Synthetic Differential Geometry

By: Anders Kock

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Paperback | 22 June 2006 | Edition Number 2

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246 Pages

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Revised

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22.86 x 15.24 x 1.42

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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.

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Preface to the Second Edition (2006)	p. vii
Preface to the First Edition (1981)	p. ix
The synthetic theory	p. 1
Basic structure on the geometric line	p. 2
Differential calculus	p. 6
Higher Taylor formulae (one variable)	p. 9
Partial derivatives	p. 12
Higher Taylor formulae in several variables. Taylor series	p. 15
Some important infinitesimal objects	p. 18
Tangent vectors and the tangent bundle	p. 23
Vector fields and infinitesimal transformations	p. 28
Lie bracket - commutator of infinitesimal transformations	p. 32
Directional derivatives	p. 36
Functional analysis. Application to proof of Jacobi identity	p. 40
The comprehensive axiom	p. 43
Order-and integration	p. 48
Forms and currents	p. 52
Currents defined using integration. Stokes' Theorem	p. 58
Weil algebras	p. 61
Formal manifolds	p. 68
Differential forms in terms of simplices	p. 75
Open covers	p. 82
Differential forms as quantities	p. 87
Pure geometry	p. 90
Categorical logic	p. 97
Generalized elements	p. 98
Satisfaction (1)	p. 99
Extensions and descriptions	p. 103
Semantics of function objects	p. 108
Axiom 1 revisited	p. 113
Comma categories	p. 115
Dense class of generators	p. 121
Satisfaction (2)	p. 123
Geometric theories	p. 127
Models	p. 131
Models for Axioms 1, 2, and 3	p. 131
Models for ?-stable geometric theories	p. 138
Axiomatic theory of well-adapted models (1)	p. 143
Axiomatic theory of well-adapted models (2)	p. 148
The algebraic theory of smooth functions	p. 154
Germ-determined T_?-algebras	p. 164
The open cover topology	p. 170
Construction of well-adapted models	p. 175
W-determined algebras, and manifolds with boundary	p. 181
A field property of R and the synthetic role of germ algebras	p. 192
Order and integration in the Cahiers topos	p. 198
Appendices	p. 207
Bibliography	p. 223
Index	p. 231
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Synthetic Differential Geometry

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