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Two Archimedean Models for Synthetic Calculus

In: Models for Smooth Infinitesimal Analysis

Author

Listed:
  • Ieke Moerdijk

    (University of Utrecht, Mathematical Institute)

  • Gonzalo E. Reyes

    (University of Montreal, Department of Mathematics)

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 $$M \to Set{s^{{L^{op}}}}$$ .

Suggested Citation

  • Ieke Moerdijk & Gonzalo E. Reyes, 1991. "Two Archimedean Models for Synthetic Calculus," Springer Books, in: Models for Smooth Infinitesimal Analysis, chapter 0, pages 97-131, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4757-4143-8_4
    DOI: 10.1007/978-1-4757-4143-8_4
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