##
**Models for smooth infinitesimal analysis.**
*(English)*
Zbl 0715.18001

New York etc.: Springer-Verlag. x, 399 p. DM 148.00 (1991).

This book systematically uses topos theory to develop models, called smooth toposes, for studying differential geometry. These models have the usual category \({\mathcal M}\) of manifolds embedded in them, however the topos structure provides one with function spaces, inverse limits and various non-classical infinitesimal spaces. Moreover, the internal intuitionistic logic of a topos supplies a language allowing for the rigorous treatment of infinitesimals (both invertible and nilpotent) and a framework for carrying out formally the synthetic arguments used by the great geometers of the last century such as Cartan and Lie.

The possibility of doing this goes back to the pioneering lectures of F. W. Lawvere in 1967 [reproduced later as “Categorical dynamics” in Var. Publ. Ser., Aarhus Univ. 30, 1-28 (1979; Zbl 0403.18005)]. In these lectures he advocated applying to differential geometry Grothendieck’s method from algebraic geometry, which involved using nilpotents to deal with infinitesimal structures. This could be done by replacing the algebraic theory of rings with its polynomial operations by the algebraic theory of \(C^{\infty}\)-rings where the operations are given by the smooth functions \({\mathbb{R}}^ n\to {\mathbb{R}}\). These ideas led to the development of synthetic differential geometry (SDG), which has been an active area of research for the past 15 years. The first book on the subject was by A.Kock [Synthetic differential geometry (1981; Zbl 0466.51008)]. The book developed the axiomatic theory, while also discussing categorical logic and some of the models of SDG. The more recent text by R. Lavendhomme [Leçons de géometrie différentielle synthétique naive (1987; Zbl 0688.18006)] concentrated on an axiomatic development of SDG using nilpotents, and it contained no discussion of models.

The book under review is thus the first text to concentrate on the models of SDG and the connections with the classical theory, while emphasizing the role of both nilpotent and invertible infinitesimals. The basic construction of the models proceeds as follows. The category \({\mathbb{L}}\) of “formal loci” is defined to be the opposite category of the category of finitely generated \(C^{\infty}\)-rings and \(C^{\infty}\)-homomorphisms. Subcategories \({\mathbb{C}}\) of \({\mathbb{L}}\) are considered where \({\mathbb{C}}\) contains the category of manifolds \({\mathcal M}\) (more precisely the \(C^{\infty}\)-rings \(C^{\infty}(M)\), where M is a manifold). Then, \({\mathbb{C}}\) is made into a site by endowing it with a Grothendieck topology and a model is obtained by taking sheaves on the site \({\mathbb{C}}\), resulting in the topos sh(\({\mathbb{C}})\). (The book has appendices discussing sites and sheaves for the uninitiated reader.)

