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because of evident rules for calculating with function sets; more gener- ally, we similarly get. R2n ∼. = RDn . (4.2). If we want to work out the description of this isomorphism, it is more convenient to use Axiom 1 in the elementwise formulation, and we will get. Proposition 4.1. For any τ : Dn → R, there exists a unique 2n-tuple.
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i SDG, version of March 2006 CUP, page i

ii CUP page ii

Synthetic Differential Geometry Second Edition

Anders Kock Aarhus University


Preface to the Second Edition (2006) Preface to the First Edition (1981) I

The I.1 I.2 I.3 I.4 I.5 I.6 I.7 I.8 I.9 I.10 I.11 I.12 I.13 I.14 I.15 I.16 I.17 I.18 I.19 I.20 I.21

page vii ix

synthetic theory Basic structure on the geometric line Differential calculus Higher Taylor formulae (one variable) Partial derivatives Higher Taylor formulae in several variables. Taylor series Some important infinitesimal objects Tangent vectors and the tangent bundle Vector fields and infinitesimal transformations Lie bracket – commutator of infinitesimal transformations Directional derivatives Functional analysis. Application to proof of Jacobi identity The comprehensive axiom Order and integration Forms and currents Currents defined using integration. Stokes’ Theorem Weil algebras Formal manifolds Differential forms in terms of simplices Open covers Differential forms as quantities Pure geometry

1 2 6 9 12 15 18 23 28 32 36 40 43 48 52 58 61 68 75 82 87 90 v

vi II


Contents Categorical logic II.1 Generalized elements II.2 Satisfaction (1) II.3 Extensions and descriptions II.4 Semantics of function objects II.5 Axiom 1 revisited II.6 Comma categories II.7 Dense class of generators II.8 Satisfaction (2) II.9 Geometric theories

Models III.1 Models for Axioms 1, 2, and 3 III.2 Models for -stable geometric theories III.3 Axiomatic theory of well-adapted models (1) III.4 Axiomatic theory of well-adapted models (2) III.5 The algebraic theory of smooth functions III.6 Germ-determined T∞ -algebras III.7 The open cover topology III.8 Construction of well-adapted models III.9 W-determined algebras, and manifolds with boundary III.10 A field property of R and the synthetic role of germ algebras III.11 Order and integration in the Cahiers topos Appendices Bibliography Index

96 97 98 102 107 112 114 120 122 126 129 129 136 141 146 152 162 168 173 179 190 196 204 220 227

Preface to the Second Edition (2006)

The First Edition (1981) of “Synthetic Differential Geometry” has been out of print since the early 1990s. I felt that there was still a need for the book, even though other accounts of the subject have in the meantime come into existence. Therefore I decided to bring out this Second Edition. It is a compromise between a mere photographic reproduction of the First Edition, and a complete rewriting of it. I realized that a rewriting would quickly lead to an almost new book. I do indeed intend to write a new book, but prefer it to be a sequel to the old one, rather than a rewriting of it. For the same reason, I have refrained from attempting an account of all the developments that have taken place since the First Edition; only very minimal and incomplete pointers to the newer literature (1981– 2006) have been included as “Notes 2006” at the end of each of the Parts of the book. Most of the basic notions of synthetic differential geometry were already in the 1981 book; the main exception being the general notion of “strong infinitesimal linearity” or “microlinearity”, which came into being just too late to be included. A small Appendix D on this notion is therefore added. Otherwise, the present edition is a re-typing of the old one, with only minor corrections, where necessary. In particular, the numberings of Parts, equations, etc. are unchanged. The bibliography consists of two parts: the first one (entries [1] to [81]) is identical to the bibliography from the 1981 edition, the second one (from entry [82] onwards) contains later literature, as referred to in the end-notes (so it is not meant to be complete; I hope in a possible forthcoming Second Book to be able to survey the field more completely). Besides the thanks that are expressed in the Preface to the 1981 edivii