Sewing lemma and knitting lemma for metric spaces

TL;DR

This paper generalizes the sewing and knitting lemmas to metric spaces, providing new tools for analyzing Lipschitz paths and homotopy groupoids in various metric contexts.

Contribution

It introduces a generalized sewing lemma for metric spaces and extends it to a two-dimensional knitting lemma, broadening the scope of path and homotopy analysis.

Findings

A sewing lemma for families of complete metric spaces.

A generalization to metric parameter spaces beyond intervals.

A two-dimensional knitting lemma for Lipschitz homotopy groupoids.

Abstract

We state and prove a sewing lemma in the general context of families of complete metric spaces indexed by an interval of the real line, encompassing the flow sewing lemma proved by I. Bailleul in 2015. A further generalisation to other metric parameter spaces P than intervals is moreover proposed, leading to a representation of the groupoid of thin-equivalent Lipschitz paths on P . Under a stronger hypothesis, we finally prove a two-dimensional version, the knitting lemma, which gives rise to a representation of the Lipschitz homotopy groupoid of the parameter space, without thinness condition.