Deformations of normed groupoids and differential calculus. First part

TL;DR

This paper explores the algebraic structures called normed groupoids with delta-structures, revealing their deep connection to differential calculus on metric spaces and extending these ideas beyond metric spaces using groupoids.

Contribution

It establishes a formal link between algebraic properties of normed groupoids with dilatation structures and differential calculus on metric spaces, extending the framework beyond metric spaces.

Findings

Algebraic properties of normed groupoids correspond to differential calculus on metric spaces

Dilatation structures in normed groups relate to metric space calculus

Results extend to non-metric spaces via groupoid language

Abstract

Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with $\delta$-structures. Algebraic study of normed groups endowed with dilatation structures is contained in the differential calculus on metric spaces. Thus all algebraic properties of the small world of normed groups with dilatation structures have equivalent formulations (of comparable complexity) in the big world of metric spaces admitting a differential calculus. Moreover these results non trivially extend beyond metric spaces, by using the language of groupoids.