We shall now summarize the content of the chapters in the book. Chapter I - \(``C^{\infty}\)-rings”: This chapter contains a thorough introduction to \(C^{\infty}\)-rings including the \(C^{\infty}\)-rings \(C^{\infty}(M)\), where M is a manifold, a discussion of local \(C^{\infty}\)-rings and Weil algebras, as well as a look at ideals of smooth functions. Chapter II - \(``C^{\infty}\)-rings as variable spaces”: This chapter studies the topos \(Sets^{{\mathbb{L}}^{op}}\) of presheaves on \({\mathbb{L}}\). The goal is to introduce the reader to the functorial approach and to familiarize them with working inside a topos. In that sense, this chapter serves as a prelude to the study of the actual models in the next chapter. Chapter III - “Two Archimedean models for synthetic calculus”: Two models are constructed in this chapter. First, one considers the subcategory \({\mathbb{G}}\) of \({\mathbb{L}}\), based on looking at germ-determined ideals. This is made into a site by endowing it with the so-called open cover topology and the result is the topos \({\mathcal G}\), often referred to as the Dubuc topos [see E. J. Dubuc, Am. J. Math. 103, 683-690 (1981; Zbl 0483.58003)]. The second topos \({\mathcal F}\) is obtained from a site \({\mathbb{F}}\), where \({\mathbb{F}}\) is a subcategory of \({\mathbb{L}}\) involving consideration of closed ideals rather than germ determined ones. The book proceeds to discuss some properties of the embedding of the category of manifolds \({\mathcal M}\) into the toposes \({\mathcal G}\) and \({\mathcal F}\). Chapter IV - “Cohomology and integration”: The purpose of this chapter is to show synthetic differential geometry in action by developing the theory of de Rham cohomology and looking at it in the models \({\mathcal F}\) and \({\mathcal G}\), as well as relating it to the classical de Rham theory. The power of the sheaf theoretic approach is illustrated by the fact that one in fact can obtain by synthetic methods a generalized de Rham theorem with parameters. Chapter V - “Connections on microlinear spaces”: The utility of having available the “non- classical” infinitesimal spaces provided by the models is investigated by studying connections, vector fields, etc. in the synthetic setting. This includes proofs of the Ambrose-Palais-Singer theorem on sprays and affine connections, as well as the two dimensional Gauss-Bonnet theorem. Chapter VI - “Models with invertible infinitesimals”: The smooth Zariski topos \({\mathcal Z}\) is introduced (its construction is modelled after that of the usual Zariski topos in algebraic geometry) by considering the finite cover topology on \({\mathbb{L}}\). In \({\mathcal Z}\) one has available invertible infinitesimals in addition to the nilpotents. A slight variation of \({\mathcal Z}\), the Basel topos \({\mathcal B}\), is then developed to ensure that there exist “globally defined” invertible infinitesimals. Chapter VII - “Smooth infinitesimal analysis”: The goal of this concluding chapter is to set up a formal system allowing for the rigorous development of smooth infinitesimal analysis. The Basel topos \({\mathcal B}\) forms a natural model for this system. Several sections indicate how to use this formal system and included is a discussion of distributions and the Dirac \(\delta\)-function. Finally, a transfer principle between the toposes \({\mathcal B}\) and \({\mathcal G}\) is derived, which allows to verify a large class of statements for \({\mathcal B}\) by checking their validity in the model \({\mathcal G}\), which is much easier to work with.

This is a well written, carefully thought out book, which is a most welcome addition to the existing literature. It clearly indicates the utility of the topos theoretic approach in clarifying certain mathematical ideas and highlights the interplay between the internal logic of a topos and the idea of a topos as a category of generalized spaces, in this case extending the category \({\mathcal M}\) of manifolds.

The possibility of doing this goes back to the pioneering lectures of F. W. Lawvere in 1967 [reproduced later as “Categorical dynamics” in Var. Publ. Ser., Aarhus Univ. 30, 1-28 (1979; Zbl 0403.18005)]. In these lectures he advocated applying to differential geometry Grothendieck’s method from algebraic geometry, which involved using nilpotents to deal with infinitesimal structures. This could be done by replacing the algebraic theory of rings with its polynomial operations by the algebraic theory of \(C^{\infty}\)-rings where the operations are given by the smooth functions \({\mathbb{R}}^ n\to {\mathbb{R}}\). These ideas led to the development of synthetic differential geometry (SDG), which has been an active area of research for the past 15 years. The first book on the subject was by A.Kock [Synthetic differential geometry (1981; Zbl 0466.51008)]. The book developed the axiomatic theory, while also discussing categorical logic and some of the models of SDG. The more recent text by R. Lavendhomme [Leçons de géometrie différentielle synthétique naive (1987; Zbl 0688.18006)] concentrated on an axiomatic development of SDG using nilpotents, and it contained no discussion of models.

The book under review is thus the first text to concentrate on the models of SDG and the connections with the classical theory, while emphasizing the role of both nilpotent and invertible infinitesimals. The basic construction of the models proceeds as follows. The category \({\mathbb{L}}\) of “formal loci” is defined to be the opposite category of the category of finitely generated \(C^{\infty}\)-rings and \(C^{\infty}\)-homomorphisms. Subcategories \({\mathbb{C}}\) of \({\mathbb{L}}\) are considered where \({\mathbb{C}}\) contains the category of manifolds \({\mathcal M}\) (more precisely the \(C^{\infty}\)-rings \(C^{\infty}(M)\), where M is a manifold). Then, \({\mathbb{C}}\) is made into a site by endowing it with a Grothendieck topology and a model is obtained by taking sheaves on the site \({\mathbb{C}}\), resulting in the topos sh(\({\mathbb{C}})\). (The book has appendices discussing sites and sheaves for the uninitiated reader.)

