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We also introduce <jats:inline-formula>\n              <jats:tex-math>$$\\omega $$<\/jats:tex-math>\n            <\/jats:inline-formula>-quantales that generalise the <jats:inline-formula>\n              <jats:tex-math>$$\\omega $$<\/jats:tex-math>\n            <\/jats:inline-formula>-Kleene algebras recently proposed for algebraic coherence proofs in higher-dimensional rewriting. We then establish correspondences between <jats:inline-formula>\n              <jats:tex-math>$$\\omega $$<\/jats:tex-math>\n            <\/jats:inline-formula>-catoids and convolution <jats:inline-formula>\n              <jats:tex-math>$$\\omega $$<\/jats:tex-math>\n            <\/jats:inline-formula>-quantales. These are related to J\u00f3nsson-Tarski-style dualisms between relational structures and lattices with operators. We extend these correspondences to <jats:inline-formula>\n              <jats:tex-math>$$(\\omega,p)$$<\/jats:tex-math>\n            <\/jats:inline-formula>-catoids, catoids with a groupoid structure above some dimension, and convolution <jats:inline-formula>\n              <jats:tex-math>$$(\\omega,p)$$<\/jats:tex-math>\n            <\/jats:inline-formula>-quantales, using Dedekind quantales above some dimension to capture homotopic constructions and proofs in higher-dimensional rewriting. We also specialise them to finitely decomposable <jats:inline-formula>\n              <jats:tex-math>$$(\\omega, p)$$<\/jats:tex-math>\n            <\/jats:inline-formula>-catoids, an appropriate setting for defining <jats:inline-formula>\n              <jats:tex-math>$$(\\omega, p)$$<\/jats:tex-math>\n            <\/jats:inline-formula>-semirings and <jats:inline-formula>\n              <jats:tex-math>$$(\\omega, p)$$<\/jats:tex-math>\n            <\/jats:inline-formula>-Kleene algebras. These constructions support the systematic development and justification of <jats:inline-formula>\n              <jats:tex-math>$$\\omega $$<\/jats:tex-math>\n            <\/jats:inline-formula>-Kleene algebra and <jats:inline-formula>\n              <jats:tex-math>$$\\omega $$<\/jats:tex-math>\n            <\/jats:inline-formula>-quantale axioms, improving on the recent approach mentioned, where axioms for <jats:inline-formula>\n              <jats:tex-math>$$\\omega $$<\/jats:tex-math>\n            <\/jats:inline-formula>-Kleene algebras have been introduced in an ad hoc fashion.<\/jats:p>","DOI":"10.1007\/s10485-025-09817-z","type":"journal-article","created":{"date-parts":[[2025,6,27]],"date-time":"2025-06-27T12:33:01Z","timestamp":1751027581000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Higher Catoids, Higher Quantales and their Correspondences"],"prefix":"10.1007","volume":"33","author":[{"given":"Cameron","family":"Calk","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Philippe","family":"Malbos","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Damien","family":"Pous","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Georg","family":"Struth","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,6,27]]},"reference":[{"key":"9817_CR1","unstructured":"Bezem, M., Klop, J.W., de Vrijer, R. 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