Smooth spaces versus continuous spaces in models for synthetic differential geometry (Q790924)

This paper considers the Dubuc model \({\mathcal G}\) for synthetic differential geometry [see \textit{E. J. Dubuc}, \(C^{\infty}\)-schemes, Am. J. Math. 103, 683-690 (1981; Zbl 0483.58003) or \textit{A. Kock}, Synthetic differential geometry, Lond. Math. Soc. Lect. Note Ser. 51 (1981; Zbl 0466.51008)] and the extent to which it serves as a model for intuitionistic analysis. This involves studying the geometric line R and comparing it with the Dedekind reals \({\mathbb{R}}\) in \({\mathcal G}\). The main tool used is the notion of an adjoint retraction of toposes, in particular a retraction between \({\mathcal G}\) and the Euclidean topos \({\mathcal E}\). \({\mathcal E}\) is the topos of sheaves on the topological site \({\mathbb{E}}\) consisting of locally closed subspaces of \({\mathbb{R}}^ n\) (\(n\in {\mathbb{N}})\) with \(C^{\infty}\)-maps as morphisms and the open cover Grothendieck topology. \({\mathcal G}\) is a topos of sheaves on \({\mathbb{G}}\), the dual of the category of germ-determined \(C^{\infty}\)-rings (for more details, see the references listed above). There is an inclusion \({\mathbb{E}}\to^{i}{\mathbb{G}}\) given by \(i(X)=C^{\infty}(X)\) and this has a right adjoint. These functors give rise to geometric morphisms \(G\to^{\rho}E\to^{\lambda}G\) with \(\lambda^*\dashv \lambda_*=\rho^*\dashv \rho_*\) and with \(\rho\) a surjection, \(\lambda\) an inclusion and \(\rho\lambda\simeq 1\). (This is the notion of an adjoint retraction.) This is used to study the geometric line R and the Dedekind reals \({\mathbb{R}}\) in \({\mathcal G}\). Objects such as \({\mathbb{R}}^{{\mathbb{R}}}\), \({\mathbb{R}}^ R\) together with properties such as ''all functions from \({\mathbb{R}}\) to \({\mathbb{R}}\) are continuous'', ''[0,1] is compact in R'' and certain principles of intuitionistic analysis are examined. The last section of the paper points out that a key property of \({\mathcal G}\) is the fact that the geometric line R is local and Archimedean and many ''good'' properties of analysis on \({\mathbb{R}}\) (i.e. intuitionistic analysis) may fail in other models. The topos \({\mathcal G}_{fin}\) of sheaves on \({\mathbb{G}}\) with the finite cover topology is considered, where for example ''all functions from \({\mathbb{R}}\) to \({\mathbb{R}}\) are continuous'' does not hold. As a final remark, the paper does contain a section on \(C^{\infty}\)- rings and germ-determined ideals, which is helpful to the reader unfamiliar with synthetic differential geometry. However, as the authors themselves remark, some knowledge of SDG would provide a better perspective on the results of the paper.