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  1.  29
    What Is the Validity Domain of Einstein’s Equations? Distributional Solutions Over Singularities and Topological Links in Geometrodynamics.Elias Zafiris - 2016 - 100 Years of Chronogeometrodynamics: The Status of the Einstein's Theory of Gravitation in Its Centennial Year.
    The existence of singularities alerts that one of the highest priorities of a centennial perspective on general relativity should be a careful re-thinking of the validity domain of Einstein’s field equations. We address the problem of constructing distinguishable extensions of the smooth spacetime manifold model, which can incorporate singularities, while retaining the form of the field equations. The sheaf-theoretic formulation of this problem is tantamount to extending the algebra sheaf of smooth functions to a distribution-like algebra sheaf in which the (...)
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  2.  65
    A Categorial Semantic Representation of Quantum Event Structures.Elias Zafiris & Vassilios Karakostas - 2013 - Foundations of Physics 43 (9):1090-1123.
    The overwhelming majority of the attempts in exploring the problems related to quantum logical structures and their interpretation have been based on an underlying set-theoretic syntactic language. We propose a transition in the involved syntactic language to tackle these problems from the set-theoretic to the category-theoretic mode, together with a study of the consequent semantic transition in the logical interpretation of quantum event structures. In the present work, this is realized by representing categorically the global structure of a quantum algebra (...)
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  3.  5
    Foundations of Relational Realism: A Topological Approach to Quantum Mechanics and the Philosophy of Nature.Michael Epperson & Elias Zafiris - 2013 - Lexington Books.
    Foundations of Relational Realism presents an intuitive interpretation of quantum mechanics, based on a revised decoherent histories interpretation, structured within a category theoretic topological formalism. -/- If there is a central conceptual framework that has reliably borne the weight of modern physics as it ascends into the twenty-first century, it is the framework of quantum mechanics. Because of its enduring stability in experimental application, physics has today reached heights that not only inspire wonder, but arguably exceed the limits of intuitive (...)
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  4.  5
    Rosen's Modelling Relations Via Categorical Adjunctions.Elias Zafiris - 2012 - International Journal of General Systems 41 (5):439-474.
    Rosen's modelling relations constitute a conceptual schema for the understanding of the bidirectional process of correspondence between natural systems and formal symbolic systems. The notion of formal systems used in this study refers to information structures constructed as algebraic rings of observable attributes of natural systems, in which the notion of observable signifies a physical attribute that, in principle, can be measured. Due to the fact that modelling relations are bidirectional by construction, they admit a precise categorical formulation in terms (...)
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  5.  4
    Sheaf-Theoretic Representation of Quantum Measure Algebras.Elias Zafiris - 2006 - Journal of Mathematical Physics 47 (9).
    We construct a sheaf-theoretic representation of quantum probabilistic structures, in terms of covering systems of Boolean measure algebras. These systems coordinatize quantum states by means of Boolean coefficients, interpreted as Boolean localization measures. The representation is based on the existence of a pair of adjoint functors between the category of presheaves of Boolean measure algebras and the category of quantum measure algebras. The sheaf-theoretic semantic transition of quantum structures shifts their physical significance from the orthoposet axiomatization at the level of (...)
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  6. Generalized Topological Covering Systems on Quantum Events' Structures.Elias Zafiris - 2006 - Journal of Physics A: Mathematics and Applications 39 (6):1485-1505.
    Homologous operational localization processes are effectuated in terms of generalized topological covering systems on structures of physical events. We study localization systems of quantum events' structures by means of Gtothendieck topologies on the base category of Boolean events' algebras. We show that a quantum events algebra is represented by means of a Grothendieck sheaf-theoretic fibred structure, with respect to the global partial order of quantum events' fibres over the base category of local Boolean frames.
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  7.  4
    Quantum Observables Algebras and Abstract Differential Geometry: The Topos-Theoretic Dynamics of Diagrams of Commutative Algebraic Localizations.Elias Zafiris - 2007 - International Journal of Theoretical Physics 46 (2):319-382.
    We construct a sheaf-theoretic representation of quantum observables algebras over a base category equipped with a Grothendieck topology, consisting of epimorphic families of commutative observables algebras, playing the role of local arithmetics in measurement situations. This construction makes possible the adaptation of the methodology of Abstract Differential Geometry (ADG), à la Mallios, in a topos-theoretic environment, and hence, the extension of the “mechanism of differentials” in the quantum regime. The process of gluing information, within diagrams of commutative algebraic localizations, generates (...)
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  8.  3
    Interpreting Observables in a Quantum World From the Categorial Standpoint.Elias Zafiris - 2004 - International Journal of Theoretical Physics 43 (1):265-298.
