The Hopf algebra of Möbius intervals

An unpublished result by the first author states that there exists a Hopf algebra H such that for any Moebius category C (in the sense of Leroux) there exists a canonical algebra morphism from the dual H* of H to the incidence algebra of C. Moreover, the Moebius inversion principle in incidence algebras follows from a `master´ inversion result in H*. The underlying module of H was originally defined as the free module on the set of iso classes of Moebius intervals, i.e. Moebius categories with initial and terminal objects. Here we consider a category of Moebius intervals and construct the Hopf algebra via the objective approach applied to a monoidal extensive category of combinatorial objects, with the values in appropriate rings being abstracted from combinatorial functors on the objects. The explicit consideration of a category of Moebius intervals leads also to two new characterizations of Moebius categories.Fil: Lawvere, F. W.. No especifíca;Fil: Menni, Matías. Ministerio de Educación, Cultura, Ciencia y Tecnología. Secretaria de Gobierno de Ciencia Tecnología e Innovación Productiva. Agencia Nacional de Promoción Científica y Tecnológica. Fondo Argentino Sectorial; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

The Hopf algebra of Möbius intervals

An unpublished result by the first author states that there exists a Hopf algebra H such that for any Moebius category C (in the sense of Leroux) there exists a canonical algebra morphism from the dual H* of H to the incidence algebra of C. Moreover, the Moebius inversion principle in incidence algebras follows from a `master´ inversion result in H*. The underlying module of H was originally defined as the free module on the set of iso classes of Moebius intervals, i.e. Moebius categories with initial and terminal objects. Here we consider a category of Moebius intervals and construct the Hopf algebra via the objective approach applied to a monoidal extensive category of combinatorial objects, with the values in appropriate rings being abstracted from combinatorial functors on the objects. The explicit consideration of a category of Moebius intervals leads also to two new characterizations of Moebius categories.Fil: Lawvere, F. W.. No especifíca;Fil: Menni, Matías. Ministerio de Educación, Cultura, Ciencia y Tecnología. Secretaria de Gobierno de Ciencia Tecnología e Innovación Productiva. Agencia Nacional de Promoción Científica y Tecnológica. Fondo Argentino Sectorial; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Deriving Bisimulation Congruences: 2-categories vs precategories

G-relative pushouts (GRPOs) have recently been proposed by the authors as a new foundation for Leifer and Milner’s approach to deriving labelled bisimulation congruences from reduction systems. This paper develops the theory of GRPOs further, arguing that they provide a simple and powerful basis towards a comprehensive solution. As an example, we construct GRPOs in a category of ‘bunches and wirings.’ We then examine the approach based on Milner’s precategories and Leifer’s functorial reactive systems, and show that it can be recast in a much simpler way into the 2-categorical theory of GRPOs

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

Kripke Semantics for Martin-L\"of's Extensional Type Theory

It is well-known that simple type theory is complete with respect to non-standard set-valued models. Completeness for standard models only holds with respect to certain extended classes of models, e.g., the class of cartesian closed categories. Similarly, dependent type theory is complete for locally cartesian closed categories. However, it is usually difficult to establish the coherence of interpretations of dependent type theory, i.e., to show that the interpretations of equal expressions are indeed equal. Several classes of models have been used to remedy this problem. We contribute to this investigation by giving a semantics that is standard, coherent, and sufficiently general for completeness while remaining relatively easy to compute with. Our models interpret types of Martin-L\"of's extensional dependent type theory as sets indexed over posets or, equivalently, as fibrations over posets. This semantics can be seen as a generalization to dependent type theory of the interpretation of intuitionistic first-order logic in Kripke models. This yields a simple coherent model theory, with respect to which simple and dependent type theory are sound and complete

Bisimilarity congruences for open terms and term graphs via tile logic

The definition of sos formats ensuring that bisimilarity on closed terms is a congruence has received much attention in the last two decades. For dealing with open terms, the congruence is usually lifted from closed terms by instantiating the free variables in all possible ways; the only alternatives considered in the literature are Larsen and Xinxin’s context systems and Rensink’s conditional transition systems. We propose an approach based on tile logic, where closed and open terms are managed uniformly, and study the ‘bisimilarity as congruence’ property for several tile formats, accomplishing different concepts of open system

Shall We (Math and) Dance?

Can we use mathematics, and in particular the abstract branch of category theory, to describe some basics of dance, and to highlight structural similarities between music and dance? We first summarize recent studies between mathematics and dance, and between music and categories. Then, we extend this formalism and diagrammatic thinking style to dance.Comment: preprin

Topos theory and `neo-realist' quantum theory

Topos theory, a branch of category theory, has been proposed as mathematical basis for the formulation of physical theories. In this article, we give a brief introduction to this approach, emphasising the logical aspects. Each topos serves as a `mathematical universe' with an internal logic, which is used to assign truth-values to all propositions about a physical system. We show in detail how this works for (algebraic) quantum theory.Comment: 22 pages, no figures; contribution for Proceedings of workshop "Recent Developments in Quantum Field Theory", MPI MIS Leipzig, July 200

Topos Theory and Consistent Histories: The Internal Logic of the Set of all Consistent Sets

A major problem in the consistent-histories approach to quantum theory is contending with the potentially large number of consistent sets of history propositions. One possibility is to find a scheme in which a unique set is selected in some way. However, in this paper we consider the alternative approach in which all consistent sets are kept, leading to a type of `many world-views' picture of the quantum theory. It is shown that a natural way of handling this situation is to employ the theory of varying sets (presheafs) on the space \B of all Boolean subalgebras of the orthoalgebra \UP of history propositions. This approach automatically includes the feature whereby probabilistic predictions are meaningful only in the context of a consistent set of history propositions. More strikingly, it leads to a picture in which the `truth values', or `semantic values' of such contextual predictions are not just two-valued (\ie true and false) but instead lie in a larger logical algebra---a Heyting algebra---whose structure is determined by the space \B of Boolean subalgebras of \UP.Comment: 28 pages, LaTe

Quantum Picturalism

The quantum mechanical formalism doesn't support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. In this review we present steps towards a diagrammatic `high-level' alternative for the Hilbert space formalism, one which appeals to our intuition. It allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the no-cloning theorem, and phenomena such as quantum teleportation. As a logic, it supports `automation'. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required step-stone towards a deeper conceptual understanding of quantum theory, as well as its unification with other physical theories. Specific applications discussed here are purely diagrammatic proofs of several quantum computational schemes, as well as an analysis of the structural origin of quantum non-locality. The underlying mathematical foundation of this high-level diagrammatic formalism relies on so-called monoidal categories, a product of a fairly recent development in mathematics. These monoidal categories do not only provide a natural foundation for physical theories, but also for proof theory, logic, programming languages, biology, cooking, ... The challenge is to discover the necessary additional pieces of structure that allow us to predict genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures, some colo