Higher Catoids, Higher Quantales and their Correspondences
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866912454798213120 |
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| author | Calk, Cameron Malbos, Philippe Pous, Damien Struth, Georg |
| author_facet | Calk, Cameron Malbos, Philippe Pous, Damien Struth, Georg |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_09253 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Higher Catoids, Higher Quantales and their Correspondences Calk, Cameron Malbos, Philippe Pous, Damien Struth, Georg Logic in Computer Science 18A05, 18F75, 06F07, 18N30, 68Q42, 68V15 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. |
| title | Higher Catoids, Higher Quantales and their Correspondences |
| topic | Logic in Computer Science 18A05, 18F75, 06F07, 18N30, 68Q42, 68V15 |
| url | https://arxiv.org/abs/2307.09253 |