4 research outputs found
Higher Catoids, Higher Quantales and their Correspondences
We establish modal 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 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 ({\omega},p)-catoids and
convolution ({\omega},p)-quantales suitable for modelling homotopies in higher
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 higher quantale axioms relative to a previous
ad hoc approach.Comment: 46 pages, 8 figure