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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2307.09253 |
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Table of Contents:
- We introduce $ω$-catoids as generalisations of (strict) $ω$-categories and in particular the higher path categories generated by computads or polygraphs in higher-dimensional rewriting. We also introduce $ω$-quantales that generalise the $ω$-Kleene algebras recently proposed for algebraic coherence proofs in higher-dimensional rewriting. We then establish correspondences between $ω$-catoids and convolution $ω$-quantales. These are related to Jónsson-Tarski-style dualities between relational structures and lattices with operators. We extend these correspondences to $(ω,p)$-catoids, catoids with a groupoid structure above some dimension, and convolution $(ω,p)$-quantales, using Dedekind quantales above some dimension to capture homotopic constructions and proofs in higher-dimensional rewriting. We also specialise them to finitely decomposable $(ω, p)$-catoids, an appropriate setting for defining $(ω, p)$-semirings and $(ω, p)$-Kleene algebras. These constructions support the systematic development and justification of $ω$-Kleene algebra and $ω$-quantale axioms, improving on the recent approach mentioned, where axioms for $ω$-Kleene algebras have been introduced in an ad hoc fashion.