Einstein Algebras in a Categorical Context

TL;DR

This paper explores Einstein algebras within a categorical framework, analyzing their geometric properties and implications for quantum gravity and noncommutative structures, revealing how different stages affect observability and the algebraic structure.

Contribution

It introduces a categorical perspective on Einstein algebras, examining their geometricity and observability at various stages, including noncommutative cases and alternative smooth algebras.

Findings

Einstein algebras are geometric at certain stages, but infinitesimal levels are unobservable macro-level.

Noncommutative Einstein algebras exhibit different properties, potentially affecting quantum gravity models.

Using various smooth algebras as stages reveals complex structures like parametrized points and loops.

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 smooth 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 smooth algebras, for example those that describe spaces in which with ordinary points coexsist "parametrized points", for example closed curves (loops). We also discuss some other consequences of putting Einstein algebras into the conceptual environment of category theory.