From Simple to Complex and Ultra-complex Systems:\ud A Paradigm Shift Towards Non-Abelian Systems Dynamics

Atoms, molecules, organisms distinguish layers of reality because of the causal links that govern their behavior, both horizontally (atom-atom, molecule-molecule, organism-organism) and vertically (atom-molecule-organism). This is the first intuition of the theory of levels. Even if the further development of the theory will require imposing a number of qualifications to this initial intuition, the idea of a series of entities organized on different levels of complexity will prove correct. Living systems as well as social systems and the human mind present features remarkably different from those characterizing non-living, simple physical and chemical systems. We propose that super-complexity requires at least four different categorical frameworks, provided by the theories of levels of reality, chronotopoids, (generalized) interactions, and anticipation

From Simple to Complex and Ultra-complex Systems:\ud A Paradigm Shift Towards Non-Abelian Systems Dynamics

Atoms, molecules, organisms distinguish layers of reality because of the causal links that govern their behavior, both horizontally (atom-atom, molecule-molecule, organism-organism) and vertically (atom-molecule-organism). This is the first intuition of the theory of levels. Even if the further development of the theory will require imposing a number of qualifications to this initial intuition, the idea of a series of entities organized on different levels of complexity will prove correct. Living systems as well as social systems and the human mind present features remarkably different from those characterizing non-living, simple physical and chemical systems. We propose that super-complexity requires at least four different categorical frameworks, provided by the theories of levels of reality, chronotopoids, (generalized) interactions, and anticipation

Non-commutative fermion mass matrix and gravity

The first part is an introductory description of a small cross-section of the literature on algebraic methods in non-perturbative quantum gravity with a specific focus on viewing algebra as a laboratory in which to deepen understanding of the nature of geometry. This helps to set the context for the second part, in which we describe a new algebraic characterisation of the Dirac operator in non-commutative geometry and then use it in a calculation on the form of the fermion mass matrix. Assimilating and building on the various ideas described in the first part, the final part consists of an outline of a speculative perspective on (non-commutative) quantum spectral gravity. This is the second of a pair of papers so far on this project.Comment: To appear in Int. J. Mod. Phys. A Previous title: An outlook on quantum gravity from an algebraic perspective. 39 pages, 1 xy-pic figure, LaTex Reasons for new version: added references, change of title and some comments more up-to-dat

Regular obstructed categories and TQFT

A proposal of the concept of $n$-regular obstructed categories is given. The corresponding regularity conditions for mappings, morphisms and related structures in categories are considered. An n-regular TQFT is introduced. It is shown the connection of time reversibility with the regularity.Comment: 22 pages in Latex. To be published in J. Math. Phy

Categories for Dynamic Epistemic Logic

The primary goal of this paper is to recast the semantics of modal logic, and dynamic epistemic logic (DEL) in particular, in category-theoretic terms. We first review the category of relations and categories of Kripke frames, with particular emphasis on the duality between relations and adjoint homomorphisms. Using these categories, we then reformulate the semantics of DEL in a more categorical and algebraic form. Several virtues of the new formulation will be demonstrated: The DEL idea of updating a model into another is captured naturally by the categorical perspective -- which emphasizes a family of objects and structural relationships among them, as opposed to a single object and structure on it. Also, the categorical semantics of DEL can be merged straightforwardly with a standard categorical semantics for first-order logic, providing a semantics for first-order DEL.Comment: In Proceedings TARK 2017, arXiv:1707.0825

Effective Theory of Braid Excitations of Quantum Geometry in terms of Feynman Diagrams

We study interactions amongst topologically conserved excitations of quantum theories of gravity, in particular the braid excitations of four-valent spin networks. These have been shown previously to propagate and interact under evolution rules of spin foam models. We show that the dynamics of these braid excitations can be described by an effective theory based on Feynman diagrams. In this language, braids which are actively interacting are analogous to bosons, in that the topological conservation laws permit them to be singly created and destroyed. Exchanges of these excitations give rise to interactions between braids which are charged under the topological conservation rules.Comment: 23 pages, 7 figures. Accepted by Nucl. Phys.

Remarks on Quantum Statistics II

Some problems related to an algebraic approach to quantum statistics are discussed. Generalized quantum statistics is described as a result of interactions. The Fock space representation is discussed. The problem of existence of well--defined scalar product is considered. An example of physical effect in system with generalized statistics is also given.Comment: 14 pages in Latex, This paper is an revisited version of the paper in Proceedings of the Conference "Particles, Fields and Gravitation", ed. by J. Rembielinski, World Scientific, Singapore 1998, April 15 - 19, (1998), Lodz, Polan