An isomorphism in a category with a ring object (Q699910)

This article studies one of the fundamental models of synthetic differential geometry (SDG), namely the presheaf category \(\mathcal S ets^{\mathcal A}\), where \(\mathcal A\) is the category of finitely presented rings. For more on SDG and its models see \textit{A. Kock} [``Synthetic differential geometry'', Lond. Math. Lect. Note Ser. 51 (1981; Zbl 0466.51008) and \textit{R. Lavendhomme} [``Basic concepts of synthetic differential geometry'', Kluwer (1996; Zbl 0866.58001)]. The main result of the paper is that the object of units of the ``real line'' in this topos model is isomorphic to the automorphism group \(\text{Aut}(D)\), where \(D\) is the standard object of nilpotents (infinitesimals) of order 2. The brevity of the introduction may be of necessity due to constraints of the journal, however, some more words about SDG, its topos theoretical models and goals would have been helpful for the general reader. As it is the article is accessible only to those well versed in these matters.