Diffeological, Fr\"{o}licher, and Differential Spaces

TL;DR

This paper explores and compares various generalizations of differential calculus on sets, focusing on diffeological, Fr"{o}licher, and differential spaces, illustrating their relationships through examples.

Contribution

It clarifies the relationships between diffeological, Fr"{o}licher, and differential structures with illustrative examples.

Findings

Demonstrates the connections between different generalized differential structures.

Provides examples illustrating the relationships among these structures.

Clarifies the conceptual framework for generalized differential calculus.

Abstract

Differential calculus on Euclidean spaces has many generalisations. In particular, on a set $X$, a diffeological structure is given by maps from open subsets of Euclidean spaces to $X$, a differential structure is given by maps from $X$ to $\mathbb{R}$, and a Fr\"{o}licher structure is given by maps from $\mathbb{R}$ to $X$ as well as maps from $X$ to $\mathbb{R}$. We illustrate the relations between these structures through examples.