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Zbl 0596.18005
Kock, Anders
Convenient vector spaces embed into the Cahiers topos. (English)
[J] Cah. Topol. Géom. Différ. Catégoriques 27, No.1, 3-17 (1986). ISSN 0008-0004

The category of "convenient" vector spaces of Frölicher and Kriegl [cf. {\it A. Frölicher}, {\it B. Gisin} and {\it A. Kriegl}, General differentiation theory, in Category theoretic methods in geometry, Var. Publ. Ser. 35, Aarhus, 125-153 (1983)], although not a model of synthetic differential geometry [in the sense of the author, Synthetic differential geometry (1981; Zbl 0466.51008)] in that it contains no infinitesimals, presents, as does the latter, a solution to the problem of repairing the lack of good categorical properties (e.g., Cartesian closedness) of the category of smooth manifolds and smooth maps. For this reason, it is of interest to find out the relationship - if any - between these two approaches to the same problem. \par The present paper claims to construct a full embedding of the category of Frölicher and Kriegl into the so-called "Cahiers topos" [cf. {\it E. Dubuc}, Cah. Topologie Géom. Différ. 20, 231-279 (1979; Zbl 0473.18008)], preserving finite products and exponentiation. However, as announced by the author later [cf. the author, Erratum to "Convenient vector spaces embed into the Cahiers topos", ibid. 27, 23-24 (1986)], the site of definition given in the paper being reviewed does not, as claimed, give the Cahiers topos, although the basic "Weil prolongation" construction provided therein does indeed give the desired result, with some additional work. These additional considerations are contained in a forthcoming paper, jointly with {\it G. E. Reyes} (ibid., to appear).
[M.Bunge]

MSC 2000:: *18F15 Abstract manifolds, etc.
18D15 Closed categories
58C99 Calculus on manifolds
18B25 Topoi

Keywords: topos theory; categories in differential geometry; Cartesian closed; categories; convenient vector spaces; Cahiers topos

Citations: Zbl 0466.51008; Zbl 0473.18008

Cited in: Zbl 0634.18007 Zbl 0601.18006

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