The Fermat functors (Q2826241)

The theory of Fermat reals [\textit{P. Giordano}, Adv.\ Math.\ 225, No. 4, 2050--2075 (2010; Zbl 1205.26051)] is a theory of infinitesimal numbers in which infinitesimals are nilpotent (as in synthetic differential geometry), but is still compatible with classical logic. The paper under review provides some fundamentals for doing differential geometry using diffeological spaces and infinitesimals.NEWLINENEWLINEThe author defines a Grothendieck site, called the Fermat site, where objects are open subsets of \(\mathbb R^n\) enlarged with Fermat infinitesimals. Then the goal is to construct comparison functors between concrete sheaves on this site, called \textit{Fermat spaces}, and diffeological spaces in the sense of Souriau. These functors are called ``adding infinitesimals'' and ``deleting infinitesimals''. The author establishes the basic properties of these functors and computes some examples, showing in particular that in his approach a \(1\)-dimensional irrational torus yields an interesting Fermat space, solving a deficiency of Giordano's earlier approach.