逻辑如何与几何相互作用：范畴空间的无穷小曲率
How Logic Interacts with Geometry: Infinitesimal Curvature of Categorical Spaces
摘要：
在范畴论中，逻辑和几何相互合作，产生了所谓的综合微分几何（SDG）。SDG和标准微分几何的主要区别在于SDG的直觉主义逻辑强制了无穷小对象的存在，这些对象本质上修改了SDG中考虑的空间的局部结构。我们研究了SDG的一个“无穷小版本”，一个无穷小的$n$维形式流形，并在其上发展了微分几何，特别地，我们证明了无穷小水平上的Riemann曲率张量本身也是无穷小的.我们构造了一个启发式模型$S^3 \times \mathbb{R} \subset \mathbb{R}^4$，并从两个角度研究它：范畴SET的角度和所谓的拓扑$\mathcal{G}$的胚决定的理想。我们表明，在这个模型中的曲率张量是无穷小的事实（在$\mathcal{G}$的角度）消除了现有的奇异性。一个令人惊讶的效果是，混合几何的基础上存在的无穷小和SET水平产生一个奇异的光滑结构的$\mathbb{R}^4$。我们简要地讨论了所获得的结果，并指出其可能的应用。
英文摘要：
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.
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