Sewing lemma and knitting lemma for metric spaces

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.