Axiomatic Differential Geometry II-1 Vector Fields

TL;DR

This paper develops an axiomatic, category-theoretic framework for differential geometry, demonstrating that vector fields on certain objects form a Lie algebra, advancing the theoretical foundations of the field.

Contribution

It introduces a categorical approach to differential geometry and proves that vector fields form a Lie algebra within this framework.

Findings

Vector fields on microlinear Weil exponential objects form a Lie algebra.

Provides a categorical axiomatic foundation for differential geometry.

Extends previous work to include vector fields in the framework.

Abstract

In our previous paper entitled "Axiomatic differential geometry -towards model categories of differential geometry-, we have given a category-theoretic framework of differential geometry. As the first part of our series of papers concerned with differential-geometric developments within the above axiomatic scheme, this paper is devoted to vector fields. The principal result is that the totality of vector fields on a microlinear and Weil exponential object forms a Lie algebra.