Synthetic aspects of \(C^\infty\)-mappings. (English) Zbl 0531.18005

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 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 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 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 [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 \(R^ n\to R''\), however in a different topos, namely \({\mathcal E}/\Delta(r)\). This allows for a simpler definition of equivalence for unfoldings than the classical one. Perhaps the synthetic viewpoint can provide a fresh perspective and insight into some of these difficult topics.
Reviewer: K.I.Rosenthal


18F15 Abstract manifolds and fiber bundles (category-theoretic aspects)
18B25 Topoi
18F10 Grothendieck topologies and Grothendieck topoi
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