Logical opens and real numbers in topoi
This paper studies open subobjects and their relationship with real numbers in the context of gross topoi whose objects themselves have some geometrical content, e.g. models of synthetic differential geometry. Let \({\mathcal E}\to^{p}{\mathcal S}\) be a Grothendieck topos, where \({\mathcal S}\) is the topos of classical sets. Let \({\mathfrak C}\) be a category with finite limits and \({\mathfrak L}\) be a class of maps of \({\mathfrak C}\) closed under isomorphisms and composition, and stable under pullback by an arbitrary map in \({\mathfrak C}\). Furthermore, a family \(X_{\alpha}\to X\) of maps in \({\mathfrak L}\) is an \({\mathfrak L}\)-cover iff its family of global sections is a surjective family of sets. Assume that \({\mathcal E}\) is defined by the site \({\mathfrak C}\) furnished with the \({\mathfrak L}\)-cover topology. One can always define the notion of a Penon open subobject of X (U\(\hookrightarrow X\) is Penon open iff given \(p\in U\), \(\neg (q=p)\vee (q\in U)\) holds for all \(q\in X)\), however in this context the author also defines the notion of an \({\mathfrak L}\)-open subobject. Every Penon open subobject of X is \({\mathfrak L}\)-open and the converse holds if X is \({\mathfrak L}\)-Hausdorff (\(\neg \Delta \hookrightarrow X\times X\) is \({\mathfrak L}\)-open). The Penon opens and \({\mathfrak L}\)-opens of X both form locales and using them one can define topologies, denoted Base(X) and Logical(X) respectively, on the set of points \(p_*(X).\) After some detailed study of these notions, the main result is proved: it states that if \({\mathbb{R}}_ E\) is the sheaf of Dedekind reals in \({\mathcal E}\) and if \(X\in {\mathfrak C}\), then \({\mathbb{R}}_ E(X)=Cont(Base(X), {\mathbb{R}})\), that is, it is the sheaf of continuous real valued functions on Base(X). In particular, for well adapted models of SDG, one can show that \(Base({\mathbb{R}}_ E)\) is \({\mathbb{R}}\) with the usual topology. These connections with real numbers are made possible by the extensive investigation of the relationship between Penon opens and real numbers objects, which takes up the first half of the paper.
scientific article; zbMATH DE number 3983449
scientific article