Calk, C., Malbos, P., Pous, D. et al. (1 more author) (2025) Higher catoids, higher quantales and their correspondences. Applied Categorical Structures, 33 (4). 25. ISSN 0927-2852
Abstract
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 dualisms 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
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © The Author(s) 2025. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. |
Keywords: | Higher catoids; Higher quantales; Multisemigroups; Convolution algebras; Categorification; Higher rewriting |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Engineering (Sheffield) > Department of Computer Science (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 30 Jun 2025 14:46 |
Last Modified: | 30 Jun 2025 14:46 |
Status: | Published |
Publisher: | Springer Science and Business Media LLC |
Refereed: | Yes |
Identification Number: | 10.1007/s10485-025-09817-z |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:228554 |