Synthetic aspects of \(C^\infty\)-mappings
type: Item
instance of: scholarly article
title: Synthetic aspects of \(C^\infty\)-mappings
zbMATH Open document ID: 0531.18005
DOI: 10.1016/0022-4049(83)90071-3
published in: Journal of Pure and Applied Algebra
publication date: 1983-01-01T00:00:00Z
review text: In this paper, several of the fundamental results about smooth functions in differential topology are formulated and proved in the context of synthetic differential geometry. The setting is a topos (model) \({\mathcal E}\) with a ring object R of line type in the strong sense [see \textit{A. Kock}, Synthetic differential geometry (1981; Zbl 0466.51008)]. The author points out that a particular model in which the results of this paper can be interpreted is the Dubuc model [see above reference or \textit{E. Dubuc}, Am. J. Math. 103, 683-690 (1981; Zbl 0483.58003)]. One of the key ideas of the paper is the development of a synthetic theory of germs based on the notion of infinitesimal nbhd. of a point proposed by \textit{J. Penon} [Cah. Topologie Géom. Différ. 22, 67-72 (1981; Zbl 0463.18005)]. Let \(\Delta(n)=\neg \neg \{0\}\hookrightarrow R^ n\). Given \(x\in R^ n\), a map \(\neg \neg \{x\}=x+\Delta(n)\to R^ p\) is a germ at x ``of a map from \(R^ n\) to \(R^ p\)''. The simplification provided by the synthetic viewpoint is that a germ is an actual map rather than an equivalence class of maps. This is also true for the theory of jets [\textit{A. Kock}, Cah. Topologie Géom. Différ. 21, 227-246 (1980; Zbl 0434.18012)] and Kock's notion of a formal manifold can be strengthened by using \(\Delta\) (n) as the model nbhd. of a point for an n-dim. manifold. Using germs of maps, a proof of the 'preimage theorem' is presented following a discussion of submersions. Transversality and a version of Thom's transversality theorem are discussed as well as a version of Sard's theorem (true in the Dubuc model), which is used to prove the density of immersions \(R^ n\to R^ p\) for \(p\geq 2n\). The last two sections of the paper, which discuss stability and unfoldings, hint at the possible simplification the synthetic viewpoint offers in considering these notions. For example, an r-unfolding of a germ (in \({\mathcal E})\) ''of a map \(R^ n\to R''\) is again a germ ''of a map \(R^ n\to…
Mathematics Subject Classification ID: 18F15
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18F10
zbMATH DE Number: 3842102
zbMATH Keywords: topos with ring object
zbMATH Keywords: Sard theorem
zbMATH Keywords: smooth functions
zbMATH Keywords: synthetic differential geometry
zbMATH Keywords: model
zbMATH Keywords: Dubuc model
zbMATH Keywords: synthetic theory of germs
zbMATH Keywords: jets
zbMATH Keywords: formal manifold
zbMATH Keywords: preimage theorem
zbMATH Keywords: submersions
zbMATH Keywords: Transversality
zbMATH Keywords: stability
zbMATH Keywords: unfoldings
author: Marta Bunge
reviewed by: K. I. Rosenthal
MaRDI profile type: Publication
full work available at URL: 0022-4049(83)90071-3
OpenAlex ID: W2064980117
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cites work: Universal projective geometry via topos theory
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description: scientific article; zbMATH DE number 3842102