A perspective on non-commutative frame theory

This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over by \'etale localic categories. This involves ideas from quantale theory and from semigroup theory, specifically Ehresmann semigroups, restriction semigroups and inverse semigroups. We establish a duality between the category of complete restriction monoids and the category of \'etale localic categories. The relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames. This result builds on the work of Pedro Resende on the connection between pseudogroups and \'etale localic groupoids but in the process we both generalize and simplify: for example, we do not require involutions and, in addition, we render his result functorial. We also project down to topological spaces and, as a result, extend the classical adjunction between locales and topological spaces to an adjunction between \'etale localic categories and \'etale topological categories. In fact, varying morphisms, we obtain several adjunctions. Just as in the commutative case, we restrict these adjunctions to spatial-sober and coherent-spectral equivalences. The classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups. Consequently, we deduce several dualities between distributive restriction semigroups and spectral \'etale topological categories. We also specialize these dualities for the setting where the topological categories are cancellative or are groupoids. Our approach thus links, unifies and extends the approaches taken in the work by Lawson and Lenz and by Resende.Comment: 69 page

Measurement spaces

The question of what should be meant by a measurement is tackled from a mathematical perspective whose physical interpretation is that a measurement is a process via which a finite amount of classical information is generated. This motivates a mathematical definition of space of measurements that consists of a topological stably Gelfand quantale whose open sets represent measurable physical properties. It also accounts for the distinction between quantum and classical measurements, and for the emergence of "classical observers." The latter have a relation to groupoid C*-algebras, and link naturally to Schwinger's notion of selective measurement

Quantalic spectra of semirings

Spectrum constructions appear throughout mathematics as a way of constructing topological spaces from algebraic data. Given a localic semiring R (the pointfree analogue of a topological semiring), we define a spectrum of R which generalises the Stone spectrum of a distributive lattice, the Zariski spectrum of a commutative ring, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann–Lawson spectrum of a continuous frame. We then provide an explicit construction of this spectrum under conditions on R which are satisfied by our main examples. Our results are constructively valid and hence admit interpretation in any elementary topos with natural number object. For this reason the spectrum we construct should actually be a locale instead of a topological space. A simple modification to our construction gives rise to a quantic spectrum in the form of a commutative quantale. Such a quantale contains 'differential' information in addition to the purely topological information of the localic spectrum. In the case of a discrete ring, our construction produces the quantale of ideals. This prompts us to study the quantale of ideals in more detail. We discuss some results from abstract ideal theory in the setting of quantales and provide a tentative definition for what it might mean for a quantale to be nonsingular by analogy to commutative ring theory

Enriched Stone-type dualities

A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces,the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem to categories including all compact Hausdorff spaces. To achieve this goal, we combine duality theory and quantale-enriched category theory. Our main idea is that, when passing from the two-element discrete space to a cogenerator of the category of compact Hausdorff spaces, all other involved structures should be substituted by corresponding enriched versions. Accordingly, we work with the unit interval $[0,1]$ and present duality theory for ordered and metric compact Hausdorff spaces and (suitably defined) finitely cocomplete categories enriched in $[0,1]$

Conference Program

Document provides a list of the sessions, speakers, workshops, and committees of the 32nd Summer Conference on Topology and Its Applications

Convolution algebras: Relational convolution, generalised modalities and incidence algebras

Convolution is a ubiquitous operation in mathematics and computing. The Kripke semantics for substructural and interval logics motivates its study for quantale-valued functions relative to ternary relations. The resulting notion of relational convolution leads to generalised binary and unary modal operators for qualitative and quantitative models, and to more conventional variants, when ternary relations arise from identities over partial semigroups. Convolution-based semantics for fragments of categorial, linear and incidence (segment or interval) logics are provided as qualitative applications. Quantitative examples include algebras of durations and mean values in the duration calculus