How Logic Interacts with Geometry: Infinitesimal Curvature of Categorical Spaces

In category theory, logic and geometry cooperate with each other producing what is known under the name Synthetic Differential Geometry (SDG). The main difference between SDG and standard differential geometry is that the intuitionistic logic of SDG enforces the existence of infinitesimal objects which essentially modify the local structure of spaces considered in SDG. We focus on an "infinitesimal version" of SDG, an infinitesimal $n$-dimensional formal manifold, and develop differential geometry on it. In particular, we show that the Riemann curvature tensor on infinitesimal level is itself infinitesimal. We construct a heuristic model $S^3 \times \mathbb{R} \subset \mathbb{R}^4$ and study it from two perspectives: the perspective of the category SET and that of the so-called topos $\mathcal{G}$ of germ-determined ideals. We show that the fact that in this model the curvature tensor is infinitesimal (in $\mathcal{G}$-perspective) eliminates the existing singularity. A surprising effect is that the hybrid geometry based on the existence of the infinitesimal and the SET levels generates an exotic smooth structure on $\mathbb{R}^4$. We briefly discuss the obtained results and indicate their possible applications.