Smooth Manifolds vs Differential triads

TL;DR

This paper explores the concept of differentiable maps within Abstract Differential Geometry, establishing conditions for the uniqueness of differentials and showing that smooth manifolds are a full subcategory of differential triads, with implications for physics.

Contribution

It proves that smooth maps between smooth manifolds have a unique differential in the setting of Abstract Differential Geometry, linking classical and abstract frameworks.

Findings

Smooth maps admit a unique differential in the abstract setting

Smooth manifolds form a full subcategory of differential triads

Results have implications for physical theories

Abstract

We consider differentiable maps in the setting of Abstract Differential Geometry and we study the conditions that ensure the uniqueness of differentials in this setting. In particular, we prove that smooth maps between smooth manifolds admit a unique differential, coinciding with the usual one. Thus smooth manifolds form a full subcategory of the category of differential triads, a result with physical implications.