Top-Loc adjunction through positive topologies

We characterize the category of Sambin's positive topologies as a fibration over the category of locales Loc. The fibration is obtained by applying the Grothendieck construction to a doctrine over Loc. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the newly constructed adjunction.

math.GN↗

arXiv2012

Convergence in Formal Topology: a unifying notion

Several variations on the definition of a Formal Topology exist in the literature. They differ on how they express convergence, the formal property corresponding to the fact that open subsets are closed under finite intersections. We introduce a general notion of convergence of which any previous definition is a special case. This leads to a predicative presentation and inductive generation of locales (formal covers), commutative quantales (convergent covers) and suplattices (basic covers) in a uniform way. Thanks to our abstract treatment of convergence, we are able to specify categorically the precise sense according to which our inductively generated structures are free, thus refining Johnstone's coverage theorem. We also obtain a natural and predicative version of a fundamental result by Joyal and Tierney: convergent covers (commutative quantales) correspond to commutative co-semigroups over the category of basic covers (suplattices).

math.LO↗

arXiv2012

A constructive Galois connection between closure and interior

We construct a Galois connection between closure and interior operators on a given set. All arguments are intuitionistically valid. Our construction is an intuitionistic version of the classical correspondence between closure and interior operators via complement.

math.LO↗

arXiv2012

Constructive version of Boolean algebra

The notion of overlap algebra introduced by G. Sambin provides a constructive version of complete Boolean algebra. Here we first show some properties concerning overlap algebras: we prove that the notion of overlap morphism corresponds classically to that of map preserving arbitrary joins; we provide a description of atomic set-based overlap algebras in the language of formal topology, thus giving a predicative characterization of discrete locales; we show that the power-collection of a set is the free overlap algebra join-generated from the set. Then, we generalize the concept of overlap algebra and overlap morphism in various ways to provide constructive versions of the category of Boolean algebras with maps preserving arbitrary existing joins.

math.LO↗