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Strong infinitesimal linearity, with applications to strong difference and affine connections. (English) Zbl 0564.18009
The notions of strong infinitesimal linearity, first considered by F. Bergeron [Objet infinitésimalement linéaire dans un modèle adapté de géométrie différentielle synthétique, Rapport de recherches, DMS 80-12, Univ. de Montréal (1980)], and of strong difference [cf.: I. Kolař, Tensor, New Ser. 38, 98-102 (1982; Zbl 0512.58002); J. E. White, The method of iterated tangents with applications in local Riemannian geometry (1982; Zbl 0478.58002)] are used in the context of synthetic differential geometry to provide with simple proofs, independent of coordinates, certain equalities on affine connections, and to get a version of the Ambrose-Palais-Singer theorem [see A. Kock, Var. Publ. Ser., Aarhus Univ. 35, 192-202 (1983; Zbl 0552.58001); also M. Bunge and P. Sawyer, Cah. Topologie Géom. Différ. 25, 221-258 (1984)]. For motivation, one of the properties of a commutative K-algebra object R in a topos E, which satisfies the central axiom $$1^ W$$, proposed by F. W. Lawvere and the first named author, is taken.
Reviewer: G.F.Nassopoulos

##### MSC:
 18F15 Abstract manifolds and fiber bundles (category-theoretic aspects) 51K10 Synthetic differential geometry 18B25 Topoi 53C05 Connections (general theory) 58A20 Jets in global analysis
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##### References:
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