    We develop a relativistic perspective on structures of quantum observables, in terms of localization systems of Boolean coordinatizing charts. This perspective implies that the quantum world is comprehended via Boolean reference frames for measurement of observables, pasted together along their overlaps. The scheme is formalized categorically, as an instance of the adjunction concept. The latter is used as a framework for the specification of a categorical equivalence signifying an invariance in the translational code of communication between Boolean localizing contexts and (...)
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  9. Complex Systems From the Perspective of Category Theory: II. Covering Systems and Sheaves.Elias Zafiris - 2005 - Axiomathes 15 (2):181-190.
    Using the concept of adjunction, for the comprehension of the structure of a complex system, developed in Part I, we introduce the notion of covering systems consisting of partially or locally defined adequately understood objects. This notion incorporates the necessary and sufficient conditions for a sheaf theoretical representation of the informational content included in the structure of a complex system in terms of localization systems. Furthermore, it accommodates a formulation of an invariance property of information communication concerning the analysis of (...)
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  10. Complex Systems From the Perspective of Category Theory: I. Functioning of the Adjunction Concept.Elias Zafiris - 2005 - Axiomathes 15 (1):147-158.
    We develop a category theoretical scheme for the comprehension of the information structure associated with a complex system, in terms of families of partial or local information carriers. The scheme is based on the existence of a categorical adjunction, that provides a theoretical platform for the descriptive analysis of the complex system as a process of functorial information communication.
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  11.  5
    On Quantum Event Structures. III. Object of Truth Values.Elias Zafiris - 2004 - Foundations Of Physics Letters 17 (5):403-432.
    In this work we expand the foundational perspective of category theory on quantum event structures by showing the existence of an object of truth values in the category of quantum event algebras, characterized as subobject classifier. This object plays the corresponking role that the two-valued Boolean truth values object plays in a classical event structure. We construct the object of quantum truth values explicitly and argue that it constitutes the appropriate choice for the valuation of propositions describing the behavior of (...)
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  12.  5
    On Quantum Event Structures. II. Interpretational Aspects.Elias Zafiris - 2001 - Foundations Of Physics Letters 14 (2):167-177.
    In this paper we analyze the physical semantics and propose an interpretation of quantum event structures from the perspective offered by the categorical scheme of Part I.
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  13.  21
    Quantum Event Structures From the Perspective of Grothendieck Topoi.Elias Zafiris - 2004 - Foundations of Physics 34 (7):1063-1090.
  14.  3
    Category-Theoretic Analysis of the Notion of Complementarity for Quantum Systems.Elias Zafiris - 2006 - International Journal of General Systems 35 (1):69-89.
    In this paper we adopt a category-theoretic viewpoint in order to analyze the semantics of complementarity for quantum systems. Based on the existence of a pair of adjoint functors between the topos of presheaves of the Boolean kind of structure and the category of the quantum kind of structure, we establish a twofold complementarity scheme which constitutes an instance of the concept of adjunction. It is further argued that the established scheme is inextricably connected with a realistic philosophical attitude, although (...)
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  15.  26
    Contextual Semantics in Quantum Mechanics From a Categorical Point of View.Vassilios Karakostas & Elias Zafiris - 2017 - Synthese 194 (3).
    The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valuation in quantum mechanics as exemplified, in particular, by Kochen–Specker’s theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event algebras. We show explicitly that the latter category is equipped with an object of truth values, (...)
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  16.  2
    Boolean Information Sieves: A Local-to-Global Approach to Quantum Information.Elias Zafiris - 2010 - International Journal of General Systems 39 (8):873-895.
    We propose a sheaf-theoretic framework for the representation of a quantum observable structure in terms of Boolean information sieves. The algebraic representation of a quantum observable structure in the relational local terms of sheaf theory effectuates a semantic transition from the axiomatic set-theoretic context of orthocomplemented partially ordered sets, la Birkhoff and Von Neumann, to the categorical topos-theoretic context of Boolean information sieves, la Grothendieck. The representation schema is based on the existence of a categorical adjunction, which is used as (...)
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  17.  2
    Probing Quantum Structure with Boolean Localization Systems.Elias Zafiris - 2000 - International Journal of Theoretical Physics 39 (12):2761-2778.
    In an attempt to probe the objects belonging to the quantum species of structure,we develop the idea of using observables of the Boolean species of structures,as coordinatizing objects in the quantum world. This results in a contextualisticperspective on the latter through local Boolean measurement reference frames.The semantics of this representation is discussed extensively.
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  18.  9
    Boolean Localization Of Quantum Events: A Processual Sheaf-Theoretic Approach.Elias Zafiris - 2016 - In David Ray Griffin, Michael Epperson & Timothy E. Eastman (eds.), Physics and Speculative Philosophy: Potentiality in Modern Science. De Gruyter. pp. 107-126.