We shall now summarize the content of the chapters in the book. Chapter I - \(``C^{\infty}\)-rings”: This chapter contains a thorough introduction to \(C^{\infty}\)-rings including the \(C^{\infty}\)-rings \(C^{\infty}(M)\), where M is a manifold, a discussion of local \(C^{\infty}\)-rings and Weil algebras, as well as a look at ideals of smooth functions. Chapter II - \(``C^{\infty}\)-rings as variable spaces”: This chapter studies the topos \(Sets^{{\mathbb{L}}^{op}}\) of presheaves on \({\mathbb{L}}\). The goal is to introduce the reader to the functorial approach and to familiarize them with working inside a topos. In that sense, this chapter serves as a prelude to the study of the actual models in the next chapter. Chapter III - “Two Archimedean models for synthetic calculus”: Two models are constructed in this chapter. First, one considers the subcategory \({\mathbb{G}}\) of \({\mathbb{L}}\), based on looking at germ-determined ideals. This is made into a site by endowing it with the so-called open cover topology and the result is the topos \({\mathcal G}\), often referred to as the Dubuc topos [see E. J. Dubuc, Am. J. Math. 103, 683-690 (1981; Zbl 0483.58003)]. The second topos \({\mathcal F}\) is obtained from a site \({\mathbb{F}}\), where \({\mathbb{F}}\) is a subcategory of \({\mathbb{L}}\) involving consideration of closed ideals rather than germ determined ones. The book proceeds to discuss some properties of the embedding of the category of manifolds \({\mathcal M}\) into the toposes \({\mathcal G}\) and \({\mathcal F}\). Chapter IV - “Cohomology and integration”: The purpose of this chapter is to show synthetic differential geometry in action by developing the theory of de Rham cohomology and looking at it in the models \({\mathcal F}\) and \({\mathcal G}\), as well as relating it to the classical de Rham theory. The power of the sheaf theoretic approach is illustrated by the fact that one in fact can obtain by synthetic methods a generalized de Rham theorem with parameters. Chapter V - “Connections on microlinear spaces”: The utility of having available the “non- classical” infinitesimal spaces provided by the models is investigated by studying connections, vector fields, etc. in the synthetic setting. This includes proofs of the Ambrose-Palais-Singer theorem on sprays and affine connections, as well as the two dimensional Gauss-Bonnet theorem. Chapter VI - “Models with invertible infinitesimals”: The smooth Zariski topos \({\mathcal Z}\) is introduced (its construction is modelled after that of the usual Zariski topos in algebraic geometry) by considering the finite cover topology on \({\mathbb{L}}\). In \({\mathcal Z}\) one has available invertible infinitesimals in addition to the nilpotents. A slight variation of \({\mathcal Z}\), the Basel topos \({\mathcal B}\), is then developed to ensure that there exist “globally defined” invertible infinitesimals. Chapter VII - “Smooth infinitesimal analysis”: The goal of this concluding chapter is to set up a formal system allowing for the rigorous development of smooth infinitesimal analysis. The Basel topos \({\mathcal B}\) forms a natural model for this system. Several sections indicate how to use this formal system and included is a discussion of distributions and the Dirac \(\delta\)-function. Finally, a transfer principle between the toposes \({\mathcal B}\) and \({\mathcal G}\) is derived, which allows to verify a large class of statements for \({\mathcal B}\) by checking their validity in the model \({\mathcal G}\), which is much easier to work with.

This is a well written, carefully thought out book, which is a most welcome addition to the existing literature. It clearly indicates the utility of the topos theoretic approach in clarifying certain mathematical ideas and highlights the interplay between the internal logic of a topos and the idea of a topos as a category of generalized spaces, in this case extending the category \({\mathcal M}\) of manifolds.

Reviewer: K.I.Rosenthal

### MSC:

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

18F15 | Abstract manifolds and fiber bundles (category-theoretic aspects) |

18B25 | Topoi |

51K10 | Synthetic differential geometry |

18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |

18F10 | Grothendieck topologies and Grothendieck topoi |