According to the basic idea of category theory, any Einstein algebra, essentially an algebraic formulation of general relativity, can be considered from the point of view of any object of the category of C^∞-algebras; such an object is then called a stage. If we contemplate a given Einstein algebra from the point of view of the stage, which we choose to be an “algebra with infinitesimals” (Weil algebra), then we can suppose it penetrates a submicroscopic level, on which quantum gravity might function. We apply Vinogradov's notion of geometricity (adapted to this situation), and show that the corresponding algebra is geometric, but then the infinitesimal level is unobservable from the macro-level. However, the situation can change if a given algebra is noncommutative. An analogous situation occurs when as stages, instead of Weil algebras, we take many other C^∞-algebras, for example those that describe spaces in which with ordinary points coexist “parametrised points”, for example closed curves (loops). We also discuss some other consequences of putting Einstein algebras into the conceptual environment of category theory.

根据范畴论的基本思想，任何爱因斯坦代数，本质上都是广义相对论的代数表述，都可以从C ^∞范畴的任何对象的角度来考虑-代数；这样的对象称为舞台。如果我们从阶段的角度考虑一个给定的爱因斯坦代数，我们选择它为“无穷小代数”（韦尔代数），那么我们可以假设它渗透到亚微观层面，量子引力可能在这个层面上发挥作用。我们应用维诺格拉多夫的几何概念（适应这种情况），并证明相应的代数是几何的，但无穷小层面从宏观层面是不可观察的。然而，如果给定的代数是不可交换的，情况就会改变。当我们采用许多其他C ^∞作为阶段而不是 Weil 代数时，就会出现类似的情况-代数，例如那些描述与普通点共存“参数化点”的空间的代数，例如闭合曲线（环）。我们还讨论了将爱因斯坦代数放入范畴论概念环境中的一些其他后果。