Synthetic Differential Topology

Marta Bunge; Felipe Gago; Ana María San Luis

doi:10.1017/9781108553490

This book formally introduces synthetic differential topology, a natural extension of the theory of synthetic differential geometry which captures classical concepts of differential geometry and topology by means of the rich categorical structure of a necessarily non-Boolean topos and of the systematic use of logical infinitesimal objects in it. Beginning with an introduction to those parts of topos theory and synthetic differential geometry necessary for the remainder, this clear and comprehensive text covers the general theory of synthetic differential topology and several applications of it to classical mathematics, including the calculus of variations, Mather's theorem, and Morse theory on the classification of singularities. The book represents the state of the art in synthetic differential topology and will be of interest to researchers in topos theory and to mathematicians interested in the categorical foundations of differential geometry and topology.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-vi
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Preface
pp vii-viii
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Acknowledgements
pp ix-x
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Introduction
pp 1-6
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PART I - TOPOSES AND DIFFERENTIAL GEOMETRY
pp 7-8
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1 - Topos Theory
pp 9-29
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2 - Synthetic Differential Geometry
pp 30-50
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PART II - TOPICS IN SDG
pp 51-52
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3 - The Ambrose-Palais-Singer Theorem in SDG
pp 53-70
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4 - Calculus of Variations in SDG
pp 71-86
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PART III - TOPOSES AND DIFFERENTIAL TOPOLOGY
pp 87-88
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5 - Local Concepts in SDG
pp 89-105
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6 - Synthetic Differential Topology
pp 106-124
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PART IV - TOPICS IN SDT
pp 125-126
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7 - Stable Mappings and Mather's Theorem in SDT
pp 127-149
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8 - Morse Theory in SDT
pp 150-158
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PART V - SDT AND DIFFERENTIAL TOPOLOGY
pp 159-160
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9 - Well Adapted Models of SDT
pp 161-176
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10 - An Application to Unfoldings
pp 177-180
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PART VI - A WELL ADAPTED MODEL OF SDT
pp 181-182
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11 - The Dubuc Topos G
pp 183-201
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12 - G as a Model of SDT
pp 202-214
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References
pp 215-220
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Index
pp 221-223
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Synthetic Differential Topology

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Contents

Frontmatter
pp i-iv

Contents
pp v-vi

Preface
pp vii-viii

Acknowledgements
pp ix-x

Introduction
pp 1-6

PART I - TOPOSES AND DIFFERENTIAL GEOMETRY
pp 7-8

1 - Topos Theory
pp 9-29

2 - Synthetic Differential Geometry
pp 30-50

PART II - TOPICS IN SDG
pp 51-52

3 - The Ambrose-Palais-Singer Theorem in SDG
pp 53-70

4 - Calculus of Variations in SDG
pp 71-86

PART III - TOPOSES AND DIFFERENTIAL TOPOLOGY
pp 87-88

5 - Local Concepts in SDG
pp 89-105

6 - Synthetic Differential Topology
pp 106-124

PART IV - TOPICS IN SDT
pp 125-126

7 - Stable Mappings and Mather's Theorem in SDT
pp 127-149

8 - Morse Theory in SDT
pp 150-158

PART V - SDT AND DIFFERENTIAL TOPOLOGY
pp 159-160

9 - Well Adapted Models of SDT
pp 161-176

10 - An Application to Unfoldings
pp 177-180

PART VI - A WELL ADAPTED MODEL OF SDT
pp 181-182

11 - The Dubuc Topos G
pp 183-201

12 - G as a Model of SDT
pp 202-214

References
pp 215-220

Index
pp 221-223

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Synthetic Differential Topology

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