Axiomatic Differential Geometry III-1

TL;DR

This paper introduces a new model theory for axiomatic differential geometry, replacing traditional smooth manifolds with functors on Weil algebras, offering a more natural and conceptual framework.

Contribution

It proposes a geometrically natural model theory based on functors on Weil algebras, contrasting with the artificiality of synthetic differential geometry.

Findings

Replaces smooth manifolds with functors on Weil algebras

Provides a natural and conceptual model theory

Contrasts with synthetic differential geometry's complexity

Abstract

In this paper is proposed a kind of model theory for our axiomatic differential geometry. It is claimed that smooth manifolds, which have occupied the center stage in differential geometry, should be replaced by functors on the category of Weil algebras. Our model theory is geometrically natural and conceptually motivated, while the model theory of synthetic differential geometry is highly artificial and exquisitely technical.