We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding a ring of smooth functions into the category of loci. This permits us to define a category of functorial differential spaces. By suitably choosing various algebras as "stages" in this category, one obtains various classes of differential spaces, both known from the literature and many so far unknown. In particular, if one chooses a Weil algebra, infinitesimals are produced. We study the case with some Weil algebra which allows us to fully develop the corresponding differential geometry with infinitesimals. To test the behavior of infinitesimals, we construct a simplified RWFL cosmological model. As it should be expected, infinitesimals remain latent during the entire macroscopic evolution (regarded backwards in time), and come into play only when the universe attains infinitesimal dimensions. Then they penetrate into the structure of the initial singularity.

中文翻译：

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函数微分空间和时空的无穷小结构

我们将微分空间概念概括为发展微分几何的工具，并用无穷小丰富这种几何，使我们能够深入空间的超精细结构。这是通过 Yoneda 将平滑函数环嵌入到轨迹类别中来实现的。这允许我们定义一个泛函微分空间的范畴。通过适当地选择各种代数作为这一类中的“阶段”，人们可以获得各种类别的微分空间，既有文献中已知的，也有许多迄今为止未知的。特别是，如果选择 Weil 代数，则会产生无穷小。我们用一些 Weil 代数来研究这个案例，这使我们能够充分开发相应的无穷小微分几何。为了测试无穷小的行为，我们构建了一个简化的 RWFL 宇宙学模型。正如所料，无穷小在整个宏观演化过程中都是潜伏的（时间倒流），只有在宇宙达到无穷小维度时才会发挥作用。然后它们渗透到初始奇点的结构中。