Two Archimedean Models for Synthetic Calculus

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