Two Archimedean Models for Synthetic Calculus

  • Ieke Moerdijk
  • Gonzalo E. Reyes


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 \(M \to Set{s^{{L^{op}}}}\).


Local Ring Open Cover Inverse Limit Left Adjoint Constant Sheaf 
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Copyright information

© Springer Science+Business Media New York 1991

Authors and Affiliations

  • Ieke Moerdijk
    • 1
  • Gonzalo E. Reyes
    • 2
  1. 1.Mathematical InstituteUniversity of UtrechtUtrechtThe Netherlands
  2. 2.Department of MathematicsUniversity of MontrealMontrealCanada

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