{"title":"Matrix Taxonomy and Bourn Localization","authors":"Michael Hoefnagel, Pierre-Alain Jacqmin","doi":"10.1007/s10485-022-09692-y","DOIUrl":null,"url":null,"abstract":"<div><p>In a recent paper (Hoefnagel et al. in Theory Appl Categ 38:737–790, 2022), an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices—the so called ‘matrix properties’. In this paper we extend this algorithm to include matrix properties involving pointedness of a category, such as the properties of a category to be unital, strongly unital or subtractive, for example. Moreover, this extended algorithm can also be used to determine whether a given matrix property is the Bourn localization of another, thus leading to new characterizations of Mal’tsev, majority and arithmetical categories. Using a computer implementation of our algorithm, we can display all such properties given by matrices of fixed dimensions, grouped according to their Bourn localizations, as well as the implications between them.</p></div>","PeriodicalId":7952,"journal":{"name":"Applied Categorical Structures","volume":"30 6","pages":"1305 - 1340"},"PeriodicalIF":0.6000,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Applied Categorical Structures","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10485-022-09692-y","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 1
Abstract
In a recent paper (Hoefnagel et al. in Theory Appl Categ 38:737–790, 2022), an algorithm has been presented for determining implications between a particular kind of category theoretic property represented by matrices—the so called ‘matrix properties’. In this paper we extend this algorithm to include matrix properties involving pointedness of a category, such as the properties of a category to be unital, strongly unital or subtractive, for example. Moreover, this extended algorithm can also be used to determine whether a given matrix property is the Bourn localization of another, thus leading to new characterizations of Mal’tsev, majority and arithmetical categories. Using a computer implementation of our algorithm, we can display all such properties given by matrices of fixed dimensions, grouped according to their Bourn localizations, as well as the implications between them.
在最近的一篇论文中(Hoefnagel et al. In Theory applied Categ 38:37 - 790,2022)，提出了一种算法，用于确定由矩阵表示的特定类型的范畴论性质之间的含义，即所谓的“矩阵性质”。在本文中，我们将该算法扩展到包含涉及范畴的点性的矩阵性质，例如范畴的酉性、强酉性或相减性。此外，该扩展算法还可用于确定给定矩阵的性质是否为另一个矩阵的Bourn局部化，从而产生新的马尔切夫、多数和算术范畴的表征。使用我们算法的计算机实现，我们可以显示固定维矩阵给出的所有这些属性，根据它们的Bourn定位分组，以及它们之间的含义。
期刊介绍：
Applied Categorical Structures focuses on applications of results, techniques and ideas from category theory to mathematics, physics and computer science. These include the study of topological and algebraic categories, representation theory, algebraic geometry, homological and homotopical algebra, derived and triangulated categories, categorification of (geometric) invariants, categorical investigations in mathematical physics, higher category theory and applications, categorical investigations in functional analysis, in continuous order theory and in theoretical computer science. In addition, the journal also follows the development of emerging fields in which the application of categorical methods proves to be relevant.
Applied Categorical Structures publishes both carefully refereed research papers and survey papers. It promotes communication and increases the dissemination of new results and ideas among mathematicians and computer scientists who use categorical methods in their research.
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