The Fermat Functors, Part I: The theory

TL;DR

This paper explores the use of quasi-topos theory to define and analyze functors that add or remove infinitesimals in diffeological spaces, laying groundwork for future differential geometry development.

Contribution

It introduces and studies two novel functors involving infinitesimals in diffeological and Fermat spaces, expanding the theoretical framework for differential geometry with infinitesimals.

Findings

Defined functors adding and deleting infinitesimals in diffeological spaces

Analyzed properties of these functors using quasi-topos theory

Calculated examples illustrating the functors' behavior

Abstract

In this paper, we use some basic quasi-topos theory to study two functors: one adding infinitesimals of Fermat reals to diffeological spaces (which generalize smooth manifolds including singular spaces and infinite dimensional spaces), and the other deleting infinitesimals on Fermat spaces. We study the properties of these functors, and calculate some examples. These serve as fundamentals for developing differential geometry on diffeological spaces using infinitesimals in a future paper.