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 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.