高阶Catoids、高阶Quantales及其对应关系
Higher Catoids, Higher Quantales and their Correspondences
AI总结：
本文引入了$ω$-catoids和$ω$-quantales作为高维重写中结构的推广，建立了它们之间的对应关系，并扩展到$(ω,p)$-catoids和卷积$(ω,p)$-quantales，为$ω$-Kleene代数和$ω$-quantale公理的系统性发展提供了支持。
摘要：
我们引入了$ω$-catoids作为（严格）$ω$-范畴的推广，特别是高维重写中由computads或polygraphs生成的高阶路径范畴。我们还引入了$ω$-quantales，推广了最近提出的用于高维重写中代数相干性证明的$ω$-Kleene代数。然后，我们建立了$ω$-catoids与卷积$ω$-quantales之间的对应关系。这些对应关系与Jónsson-Tarski风格的关系结构与带算子格之间的对偶性相关。我们将这些对应关系扩展到$(ω,p)$-catoids（在某些维度以上具有群胚结构的catoids）和卷积$(ω,p)$-quantales，使用某些维度以上的Dedekind quantales来捕捉高维重写中的同伦构造和证明。我们还将它们特化到有限可分解的$(ω,p)$-catoids，这是定义$(ω,p)$-半环和$(ω,p)$-Kleene代数的适当设置。这些构造支持$ω$-Kleene代数和$ω$-quantale公理的系统性发展和证明，改进了最近提到的方法，其中$ω$-Kleene代数的公理是以特定方式引入的。
英文摘要：
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.
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