ISBN-10:
0521760364
ISBN-13:
9780521760362
Pub. Date:
11/30/2009
Publisher:
Cambridge University Press
Directed Algebraic Topology: Models of Non-Reversible Worlds

Directed Algebraic Topology: Models of Non-Reversible Worlds

by Marco Grandis

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Product Details

ISBN-13: 9780521760362
Publisher: Cambridge University Press
Publication date: 11/30/2009
Series: New Mathematical Monographs Series , #13
Pages: 444
Product dimensions: 6.20(w) x 9.00(h) x 1.20(d)

About the Author

Marco Grandis is Professor in the Department of Mathematics at the University of Genoa, Italy.

Table of Contents

Introduction 1

1 Aims and applications 1

2 Some examples 2

3 Directed spaces and other directed structures 3

4 Formal foundations for directed algebraic topology 5

5 Interactions with category theory 6

6 Interactions with non-commutative geometry 7

7 From directed to weighted algebraic topology 7

8 Terminology and notation 8

9 Acknowledgements 9

Part I First-order directed homotopy and homology 11

1 Directed structures and first-order homotopy properties 13

1.1 From classical homotopy to the directed case 14

1.2 The basic structure of the directed cylinder and cocylinder 28

1.3 First-order homotopy theory by the cylinder functor, I 40

1.4 Topological spaces with distinguished paths 50

1.5 The basic homotopy structure of d-spaces 61

1.6 Cubical sets 65

1.7 First-order homotopy theory by the cylinder functor, II 79

1.8 First-order homotopy theory by the path functor 89

1.9 Other topological settings 97

2 Directed homology and non-commutative geometry 105

2.1 Directed homology of cubical sets 106

2.2 Properties of the directed homology of cubical sets 114

2.3 Pointed homotopy and homology of cubical sets 120

2.4 Group actions on cubical sets 127

2.5 Interactions with non-commutative geometry 130

2.6 Directed homology theories 140

3 Modelling the fundamental category 145

3.1 Higher properties of homotopies of d-spaces 146

3.2 The fundamental category of a d-space 153

3.3 Future and past equivalences of categories 165

3.4 Bilateral directed equivalences of categories 177

3.5 Injective and projective models of categories 185

3.6 Minimal models of a category 193

3.7 Future invariant properties 199

3.8 Spectra and pf-equivalence of categories 206

3.9 A gallery of spectra and models 214

Part II Higher directed homotopy theory 227

4 Settings for higher order homotopy 229

4.1 Preserving homotopies and transposition 230

4.2 A strong setting for directed homotopy 239

4.3 Examples, I 250

4.4 Examples, II. Chain complexes 254

4.5 Double homotopies and the fundamental category 262

4.6 Higher properties of h-pushouts and cofibrations 269

4.7 Higher properties of cones and Puppe sequences 277

4.8 The cone monad 283

4.9 The reversible case 290

5 Categories of functors and algebras, relative settings 296

5.1 Directed homotopy of diagrams and sheaves 297

5.2 Directed homotopy in slice categories 301

5.3 Algebras for a monad and the path functor 309

5.4 Applications to d-spaces and small categories 319

5.5 The path functor of differential graded algebras 327

5.6 Higher structure and cylinder of dg-algebras 334

5.7 Cochain algebras as internal semigroups 342

5.8 Relative settings based on forgetful functors 345

6 Elements of weighted algebraic topology 351

6.1 Generalised metric spaces 352

6.2 Elementary and extended homotopies 362

6.3 The fundamental weighted category 366

6.4 Minimal models 373

6.5 Spaces with weighted paths 376

6.6 Linear and metrisable w-spaces 387

6.7 Weighted non-commutative tori 391

6.8 Tentative formal settings for the weighted case 394

Appendix A Some points of category theory 397

A1 Basic notions 397

A2 Limits and colimits 405

A3 Adjoint functors 407

A4 Monoidal categories, monads, additive categories 410

A5 Two-dimensional categories and mates 414

References 418

Glossary of symbols 424

Index 427

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