Virtual Topology and Functor Geometry: 1st Edition (Paperback) book cover

Virtual Topology and Functor Geometry

1st Edition

By Fred Van Oystaeyen

Chapman and Hall/CRC

168 pages | 51 B/W Illus.

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pub: 2007-11-15
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Description

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.

Reviews

"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

Table of Contents

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

About the Series

Lecture Notes in Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT002000
MATHEMATICS / Algebra / General
MAT012000
MATHEMATICS / Geometry / General
SCI040000
SCIENCE / Mathematical Physics