Synthetic aspects of \(C^{\infty}\)-mappings. II: Mather's theorem for infinitesimally represented germs
This paper continues the work of the first author [ibid. 28, 41-63 (1983; Zbl 0531.18005)] on studying the theory of \(C^{\infty}\)-mappings via synthetic differential geometry (SDG), and it provides further evidence of the power and insight inherent in the synthetic approach. The article begins with a discussion of various axioms which are added to the usual list of axioms of SDG. These axioms are studied in detail and their validity in the Dubuc model \({\mathcal G}\) of SDG is established. (Recall that the Dubuc model is the topos of sheaves on the open cover topology on \({\mathbb{B}}^{op}\), where \({\mathbb{B}}\) is the category of \(C^{\infty}\)- rings of finite presentation by a germ-determined ideal.) It is then shown how it is possible to develop an infinitesimal notion of stability by representing germs as actual mappings on an infinitesimal domain. This is used to prove Mather's theorem [\textit{J. N. Mather}, Ann. Math., II. Ser. 89, 254-291 (1969; Zbl 0177.260] synthetically. The classical result is recovered by interpretation in the Dubuc model \({\mathcal G}\). The paper concludes by using a particular interpretation of the synthetic Mather theorem in \({\mathcal G}\) to study Wassermann's theory of unfoldings [\textit{G. Wassermann}, Stability of unfoldings; (1974; Zbl 0288.57017)]. There is not such a direct link to unfoldings when the situation is studied classically.
scientific article; zbMATH DE number 4073250
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