Boolean Localization Research Articles

$\sigma$-locales in Formal Topology

A $\sigma$-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 $\sigma$-frames, actually $\sigma$-locales, can be seen as a branch of Formal Topology, that is, intuitionistic and predicative point-free topology. Every $\sigma$-frame $L$ is the lattice of Lindel\"of 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 $\sigma$-sublocale of a given $\sigma$-locale, thus providing a "$\sigma$-version" of a Boolean locale. Our development depends on the axiom of countable choice.

Fuzzy sets and presheaves

This paper presents a presheaf theoretic approach to the construction of fuzzy sets, which builds on Barr's description of fuzzy sets as sheaves of monomorphisms on a locale. Presheaves are used to give explicit descriptions of limit and colimit descriptions in fuzzy sets on an interval. The Boolean localization construction for sheaves on a locale specializes to a theory of stalks for sheaves and presheaves on an interval.The systemV∗(X)of Vietoris-Rips complexes for a data setXis both a simplicial fuzzy set and a simplicial sheaf in this general framework. This example is explicitly discussed through a series of examples.

Pseudocompleteness in the category of locales

We define pseudocompleteness in the category of locales in a conservative way; so that a space is pseudocomplete in the sense of Oxtoby [24] if and only if the locale it determines is pseudocomplete. We show that a pseudocomplete locale whose Gδ-sublocales are complemented (for instance if it is scattered) is a Baire locale in the sense of Isbell [20]. Our main theorem is that products of pseudocomplete locales are pseudocomplete. Whereas every discrete space is pseudocomplete, and Boolean locales generalize discrete spaces, we demonstrate that not every Boolean locale is pseudocomplete. In [27] Pichardo-Mendoza asks whether pseudocompleteness (in topological spaces) is an invariant of closed irreducible maps. We answer this in the affirmative.

Pointfree pointwise suprema in unital archimedean ℓ-groups

We generalize the concept of the pointwise supremum of real-valued functions to the pointfree setting. The concept itself admits a direct and intuitive formulation which makes no mention of points. But our aim here is to investigate pointwise suprema of subsets of RL, the family of continuous real valued functions on a locale, or pointfree space.Thus our setting is the category W of archimedean lattice-ordered groups (ℓ-groups) with designated weak order unit, with morphisms which preserve the group and lattice operations and take units to units. This is an appropriate context for this investigation because every W-object can be canonically represented as a subobject of some RL.We show that the suprema which are pointwise in the Madden representation can be characterized purely algebraically. They are precisely the suprema which are context-free, in the sense of being preserved by every W homomorphism out of G. We show that closure under such suprema characterizes the W-kernels among the convex ℓ-subgroups. Finally, we prove that all existing joins in a W-object G are pointwise iff its Madden frame L is boolean, and that all existing countable joins in G are pointwise if L is a P-frame, but not conversely.This leads up to the appropriate analog of the Nakano–Stone Theorem: a (completely regular) locale L has the feature that RL is conditionally pointwise complete (σ-complete), i.e., every bounded (countable) family from RL has a pointwise supremum in RL, iff L is boolean (a P-locale).We adopt a maximally broad definition of unconditional pointwise completeness (σ-completeness): a divisible W-object G is pointwise complete (σ-complete) if it contains a pointwise supremum for every subset which has a supremum in any extension. We show that the pointwise complete (σ-complete) W-objects are those of the form RL for L a boolean locale (P-locale). Finally, we show that a W-object G is pointwise σ-complete iff it is epicomplete.

Isocompactness in the Category of Locales

We study isocompactness in Loc defined, exactly as in Top, by requiring that every countably compact closed sublocale be compact. This is a genuine extension of the same-named topological concept since every Boolean (or, even more emphatically, every paracompact) locale is isocompact. A slightly stronger variant is defined by decreeing that the closure of every complemented countably compact sublocale be compact. Dropping the adjective “complemented” yields a formally even stronger property, which we show to be preserved by finite products. Metrizable locales (or, more generally, perfectly normal locales) do not distinguish between the three variants of isocompactness. Each of the stronger variants of isocompactness travels across a proper map of locales, and in the opposite direction if the map is a surjection in Loc.

A Categorial Semantic Representation of Quantum Event Structures

The overwhelming majority of the attempts in exploring the problems related to quantum logical structures and their interpretation have been based on an underlying set-theoretic syntactic language. We propose a transition in the involved syntactic language to tackle these problems from the set-theoretic to the category-theoretic mode, together with a study of the consequent semantic transition in the logical interpretation of quantum event structures. In the present work, this is realized by representing categorically the global structure of a quantum algebra of events (or propositions) in terms of sheaves of local Boolean frames forming Boolean localization functors. The category of sheaves is a topos providing the possibility of applying the powerful logical classification methodology of topos theory with reference to the quantum world. In particular, we show that the topos-theoretic representation scheme of quantum event algebras by means of Boolean localization functors incorporates an object of truth values, which constitutes the appropriate tool for the definition of quantum truth-value assignments to propositions describing the behavior of quantum systems. Effectively, this scheme induces a revised realist account of truth in the quantum domain of discourse. We also include an appendix, where we compare our topos-theoretic representation scheme of quantum event algebras with other categorial and topos-theoretic approaches.

