Lawvere-Tierney sheaves in algebraic set theory

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming the existing topos-theoretic results.Comment: 33 pages; a revised version of the earlier paper "A general construction of internal sheaves in algebraic set theory"; accepted for publication in the Journal of Symbolic Logi

Formal concept analysis and structures underlying quantum logics

A Hilbert space $H$ induces a formal context, the Hilbert formal context $\overline H$, whose associated concept lattice is isomorphic to the lattice of closed subspaces of $H$. This set of closed subspaces, denoted $\mathcal C(H)$, is important in the development of quantum logic and, as an algebraic structure, corresponds to a so-called ``propositional system'', that is, a complete, atomistic, orthomodular lattice which satisfies the covering law. In this paper, we continue with our study of the Chu construction by introducing the Chu correspondences between Hilbert contexts, and showing that the category of Propositional Systems, PropSys, is equivalent to the category of $\text{ChuCors}_{\mathcal H}$ of Chu correspondences between Hilbert contextsUniversidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Bigraphical modelling of architectural patterns

Selected for publication in FACS'2011 post-proceedings, to appear in Springer Lecture Notes in Computer ScienceArchery is a language for behavioural modelling of architectural patterns, supporting hierarchical composition and a type discipline. This paper extends Archery to cope with the patterns' structural dimension through a set of (re-)configuration combinators and constraints that all instances of a pattern must obey. Both types and instances of architectural patterns are semantically represented as bigraphical reactive systems and operations upon them as reaction rules. Such a bigraphical semantics provides a rigorous model for Archery patterns and reduces constraint verification in architectures to a type-checking problem.(undefined

Generalizations of Hedberg’s Theorem

Streicher’s axiom K or the uniqueness of identity proofs (UIP) is the statement that every identity type has at most one inhabitant. The groupoid interpretation by Hofmann and Streicher shows that this is not provable for an arbitrary type, but a theorem by Hedberg gives a sufficient condition. Types satisfying UIP are known as h-sets. The main contributions of this paper are: (i) Natural, more general sufficient conditions for a type being an h-set. (ii) A new characterization of h-sets. (iii) A new result inspired by these characterizations, with some applications. All the proofs have been formalized in Agda

Layer by layer - Combining Monads

We develop a method to incrementally construct programming languages. Our approach is categorical: each layer of the language is described as a monad. Our method either (i) concretely builds a distributive law between two monads, i.e. layers of the language, which then provides a monad structure to the composition of layers, or (ii) identifies precisely the algebraic obstacles to the existence of a distributive law and gives a best approximant language. The running example will involve three layers: a basic imperative language enriched first by adding non-determinism and then probabilistic choice. The first extension works seamlessly, but the second encounters an obstacle, which results in a best approximant language structurally very similar to the probabilistic network specification language ProbNetKAT

Involutive Categories and Monoids, with a GNS-correspondence

This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of Eilenberg-Moore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. The core of the so-called Gelfand-Naimark-Segal (GNS) construction is identified as a bijective correspondence between states on involutive monoids and inner products. This correspondence exists in arbritrary involutive categories

Quotient inductive-inductive types

Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define sets, such as the Cauchy reals, the partiality monad, and the internal, total syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs). Although there has been recent progress on the general theory of HITs, there isn't yet a theoretical foundation of the combination of equality constructors and induction-induction, despite having many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics and show that this is equivalent to the section induction principle, which justifies the intuitively expected elimination rules

System analysis and robustness

Software is increasingly embedded in a variety of physical contexts. This imposes new requirements on tools that support the design and analysis of systems. For instance, modeling embedded and cyberphysical systems needs to blend discrete mathematics, which is suitable for modeling digital components, with continuous mathematics, used for modeling physical components. This blending of continuous and discrete creates challenges that are absent when the discrete or the continuous setting are considered in isolation. We consider robustness, that is, the ability of an analysis of a model to cope with small amounts of imprecision in the model. Formally, we identify analyses with monotonic maps between complete lattices (a mathematical framework used for abstract interpretation and static analysis) and define robustness for monotonic maps between complete lattices of closed subsets of a metric space.</p

Higher Structures in M-Theory

The key open problem of string theory remains its non-perturbative completion to M-theory. A decisive hint to its inner workings comes from numerous appearances of higher structures in the limits of M-theory that are already understood, such as higher degree flux fields and their dualities, or the higher algebraic structures governing closed string field theory. These are all controlled by the higher homotopy theory of derived categories, generalised cohomology theories, and $L_\infty$-algebras. This is the introductory chapter to the proceedings of the LMS/EPSRC Durham Symposium on Higher Structures in M-Theory. We first review higher structures as well as their motivation in string theory and beyond. Then we list the contributions in this volume, putting them into context.Comment: 22 pages, Introductory Article to Proceedings of LMS/EPSRC Durham Symposium Higher Structures in M-Theory, August 2018, references update