Virtual Topology and Functor Geometry

ISBN-13: 9781420060560 / Angielski / Miękka / 2007 / 168 str.

Fred Van Oystaeyen; F. Van Oystaeyen

Opis

In today's research, intrinsically noncommutative spaces 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 seems necessary to have a concept of space and its geometry that is fundamentally noncommutative. Virtual Topology and Functor Geometry presents new ideas for the development of an intrinsically noncommutative geometry. Written in an easily accessible, colloquial style, the book provides numerous suggestions for possible projects, ranging from exercise level to research level. Topics covered include Grothendieck categorical representations and structure sheaves.

Recenzje

"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

Spis treści

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

O autorze

Fred Van Oystaeyen