C∞-Rings as Variable Spaces
In this chapter we will consider the geometric aspect of C ∞-rings. As a first step, we will introduce the category L of loci or formal C ∞-varieties. These are just the duals of finitely generated C ∞- rings. This category L contains the usual category of manifolds, but also other useful objects such as infinitesimal spaces. However, function spaces can in general not be constructed in L. This is only possible when the domain space is sufficiently small (the dual of a Weil algebra).
KeywordsVariable Space Open Cover Natural Transformation Weil Algebra Opposite Category
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