Higher Catoids, Higher Quantales and their Correspondences

Abstract 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.