1st Edition

Virtual Topology and Functor Geometry

By Fred Van Oystaeyen Copyright 2008
    168 Pages 51 B/W Illustrations
    by Chapman & Hall

    168 Pages
    by Chapman & Hall

    Intrinsically noncommutative spaces today are considered from the perspective of several branches of modern physics, including quantum gravity, string theory, and statistical physics. From this point of view, it is ideal to devise a concept of space and its geometry that is fundamentally noncommutative. Providing a clear introduction to noncommutative topology, Virtual Topology and Functor Geometry explores new aspects of these areas as well as more established facets of noncommutative algebra.

    Presenting the material in an easy, colloquial style to facilitate understanding, the book begins with an introduction to category theory, followed by a chapter on noncommutative spaces. This chapter examines noncommutative lattices, noncommutative opens, sheaf theory, the generalized Stone space, and Grothendieck topology. The author then studies Grothendieck categorical representations to formulate an abstract notion of "affine open". The final chapter proposes a dynamical version of topology and sheaf theory, providing at least one solution of the problem of sheafification independent of generalizations of topos theory.

    By presenting new ideas for the development of an intrinsically noncommutative geometry, this book fosters the further unification of different kinds of noncommutative geometry and the expression of observations that involve natural phenomena.

    FOREWORD
    INTRODUCTION
    PROJECTS

    A TASTE OF CATEGORY THEORY
    Basic Notions
    Grothendieck Categories
    Separable Functors

    NONCOMMUTATIVE SPACES
    Small Categories, Posets, and Noncommutative Topologies
    The Topology of Virtual Opens and Its Commutative Shadow
    Points and the Point Spectrum: Points in a Pointless World
    Presheaves and Sheaves over Noncommutative Topologies
    Noncommutative Grothendieck Topologies
    The Fundamental Examples I: Torsion Theories
    The Fundamental Examples II: L(H)
    Ore Sets in Schematic Algebras

    GROTHENDIECK CATEGORICAL REPRESENTATIONS
    Spectral Representations
    Affine Elements
    Quotient Representations
    Noncommutative Projective Space

    SHEAVES AND DYNAMICAL TOPOLOGY
    Introducing Structure Sheaves
    Dynamical Presheaves and Temporal Points
    The Spaced-Time Model

    BIBLIOGRAPHY
    INDEX

    Biography

    Fred Van Oystaeyen

    "This book has a special character. Its main theme is to describe development of new branches of non-commutative geometry on a different level of realizations, ranging from areas already fully developed to many different suggestions for possible future investigations. … the book is very inspiring and worth reading."
    EMS Newsletter, December 2009