Local concepts in synthetic differential geometry and germ representability. (English) Zbl 0658.18004

Mathematical logic and theoretical computer science, Lect. Notes Pure Appl. Math. 106, 93-159 (1987).
[For the entire collection see Zbl 0642.00005.]
The development of synthetic differential geometry (SDG) started from the idea of being able to replace the analytic calculations of classical differential geometry with algebraic reasoning using infinitesimals (nilpotents) analogously to what is done in algebraic geometry. This infinitesimal approach can be thought of as revolving around the notion of representability of jets. This survey article by two of the leading practitioners of SDG summarizes these ideas of jet representability and then in the main section of the paper the notion of germ representability is developed as a tool for studying local (as opposed to infinitesimal) concepts in SDG.
In the first section, the authors recount the development of the infinitesimal aspects of SDG from the simplest form of the Kock-Lawvere axiom for ring objects of line type through the statement of this axiom for Weil algebras, where the resulting Weil bundles can be viewed as generalized jet bundles. Notions such as infinitesimally linear object, vector field and flow are also presented from the synthetic point of view.
In order to develop “local” concepts in SDG, one should postulate that germs of functions \(R^ n\to R\) (where R is the line) are representable in a sense analogous to the case of jets. This requires developing the theory of topological structures in a ringed topos (E,R). Two such topological structures are the Zariski structure, which is based on looking at the subobject \(R^*\) of units of R, and the Euclidean topology, which is derived from the order structure on the line R. A third notion of topology comes from the internal topos structure of E. One considers \(\Delta (n)=\neg \neg \{0\}\) in \(R^ n\), which is the largest infinitesimal object smaller than all open neighborhoods of O. It turns out that the germ at 0 of a smooth function \(R^ n\to R\) is representable by a function \(\Delta\) (n)\(\to R\). This is due to J. Penon [Cah. Topologie Geom. Differ. 22, 67-72 (1981; Zbl 0463.18005)] and is connected to the topological structure of Penon opens. These three topological structures agree on a large class of objects in the so-called well adapted models of SDG. The authors show how to apply these ideas of germ representability to problems concerning flows and integration of vector fields and then the article concludes with an appendix discussing construction of the models. As a whole, the article is clearly written and achieves the aim of providing an introduction and overview to a large body of material. There is a thorough bibliography and many historical remarks. No previous knowledge of SDG is presupposed, however a reader without some knowledge and intuition from topos theory will find parts of it tough going.
Reviewer: K.I.Rosenthal


18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
18-02 Research exposition (monographs, survey articles) pertaining to category theory
51K10 Synthetic differential geometry