Synthetic Differential Geometry / Edition 2

by Anders Kock

Overview

Synthetic Differential Geometry is a method of reasoning in differential geometry and differential calculus, based on the assumption of sufficiently many nilpotent elements on the number line, in particular numbers d such that d2=0. The use of nilpotent elements allows one to replace the limit processes of calculus by purely algebraic calculations and notions. For the first half of the book, first published in 2006, familiarity with differential calculus and abstract algebra is presupposed during the development of results in calculus and differential geometry on a purely axiomatic/synthetic basis. In the second half basic notions of category theory are presumed in the construction of suitable Cartesian closed categories and the interpretation of logical formulae within them. This is a second edition of Kock's classical text from 1981. Many notes have been included, with comments on developments in the field from the intermediate years, and almost 100 new bibliographic entries have been added.

Product Details

ISBN-13:	9780521687386
Publisher:	Cambridge University Press
Publication date:	06/22/2006
Series:	London Mathematical Society Lecture Note Series , #333
Edition description:	Revised Edition
Pages:	246
Product dimensions:	5.98(w) x 9.02(h) x 0.55(d)

About the Author

Anders Kock is an Associate Professor of Mathematics at the University of Aarhus, Denmark.

Table of Contents

Preface to the Second Edition (2006) vii

Preface to the First Edition (1981) ix

I The synthetic theory 1

I.1 Basic structure on the geometric line 2

I.2 Differential calculus 6

I.3 Higher Taylor formulae (one variable) 9

I.4 Partial derivatives 12

I.5 Higher Taylor formulae in several variables. Taylor series 15

I.6 Some important infinitesimal objects 18

I.7 Tangent vectors and the tangent bundle 23

I.8 Vector fields and infinitesimal transformations 28

I.9 Lie bracket - commutator of infinitesimal transformations 32

I.10 Directional derivatives 36

I.11 Functional analysis. Application to proof of Jacobi identity 40

I.12 The comprehensive axiom 43

I.13 Order-and integration 48

I.14 Forms and currents 52

I.15 Currents defined using integration. Stokes' Theorem 58

I.16 Weil algebras 61

I.17 Formal manifolds 68

I.18 Differential forms in terms of simplices 75

I.19 Open covers 82

I.20 Differential forms as quantities 87

I.21 Pure geometry 90

II Categorical logic 97

II.1 Generalized elements 98

II.2 Satisfaction (1) 99

II.3 Extensions and descriptions 103

II.4 Semantics of function objects 108

II.5 Axiom 1 revisited 113

II.6 Comma categories 115

II.7 Dense class of generators 121

II.8 Satisfaction (2) 123

II.9 Geometric theories 127

III Models 131

III.1 Models for Axioms 1, 2, and 3 131

III.2 Models for ?-stable geometric theories 138

III.3 Axiomatic theory of well-adapted models (1) 143

III.4 Axiomatic theory of well-adapted models (2) 148

III.5 The algebraic theory of smooth functions 154

III.6 Germ-determined T_?-algebras 164

III.7 The open cover topology 170

III.8 Construction of well-adapted models 175

III.9 W-determined algebras, and manifolds with boundary 181

III.10 A field property of R and the synthetic role of germ algebras 192

III.11 Order and integration in the Cahiers topos 198

Appendices 207

Bibliography 223

Index 231