Functorial differential spaces and the infinitesimal structure of space-time

TL;DR

This paper introduces a generalized framework for differential geometry using functorial differential spaces enriched with infinitesimals, enabling detailed analysis of space-time's infinitesimal structure, especially near singularities.

Contribution

It develops a new category of functorial differential spaces incorporating infinitesimals via Weil algebras, extending existing differential geometry methods.

Findings

Infinitesimals remain latent during macroscopic evolution.

Infinitesimals become relevant at infinitesimal universe scales.

The framework allows detailed study of initial singularities.

Abstract

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.