TOPOS-THEORETIC CLASSIFICATION OF QUANTUM EVENTS STRUCTURES IN TERMS OF BOOLEAN REFERENCE FRAMES

We construct a sheaf-theoretic representation of quantum events structures, in terms of Boolean localization systems. These covering systems are constructed as ideals of structure-preserving morphisms of quantum event algebras from varying Boolean domains, identified with physical contexts of measurement. The modeling sheaf-theoretic scheme is based on the existence of a categorical adjunction between presheaves of Boolean event algebras and quantum event algebras. On the basis of this adjoint correspondence, we also prove the existence of an object of truth values in the category of quantum logics, characterized as subobject classifier. This classifying object plays the equivalent role that the two-valued Boolean truth values object plays in classical events structures. We construct the object of quantum truth values explicitly, and furthermore, demonstrate its functioning for the valuation of propositions in a typical quantum measurement situation.

Sheaf-theoretic representation of quantum measure algebras

We construct a sheaf-theoretic representation of quantum probabilistic structures, in terms of covering systems of Boolean measure algebras. These systems coordinatize quantum states by means of Boolean coefficients, interpreted as Boolean localization measures. The representation is based on the existence of a pair of adjoint functors between the category of presheaves of Boolean measure algebras and the category of quantum measure algebras. The sheaf-theoretic semantic transition of quantum structures shifts their physical significance from the orthoposet axiomatization at the level of events, to the sheaf-theoretic gluing conditions at the level of Boolean localization systems.

Higher order dissolutions and Boolean coreflections of locales

If a locale has a Boolean coreflection, then this coreflection can be constructed by iterating the dissolution functor A↦ A d until it stabilizes at a Boolean locale. In this paper we characterize α-soluble locales, i.e. locales whose αth dissolution is Boolean, for α≤4. There are examples of spaces whose chain of dissolutions stabilizes at the first, second, or third dissolution. Beyond that no examples are known. We also give a sufficient condition for insolubility which implies that the locale of rational numbers has no Boolean coreflection, and hence that the only metrizable spaces with a Boolean coreflection are the scattered ones.

Probing Quantum Structure with Boolean Localization Systems

In an attempt to probe the objects belonging to the quantum species of structure,we develop the idea of using observables of the Boolean species of structures,as coordinatizing objects in the quantum world. This results in a contextualisticperspective on the latter through local Boolean measurement reference frames.The semantics of this representation is discussed extensively.

Empirical quantum mechanics

Machida and Namiki developed a many-Hilbert-spaces formalism for dealing with the interaction between a quantum object and a measuring apparatus. Their mathematically rugged formalism was polished first by Araki from an operator-algebraic standpoint and then by Ozawa for Boolean quantum mechanics, which approaches a quantum system with a compatible family of continuous superselection rules from a notable and perspicacious viewpoint. On the other hand, Foulis and Randall set up a formal theory for the empirical foundation of all sciences, at the hub of which lies the notion of a manual of operations. They deem an operation as the set of possible outcomes and put down a manual of operations at a family of partially overlapping operations. Their notion of a manual of operations was incorporated into a category-theoretic standpoint into that of a manual of Boolean locales by Nishimura, who looked upon an operation as the complete Boolean algebra of observable events. Considering a family of Hilbert spaces not over a single Boolean locale but over a manual of Boolean locales as a whole, Ozawa's Boolean quantum mechanics is elevated into empirical quantum mechanics, which is, roughly speaking, the study of quantum systems with incompatible families of continuous superselection rules. To this end, we are obliged to develop empirical Hilbert space theory. In particular, empirical versions of the square root lemma for bounded positive operators, the spectral theorem for (possibly unbounded) self-adjoint operators, and Stone's theorem for one-parameter unitary groups are established.

Boolean localization, in practice

A new proof of the existence of the standard closed model structure for the category of simplicial presheaves on an arbitrary Grothendieck site is given. This proof uses the principle of Boolean localization.

Injectivity and injective hulls of Abelian groups in a localic topos

We prove the analogue of the Baer Criterion for injectivity in the category AbShℒ of abelian groups in a topos of sheaves on a locale, that is, we show A is injective in AbShℒ if and only if it is injective relative to all S ↣ Zℒ where Zℒ is the group of integers in Shℒ. for a well-ordered locale we describe the injective hulls in AbShℒ in terms of injective hulls in Ab. Further we show that the global functor A → AE preserves injective hulls if and only if ℒ is a finite boolean locale. Finally We characterise injectives in AbShℒ for some special locales.