Higher catoids, higher quantales and their correspondences

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