Higher Catoids, Higher Quantales and their Correspondences

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 J\'onsson-Tarski-style dualities 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.