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El. knyga: Virtual Topology and Functor Geometry

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(University of Antwerp, Belgium)
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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.

Recenzijos

"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

Foreword ix
Acknowledgments xi
Introduction xiii
Projects xvii
1 A Taste of Category Theory 1
1.1 Basic Notions
1
1.1.1 Examples and Notation
1
1.2 Grothendieck Categories
5
1.3 Separable Functors
9
2 Noncommutative Spaces 11
2.1 Small Categories, Posets, and Noncommutative Topologies
11
2.1.1 Sheaves over Posets
13
2.1.2 Directed Subsets and the Limit Poset
14
2.1.3 Poset Dynamics
16
2.2 The Topology of Virtual Opens and Its Commutative Shadow
19
2.2.1 Properties
20
2.2.2 Projects
28
2.2.2.1 More Noncommutative Topology
28
2.2.2.2 Some Dimension Theory
28
2.3 Points and the Point Spectrum: Points in a Pointless World
29
2.3.1 Projects
35
2.3.1.1 The Relation between Quantum Points and Strong Idempotents
35
2.3.1.2 Functions on Sets of Quantum Points
36
2.4 Presheaves and Sheaves over Noncommutative Topologies
36
2.4.1 Project: Quantum Points and Sheaves
39
2.5 Noncommutative Grothendieck Topologies
40
2.5.1 Warning
43
2.5.2 Projects
44
2.5.2.1 A Noncommutative Topos Theory
44
2.5.2.2 Noncommutative Probability (and Measure) Theory
44
2.5.2.3 Covers and Cohomology Theories
45
2.5.2.4 The Derived Imperative
45
2.6 The Fundamental Examples I: Torsion Theories
45
2.6.1 Project: Microlocalization in a Grothendieck Category
63
2.7 The Fundamental Examples II: L(H)
64
2.7.1 The Generalized Stone Topology
67
2.7.2 Note
69
2.7.3 Project: Noncommutative Gelfand Duality
73
2.8 Ore Sets in Schematic Algebras
73
3 Grothendieck Categorical Representations 79
3.1 Spectral Representations
79
3.2 Affine Elements
94
3.2.1 Observation and Example
96
3.3 Quotient Representations
96
3.3.1 Project: Geometrically Graded Rings
100
3.4 Noncommutative Projective Space
104
3.4.1 Project: Extended Theory for Gabriel Dimension
107
3.4.2 Properties of Gabriel Dimension
108
3.4.3 Project: General Birationality
110
4 Sheaves and Dynamical Topology 111
4.1 Introducing Structure Sheaves
111
4.1.1 Classical Example and Motivation
113
4.1.2 Abstract Noncommutative Spaces and Schemes
113
4.1.3 Project: Replacing Essential by Separable Functors
119
4.1.4 Example: Ore Sets in Schematic Algebras
119
4.2 Dynamical Presheaves and Temporal Points
121
4.2.1 Project: Monads in Bicategories
122
4.2.2 Project: Spectral Families on the Spectrum
133
4.2.3 Project: Temporal tech and Sheaf Cohomology
134
4.2.3.1 Subproject 1: Temporal Grothendieck Representations
134
4.2.3.2 Subproject 2: Temporal tech Cohomology and Sheaf Cohomology
134
4.2.4 Project: Dynamical Grothendieck Topologies
135
4.2.5 Conjecture
136
4.3 The Spaced-Time Model
137
4.3.1 Noncommutative Manifolds
137
4.3.1.1 Toward Real Noncommutative Manifolds
140
4.3.2 Food for Thought: From Physics to Philosophy
141
Bibliography 143
Index 147
University of Antwerp, Belgium
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