Sewing lemma and knitting lemma for metric spaces
Charles H. A. Curry, Dominique Manchon
TL;DR
The paper broadens the sewing lemma beyond Banach-space-valued two-variable maps to a general framework of complete metric spaces indexed by a parameter space, emphasizing a groupoid viewpoint for path concatenation. It proves a main sewing theorem in this setting, and proposes a conjectural generalization to arbitrary metric parameter spaces P via thin-equivalent Lipschitz paths. Under a stronger hypothesis, it introduces a knitting lemma that yields a representation of the Lipschitz homotopy groupoid, connecting two-dimensional concatenation with holonomy-like structure. Together, these results offer a versatile, algebraically informed toolkit for constructing flows driven by rough data across generalized spaces and for relating local approximation to global Lipschitz-path groupoid actions.
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.