Sewing on Thin Groupoids, Knitting, and Based Holonomy for Lipschitz Paths

Abstract

An approximate action of the pair groupoid of a metric space determines an action of its Lipschitz-thin groupoid on complete extended metric fibers, uniquely characterized by its local sewing estimate. We prove this assertion under a superlinear three-point defect estimate and a Lipschitz bound for finite products. As a result, we obtain a positive proof of Curry and Manchon's sewing conjecture [2], with the necessary path-scaling correction to its estimate. Further restriction to isotropy constructs the based holonomy anticipated in their Remark 4.15 as a representation of based Lipschitz loops modulo thin equivalence. Rectangular comparison estimates give full relative Lipschitz-homotopy descent, in particular under their strong knitting hypothesis and under any three point estimate of total order greater than two. We prove the corresponding flatness and basepoint-covariance statements. An area model shows that the order threshold is sharp. We also separate the exponential estimate from the general knitting assumptions and exhibit a compact disk metric for which strong knitting does not imply invariance under merely continuous homotopy. As an application, for controlled fields on a metric space, we prove that the classical rough integral descends to the thin groupoid and admits substitution with a common rough controller.

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BibTeX RIS

BibTeX

@misc{teixeira2026sewing,
  title = {{Sewing on Thin Groupoids, Knitting, and Based Holonomy for Lipschitz Paths}},
  author = {Alexandre Reggiolli Teixeira},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23831},
  eprint = {2609.23831},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23831},
}

RIS

TY  - UNPB
TI  - Sewing on Thin Groupoids, Knitting, and Based Holonomy for Lipschitz Paths
AU  - Alexandre Reggiolli Teixeira
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23831
UR  - https://arxiv.org/abs/2609.23831
AB  - An approximate action of the pair groupoid of a metric space determines an action of its Lipschitz-thin groupoid on complete extended metric fibers, uniquely characterized by its local sewing estimate. We prove this assertion under a superlinear three-point defect estimate and a Lipschitz bound for finite products. As a result, we obtain a positive proof of Curry and Manchon's sewing conjecture [2], with the necessary path-scaling correction to its estimate. Further restriction to isotropy constructs the based holonomy anticipated in their Remark 4.15 as a representation of based Lipschitz loops modulo thin equivalence. Rectangular comparison estimates give full relative Lipschitz-homotopy descent, in particular under their strong knitting hypothesis and under any three point estimate of total order greater than two. We prove the corresponding flatness and basepoint-covariance statements. An area model shows that the order threshold is sharp. We also separate the exponential estimate from the general knitting assumptions and exhibit a compact disk metric for which strong knitting does not imply invariance under merely continuous homotopy. As an application, for controlled fields on a metric space, we prove that the classical rough integral descends to the thin groupoid and admits substitution with a common rough controller.
ER  -

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