Vector fields on \({\mathbb{R}}^{{\mathbb{R}}}\) in well adapted models of synthetic differential geometry (Q1094524)

The problem mentioned in the title of this paper is probably less important than a result established on the way to it, and which deals with (not necessarily linear) operators \(G: C^{\infty}({\mathbb{R}})\to C^{\infty}({\mathbb{R}}^ p)\) which are smooth in the sense of \textit{A. Frölicher} [cf. Lect. Notes Math. 962, 69-81 (1982; Zbl 0498.58004)], namely: if I is a sufficiently good ideal in \(C^{\infty}({\mathbb{R}}^ n)\), then the parameterwise extension of G to an operator \(C^{\infty}({\mathbb{R}}^{1+n})\to C^{\infty}({\mathbb{R}}^{p+n})\) preserves congruence modulo I. [A generalization of this result, in several directions, appears in the reviewer's paper Bull. Aust. Math. Soc. 34, 395-410 (1986; Zbl 0596.18006)]. A consequence is that such operators may be identified with maps between the corresponding internal function space objects in some of the good topos models for synthetic differential geometry.