The free tangent structure

TL;DR

This paper compares two abstract frameworks for tangent structures in differential geometry, Synthetic Differential Geometry and Tangent Structures, aiming to clarify their relationship.

Contribution

It provides a precise comparison between SDG and tangent structures, elucidating their conceptual connections and differences.

Findings

Establishes formal links between SDG and tangent structures.

Clarifies the axiomatic foundations of tangent structures.

Enhances understanding of differential geometry abstractions.

Abstract

At the heart of differential geometry is the construction of the tangent bundle of a manifold. There are various abstractions of this construction, and this paper seeks to compare two of them: Synthetic Differential Geometry (SDG) and Tangent Structures. Tangent structure is defined via giving an underlying category M and a tangent functor T along with a list of natural transformations satisfying a set of axioms, then detailing the behaviour of T in the category End(M). SDG on the other hand is defined through the use of Weil algebras. The aim of this paper is to present a more precise relationship between the two approaches for describing tangent structures.