3,366 research outputs found
Positivity relations on a locale
This paper analyses the notion of a positivity relationof Formal Topology from the point of view of the theory of Locales. It is shown that a positivity relation on a locale corresponds to a suitable class of points of its lower powerlocale. In particular, closed subtopologies associated to the positivity relation correspond to overt (that is, with open domain) weakly closed sublocales. Finally, some connection is revealed between positivity relations and localic suplattices (these are algebras for the powerlocale monad)
Embedding locales and formal topologies into positive topologies
A positive topology is a set equipped with two particular relations between elements and subsets of that set: a convergent cover relation and a positivity relation. A set equipped with a convergent cover relation is a predicative counterpart of a locale, where the given set plays the role of a set of generators, typically a base, and the cover encodes the relations between generators. A positivity relation enriches the structure
of a locale; among other things, it is a tool to study some particular subobjects, namely the overt weakly closed sublocales. We relate the category of locales to that of positive topologies and we show that the former is a re\ufb02ective subcategory of the latter. We then generalize such a result to the (opposite of the) category of suplattices, which we present by means of (not necessarily convergent) cover relations. Finally, we show that the category of positive topologies also generalizes that of formal topologies, that is, overt locales
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.Comment: Postproceedings of CCC2018: Continuity, Computability,
Constructivity. Faro, Portugal, 24-28 Sep 201
The connected Vietoris powerlocale
The connected Vietoris powerlocale is defined as a strong monad Vc on the category of locales. VcX is a sublocale of Johnstone's Vietoris powerlocale VX, a localic analogue of the Vietoris hyperspace, and its points correspond to the weakly semifitted sublocales of X that are “strongly connected”. A product map ×:VcX×VcY→Vc(X×Y) shows that the product of two strongly connected sublocales is strongly connected. If X is locally connected then VcX is overt. For the localic completion of a generalized metric space Y, the points of are certain Cauchy filters of formal balls for the finite power set with respect to a Vietoris metric. \ud
Application to the point-free real line gives a choice-free constructive version of the Intermediate Value Theorem and Rolle's Theorem. \ud
\ud
The work is topos-valid (assuming natural numbers object). Vc is a geometric constructio
Localic Metric spaces and the localic Gelfand duality
In this paper we prove, as conjectured by B.Banachewski and C.J.Mulvey, that
the constructive Gelfand duality can be extended into a duality between compact
regular locales and unital abelian localic C*-algebras. In order to do so we
develop a constructive theory of localic metric spaces and localic Banach
spaces, we study the notion of localic completion of such objects and the
behaviour of these constructions with respect to pull-back along geometric
morphisms.Comment: 57 page
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.Comment: 22 page
Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory
In Riesz space theory it is good practice to avoid representation theorems
which depend on the axiom of choice. Here we present a general methodology to
do this using pointfree topology. To illustrate the technique we show that
almost f-algebras are commutative. The proof is obtained relatively
straightforward from the proof by Buskes and van Rooij by using the pointfree
Stone-Yosida representation theorem by Coquand and Spitters
Integrals and Valuations
We construct a homeomorphism between the compact regular locale of integrals
on a Riesz space and the locale of (valuations) on its spectrum. In fact, we
construct two geometric theories and show that they are biinterpretable. The
constructions are elementary and tightly connected to the Riesz space
structure.Comment: Submitted for publication 15/05/0
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