Etale groupoids and their quantales

We establish a close and previously unknown relation between quantales and groupoids, in terms of which the notion of etale groupoid is subsumed in a natural way by that of quantale. In particular, to each etale groupoid, either localic or topological, there is associated a unital involutive quantale. We obtain a bijective correspondence between localic etale groupoids and their quantales, which are given a rather simple characterization and are here called inverse quantal frames. We show that the category of inverse quantal frames is equivalent to the category of complete and infinitely distributive inverse monoids, and as a consequence we obtain a correspondence between these and localic etale groupoids that generalizes more classical results concerning inverse semigroups and topological etale groupoids. This generalization is entirely algebraic and it is valid in an arbitrary topos. As a consequence of these results we see that a localic groupoid is etale if and only if its sublocale of units is open and its multiplication map is semiopen, and an analogue of this holds for topological groupoids. In practice we are provided with new tools for constructing localic and topological etale groupoids, as well as inverse semigroups, for instance via presentations of quantales by generators and relations. The characterization of inverse quantal frames is to a large extent based on a new quantale operation, here called a support, whose properties are thoroughly investigated, and which may be of independent interest.Comment: Version 3 contains 16 additional pages (now the total is 75) and completes the characterization of etale groupoid quantales; in particular it is proved that every inverse quantal frame is multiplicativ

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

Convolution as a unifying concept: Applications in separation logic, interval calculi and concurrency

The research reported here was supported in part by Australian Research Council Grant No. DP130102901 and EPSRC Grant No. EP/J003727/1

Resource Theories as Quantale Modules

We aim to counter the tendency for specialization in science by advancing a language that can facilitate the translation of ideas and methods between disparate contexts. The methods we address relate to questions of "resource-theoretic nature". In a resource theory, one identifies resources and allowed manipulations that can be used to transform them. Some of the main questions are: How to optimize resources? What are the trade-offs between them? Can a given resource be converted to another one via the allowed manipulations? Because of the ubiquity of such questions, methods for answering them in one context can be used to tackle corresponding questions in new contexts. The translation occurs in two stages. Firstly, concrete methods are generalized to the abstract language to find under what conditions they are applicable. Then, one can determine whether potentially novel contexts satisfy these conditions. Here, we mainly focus on the first part of this two-stage process. The thesis starts with a more thorough introduction to resource theories and our perspective on them in chapter 1. Chapter 2 then provides a selection of mathematical ideas that we make heavy use of in the rest of the manuscript. In chapter 3, we present two variants of the abstract framework, whose relations to existing ones are summarized in table 1.1. The first one, universally combinable resource theories, offers a structure in which resources, desired tasks, and resource manipulations may all be viewed as "generalized resources". Blurring these distinctions, whenever appropriate, is a simplification that lets us understand the abstract results in elementary terms. It offers a slightly distinct point of view on resource theories from the traditional one, in which resources and their manipulations are considered independently. In this sense, the second framework in terms of quantale modules follows the traditional conception. Using these, we make contributions towards the task of generalizing concrete methods in chapter 4 by studying the ways in which meaningful measures of resources may be constructed. One construction expresses a notion of cost (or yield) of a resource, summarized in its generalized form in theorems 4.21 and 4.22. Among other applications, this construction may be used to extend measures from a subset of resources to a larger domain—such as from states to channels and other processes. Another construction allows the translation of resource measures between resource theories. A particularly useful version thereof is the translation of measures of distinguishability to other resource theories, which we study in detail. Special cases include resource robustness and weight measures as well as relative entropy based measures quantifying minimal distinguishability from freely available resources. We instantiate some of these ideas in a resource theory of distinguishability in chapter 5. It describes the utility of systems with probabilistic behavior for the task of distinguishing between hypotheses, which said behavior may depend on

Resource Theories as Quantale Modules

We aim to counter the tendency for specialization in science by advancing a language that can facilitate the translation of ideas and methods between disparate contexts. The methods we address relate to questions of "resource-theoretic nature". In a resource theory, one identifies resources and allowed manipulations that can be used to transform them. Some of the main questions are: How to optimize resources? What are the trade-offs between them? Can a given resource be converted to another one via the allowed manipulations? Because of the ubiquity of such questions, methods for answering them in one context can be used to tackle corresponding questions in new contexts. The translation occurs in two stages. Firstly, concrete methods are generalized to the abstract language to find under what conditions they are applicable. Then, one can determine whether potentially novel contexts satisfy these conditions. Here, we mainly focus on the first part of this two-stage process. The thesis starts with a more thorough introduction to resource theories and our perspective on them in chapter 1. Chapter 2 then provides a selection of mathematical ideas that we make heavy use of in the rest of the manuscript. In chapter 3, we present two variants of the abstract framework, whose relations to existing ones are summarized in table 1.1. The first one, universally combinable resource theories, offers a structure in which resources, desired tasks, and resource manipulations may all be viewed as "generalized resources". Blurring these distinctions, whenever appropriate, is a simplification that lets us understand the abstract results in elementary terms. It offers a slightly distinct point of view on resource theories from the traditional one, in which resources and their manipulations are considered independently. In this sense, the second framework in terms of quantale modules follows the traditional conception. Using these, we make contributions towards the task of generalizing concrete methods in chapter 4 by studying the ways in which meaningful measures of resources may be constructed. One construction expresses a notion of cost (or yield) of a resource, summarized in its generalized form in theorems 4.21 and 4.22. Among other applications, this construction may be used to extend measures from a subset of resources to a larger domain—such as from states to channels and other processes. Another construction allows the translation of resource measures between resource theories. A particularly useful version thereof is the translation of measures of distinguishability to other resource theories, which we study in detail. Special cases include resource robustness and weight measures as well as relative entropy based measures quantifying minimal distinguishability from freely available resources. We instantiate some of these ideas in a resource theory of distinguishability in chapter 5. It describes the utility of systems with probabilistic behavior for the task of distinguishing between hypotheses, which said behavior may depend on

Entailment systems for stably locally compact locales

The category SCFrU of stably continuous frames and preframe ho-momorphisms (preserving ¯nite meets and directed joins) is dual to the Karoubi envelope of a category Ent whose objects are sets and whose morphisms X ! Y are upper closed relations between the ¯nite powersets FX and FY . Composition of these morphisms is the \cut composition" of Jung et al. that interfaces disjunction in the codomains with conjunctions in the domains, and thereby relates to their multi-lingual sequent calculus. Thus stably locally compact locales are represented by \entailment systems" (X; `) in which `, a generalization of entailment relations,is idempotent for cut composition. Some constructions on stably locally compact locales are represented in terms of entailment systems: products, duality and powerlocales. Relational converse provides Ent with an involution, and this gives a simple treatment of the duality of stably locally compact locales. If A and B are stably continuous frames, then the internal preframe hom A t B is isomorphic to e A ­ B where e A is the Hofmann-Lawson dual. For a stably locally compact locale X, the lower powerlocale of X is shown to be the dual of the upper powerlocale of the dual of X