Axiomatic Differential Geometry III-2

TL;DR

This paper explores the axiomatic foundations of differential geometry within a categorical framework, demonstrating that functor categories from Weil algebras to a suitable base category are cartesian closed, thus formalizing key geometric concepts.

Contribution

It establishes that functor categories from Weil algebras to a complete, cartesian closed category are themselves cartesian closed, providing a categorical axiomatization for differential geometry.

Findings

Functor categories from Weil algebras are cartesian closed.

Axiomatization of differential geometry using Weil functors.

Formal framework for differential geometry in category theory.

Abstract

Given a complete and (locally) cartesian closed category U, it is shown that the category of functors from the category of Weil algebras to the category U is (locally, resp.) cartesian closed. The corresponding axiomatization for differential geometry based upon Weil functors is then given.