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Synthetic differential geometry. (English) Zbl 0466.51008

London Mathematical Lecture Note Series, 51. Cambridge etc.: Cambridge University Press. viii, 311 p. £10.00 (1981).
What is synthetic differential geometry? It is something that most of us have met as students, when we first learn about calculus. Infinitesimal quantities, infinitesimal vectors, infinitesimal transformations and so forth, are not strange concepts, but explanations which use them are generally accompanied by an apology for lack of rigour. We know how to make rigorous analytic definitions which bypass the question ‘what is an infinitesimal quantity?’, and yet when we want to explain something or draw pictures on the blackboard, we often fall back into the language of infinitesimals.
The preface of this book opens with the following translation of a passage from Sophus Lie’s “Allgemeine Theorie der partiellen Differentialgleichungen erster Ordnung” [Math. Ann. 9, 245–296 (1875; JFM 07.0225.01)]: “The reason why I have postponed for so long these investigations, which are basic to my other work in this field, is essentially the following. I found these theories originally by synthetic considerations. But I soon realized that, as expedient (zweckmäßig) the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations, which deal, with objects that till now have almost been considered analytically. After long vacillations, I have decided to use a half synthetic, half analytic form. I hope my work will serve to bring justification to the synthetic method besides the analytical one.”
It is important not to misunderstand what this book is about. “The aim of the present book is to describe a foundation for synthetic reasoning in differential geometry.”
It is not attempting to put differential geometry on a different footing. It is not claiming to provide new synthetic arguments. The foundation which it describes fits into a much wider picture, which is hinted at, which has a naturalness that I am sure Lie would have applauded. On the other hand it may shock some readers; for this foundation is based on challenges to two assumptions that are dinned into us from our earliest mathematical training. We meet the first at the bottom of the first page of the first section of chapter I. I quote from this page:
“The geometric line can, as soon as one chooses two distinct points on it, be made into a commutative ring, with the two points as respectively 0 and 1. This is a decisive structure on it, already known and considered by Euclid, who assumes that his reader is able to move line segments around in the plane (which gives addition), and who teaches his reader how he, with ruler and compass, can construct the fourth proportional of three line segments; taking one of these to be \([0,1]\), this defines the product of the two others, and thus the multiplication on the line. We denote the line, with its commutative ring structure (relative to some fixed choice of 0 and 1) by the letter \(R\). Also, the geometric plane can, by some of the basic structure (ruler-and-compass-constructions again), be identified with \(R\times R=R^2\) (choose a fixed pair of mutually orthogonal copies of the line \(R\) in it), and similarly, space with \(R^3\). Of course, this basic structure does not depend on having the (arithmetically constructed) real numbers \(\mathbb R\) as a mathematical model for \(R\). Euclid maintained further that \(R\) was not just a commutative ring, but actually a field. This follows because of his assumption: for any two points in the plane, either they are equal, or they determine a unique line. We cannot agree with Euclid on this point.”
There may be many readers who automatically identify the linear continuum which we abstract from experience of the world around us, with \(R\). They should stop to realize that \(\mathbb R\) is not so “concrete” a mathematical object as they suppose, and that, useful though it is for modelling some properties of the real world, there is no reason why we should expect it to be the only useful model. It is worth pointing out here that the models \(R\) employed in this book are not non-standard models of the reals, in the usual sense of non-standard arithmetic. The technique is to consider commutative rings \(R\) with nilpotent elements. Throughout the book, the object \(D\) given by \(D:= \{x\in R\mid x^2 =0\}\) plays a fundamental role. We may think of \(D\) as an infinitesimal neighbourhood of zero on the line. Axiom 1 states: For any \(g\in R^D\) (the object of mappings from \(D\) to \(R)\) there exists a unique \(b\in R\) such that \(\forall d\in D: g(d) = g(0)+d.b\). In other words, the graphs of mappings restricted to infinitesimal regions are straight lines. Three pages later we are confronted with the second challenge: “Axiom 1 is incompatible with the law of excluded middle. Either the one or the other has to leave the scene. In Part I of this book, the law of excluded middle has to leave, being incompatible with the natural synthetic reasoning on smooth geometry to be presented here. In the terms which the logicians use, this means that the logic employed is ‘constructive’ or ‘intuitionistic’.” The lessons learnt about the relationship between mathematics and the real world from the work of Lobachevsky and Bolyai apply no less strongly to logic, but I fear they have not been equally appreciated. Most practicing mathematicians are probably not accustomed to monitoring their proofs for intuitionistic validity. The readers of this book need not take alarm; it in no way resembles a textbook on logic.
The main text of the book is divided into three sections, each divided into chapters. As the book is not a complete compendium of work done in this area, each chapter is followed by exercises which point to material not covered. After the three main sections there are short chapters on loose ends and historical remarks, three technical appendixes, references and an index.
