We introduce \(\omega \) -catoids as generalisations of (strict) \(\omega \) -categories and in particular the higher path categories generated by computads or polygraphs in higher-dimensional rewriting. We also introduce \(\omega \) -quantales that generalise the \(\omega \) -Kleene algebras recently proposed for algebraic coherence proofs in higher-dimensional rewriting. We then establish correspondences between \(\omega \) -catoids and convolution \(\omega \) -quantales. These are related to Jónsson-Tarski-style dualisms between relational structures and lattices with operators. We extend these correspondences to \((\omega,p)\) -catoids, catoids with a groupoid structure above some dimension, and convolution \((\omega,p)\) -quantales, using Dedekind quantales above some dimension to capture homotopic constructions and proofs in higher-dimensional rewriting. We also specialise them to finitely decomposable \((\omega, p)\) -catoids, an appropriate setting for defining \((\omega, p)\) -semirings and \((\omega, p)\) -Kleene algebras. These constructions support the systematic development and justification of \(\omega \) -Kleene algebra and \(\omega \) -quantale axioms, improving on the recent approach mentioned, where axioms for \(\omega \) -Kleene algebras have been introduced in an ad hoc fashion.