  19.  1
    On Quantum Event Structures. I. The Categorical Scheme.Elias Zafiris - 2001 - Foundations Of Physics Letters 14 (2):147-166.
    In this paper a mathematical scheme for the analysis of quantum event structures is being proposed based on category theoretical methods. It is shown that there exists an adjunctive correspondence between Boolean presheaves of event algebras and quantum event algebras. The adjunction permits a characterization of quantum event structures as Boolean manifolds of event structures. -/- .
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  20.  1
    Loops, Projective Invariants, and the Realization of the Borromean Topological Link in Quantum Mechanics.Elias Zafiris - 2016 - Quantum Studies: Mathematics and Foundations 3 (4):337-359.
    All the typical global quantum mechanical observables are complex relative phases obtained by interference phenomena. They are described by means of some global geometric phase factor, which is thought of as the “memory” of a quantum system undergoing a “cyclic evolution” after coming back to its original physical state. The origin of a geometric phase factor can be traced to the local phase invariance of the transition probability assignment in quantum mechanics. Beyond this invariance, transition probabilities also remain invariant under (...)
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  21. On the Notion of Truth in Quantum Mechanics: A Category-Theoretic Standpoint.Vassilios Karakostas & Elias Zafiris - 2016 - In Diederik Aerts, Christian de Ronde, Hector Freytes & Roberto Giuntini (eds.), Probing the Meaning and Structure of Quantum Mechanics: Semantics, Dynamics and Identity. World Scientific. pp. 1-43.
    The category-theoretic representation of quantum event structures provides a canonical setting for confronting the fundamental problem of truth valua- tion in quantum mechanics as exemplified, in particular, by Kochen-Specker’s theorem. In the present study, this is realized on the basis of the existence of a categorical adjunction between the category of sheaves of variable local Boolean frames, constituting a topos, and the category of quantum event al- gebras. We show explicitly that the latter category is equipped with an object of (...)
     
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  22.  4
    Differential Sheaves and Connections: A Natural Approach to Physical Geometry.Anastasios Mallios & Elias Zafiris - 2015 - World Scientific.
    This unique book provides a self-contained conceptual and technical introduction to the theory of differential sheaves. This serves both the newcomer and the experienced researcher in undertaking a background-independent, natural and relational approach to "physical geometry". In this manner, this book is situated at the crossroads between the foundations of mathematical analysis with a view toward differential geometry and the foundations of theoretical physics with a view toward quantum mechanics and quantum gravity. The unifying thread is provided by the theory (...)
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  23.  10
    Concept and Formalization of Constellatory Self-Unfolding: A Novel Perspective on the Relation Between Quantum and Relativistic Physics.Albrecht Von Müller & Elias Zafiris - 2018 - Springer.
    This volume develops a fundamentally different categorical framework for conceptualizing time and reality. The actual taking place of reality is conceived as a “constellatory self-unfolding” characterized by strong self-referentiality and occurring in the primordial form of time, the not yet sequentially structured “time-space of the present.” Concomitantly, both the sequentially ordered aspect of time and the factual aspect of reality appear as emergent phenomena that come into being only after reality has actually taken place. In this new framework, time functions (...)
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  24. Categorical Modeling of Natural Complex Systems. Part I: Functorial Process of Representation.Elias Zafiris - 2008 - Advances in Systems Science and Applications 8 (2):187-200.
    We develop a general covariant categorical modeling theory of natural systems’ behavior based on the fundamental functorial processes of representation and localization-globalization. In the first part of this study we analyze the process of representation. Representation constitutes a categorical modeling relation that signifies the semantic bidirectional process of correspondence between natural systems and formal symbolic systems. The notion of formal systems is substantiated by algebraic rings of observable attributes of natural systems. In this perspective, the distinction between simple and complex (...)
     
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  25. Categorical Modeling of Natural Complex Systems. Part II: Functorial Process of Localization-Globalization.Elias Zafiris - 2008 - Advances in Systems Science and Applications 8 (3):367-387.
    We develop a general covariant categorical modeling theory of natural systems' behavior based on the fundamental functorial processes of representation and localization-globalization. In the second part of this study we analyze the semantic bidirectional process of localization-globalization. The notion of a localization system of a complex information structure bears a dual role: Firstly, it determines the appropriate categorical environment of base reference contexts for considering the operational modeling of a complex system's behavior, and secondly, it specifies the global compatibility conditions (...)
     
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  26. The Nature of Local/Global Distinctions, Group Actions and Phases: A Sheaf=Theoretic Approach to Quantum Geometric Spectra.Elias Zafiris - 2015 - In Vera Bühlmann, Ludger Hovestadt & Vahid Moosavi (eds.), Coding as Literacy - Metalithicum IV. Basel: BIRKHÄUSER. pp. 172-186.