The first part, the Synthetic Theory, explores the geometric significance of the axioms of the theory. In this part the role of logic is played down as far as possible, and a “trick” is employed to lull the reader who might be frightened by the logical aspects. The second, fairly brief part, Categorical Logic, analyzes in more detail the allowable rules for deducing theorems from the axioms. This section performs a valuable service to the mathematical public in giving a readable introduction to the ideas of categorical logic. The style of this section, in fact of the whole book, is to adopt the pregnant example rather than the most general case, to adopt the simplest and most suggestive notation, and to relate any new notion back to well known ones. For example, the chapter on geometric theories, instead of giving a general definition, looks at specific example of geometric theories in the language of commutative rings, with which the reader can be expected to be familiar. The third part, Models, looks at various models of the axioms. The chief aim, of course, is to find a model of the axioms which includes all smooth manifolds in a suitably nice way, so that the synthetic constructions and theorems should coincide with the usual analytic ones, and so that, furthermore, these constructions can be given a sensible meaning for smooth function spaces and other objects which are not strictly speaking manifolds.
Of course, this book does not represent the first attempt at such a task, but I would claim it is the most coherent. The use of nilpotent elements to represent infinitesimals has a long history, running right up to modern algebraic geometry. Inevitably the approach described in this book shares many common features with algebraic geometry, and these are pointed out in the historical notes. There is an interesting reference to Protagoras who claimed that it was evident that the intersection of a circle with a tangent line consisted of more than a single point. The idea which led to much of these development was due to Lawvere; it was the idea of the representability of the tangent bundle (or more generally, of jet bundles): A tangent at a point \(x\) of an object \(M\) is defined to be a map \(D{\overset {t} \rightarrow} M\) such that \(t(0)=x\). The object \(M^D\) can then be identified with the object of tangents to \(M\), and the map \(M{\overset {\pi} \rightarrow} M\) given by \(t\to t(0)\) defines the tangent bundle. If \(M\) satisfies a condition called infinitesimal linearity, then it becomes possible to add tangents at the same point, and \(M^D\to M\) becomes an \(R\)-module in the category of objects over \(M\). A vector field on \(M\) can now be described in three equivalent ways (by exponential adjointness, i.e. \(\lambda\)-conversion): i) as a section \(M{\overset {\xi} \rightarrow} M^D\), \(\pi\xi = 1_M\) of the tangent bundle, ii) as an infinitesimal flow \(M\times D{\overset {\xi} \rightarrow}M\), \(\xi(m,0)=m\), iii) as a tangent to the identity map \(D{\overset {\xi} \rightarrow}M^M\), \(\xi(0) = 1_M\). The advantage of this over the analytic approach, is that for \(d\in D\), \(\xi(d)\) is an actual permutation of \(M\), and this can be exploited to give easy proofs of many theorems about vector fields. The first part of the book is like an hors d’hœuvres tray; a lot of short chapters on a good many topics: elementary calculus, partial derivatives, tangent vectors, vector fields, the Lie bracket, order and integration, differential forms, currents, Stokes’ theorem, Weil algebras, formal manifolds, differential forms as quantities rather than as functionals, incidence geometry.
I have said that the first part uses a “trick” of notation to make it easier for the reader. I quote from the preface: “Most of Part I, as well as several of the papers on the bibliography which go deeper into actual geometric matters with synthetic methods, are written in the “naive” style. By this, we mean that all notions, constructions, and proofs involved are presented as if the base category were the category of sets; in particular, all constructions on the objects involved are described in terms of “elements” of them. However, it is necessary and possible to be able to understand this naive writing as referring to Cartesian closed categories. It is necessary because the basic axioms of synthetic differential geometry have no models in the category of sets (cf. I §1); and it is possible: this is what Part II is about. The method is that we have to understand by an element \(b\) of an object \(B\) a generalized element, that is, a map \(b: X\to B\), where \(X\) is an arbitrary object, called the stage of definition, or the domain of variation of the element \(b\). Elements “defined as different stages” have a long tradition in geometry. In fact, a special case of it is when the geometers say: A circle has no real points at infinity, but there are two imaginary points at infinity such that every circle passes through them. Here \(B\) and \(C\) are two different stages of mathematical knowledge, and something that does not yet exist at stage \(B\) may come into existence at the “later” or “deeper” stage \(C\).
I found one or two misprints, and ‘explicitly’ is spelled ‘explicitely’ right the way through, but these details do not obtrude. The general layout, the look of the pages, and the diagrams, are inviting and well done. It is in many ways an unusual book, with a strong individual flavour. It is both a textbook and a summary of recent research. It deals with some elementary aspects of mathematics in as simple a way as possible, and yet also involves some very subtle ideas. The first part can be read and enjoyed by a wide readership. The second part is more difficult, but is probably the most accessible primer on categorical logic yet published. The exercises and notes link the material in the book with so many other themes it is impossible in a review to do them justice.

MSC:

51K10 Synthetic differential geometry
03G30 Categorical logic, topoi
18F10 Grothendieck topologies and Grothendieck topoi
51-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to geometry
18-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory
58A10 Differential forms in global analysis

Citations:

JFM 07.0225.01