  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.
