Higher Catoids, Higher Quantales and their Correspondences
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We establish modal correspondences between omega-catoids and convolution omega-quantales. These are related to Jónsson-Tarski style-dualities between relational structures and lattices with operators. We introduce omega-catoids as generalisations of (strict) omega-categories and in particular of the higher path categories generated by polygraphs (or computads) in higher rewriting. Convolution omega-quantales generalise the powerset omega-Kleene algebras recently proposed for algebraic coherence proofs in higher rewriting to weighted variants. We extend these correspondences to (ω,p)-catoids and convolution (ω,p)-quantales suitable for modelling homotopies in higher 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 higher quantale axioms relative to a previous ad hoc approach.
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