Einstein algebras in a categorical context

Abstract

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.