Two Archimedean Models for Synthetic Calculus

Ieke Moerdijk and Gonzalo E. Reyes
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Ieke Moerdijk: University of Utrecht, Mathematical Institute
Gonzalo E. Reyes: University of Montreal, Department of Mathematics

Chapter Chapter III in Models for Smooth Infinitesimal Analysis, 1991, pp 97-131 from Springer

Abstract: Abstract In chapter II, we introduced the category of smooth functors $$Set{s^{{L^{op}}}}$$ This category has good function spaces, infinitesimal spaces, convenient exactness properties, and it contains the usual category of manifolds M. Furthermore, the embedding $$M \to Set{s^{{L^{op}}}}$$ preserves the good limits in M, namely tranversal pullbacks. Nevertheless, $$Set{s^{{L^{op}}}}$$ has pathological properties: the smooth line R, which is a commutative ring with unit, is not even a local ring. Moreover, R is not Archimedean. From a somewhat different viewpoint, one can say that, besides some good limits, M also has good colimits, such as open covers. The trouble with $$Set{s^{{L^{op}}}}$$ is that these covers are not preserved by the embedding $$M \to Set{s^{{L^{op}}}}$$ .

Keywords: Local Ring; Open Cover; Inverse Limit; Left Adjoint; Constant Sheaf (search for similar items in EconPapers)
Date: 1991
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DOI: 10.1007/978-1-4757-4143-8_4

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