{"title":"Hom weak ω-categories of a weak ω-category","authors":"Thomas Cottrell, Soichiro Fujii","doi":"10.1017/S0960129522000111","DOIUrl":null,"url":null,"abstract":"Abstract Classical definitions of weak higher-dimensional categories are given inductively, for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak n-categories for all natural numbers n, or of weak \n$\\omega$\n -categories, take more sophisticated approaches, and the nature of the ‘hom is often not immediate from the definitions’. In this paper, we focus on Leinster’s definition of weak \n$\\omega$\n -category based on an earlier definition by Batanin and construct, for each weak \n$\\omega$\n -category \n$\\mathcal{A}$\n , an underlying (weak \n$\\omega$\n -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak \n$\\omega$\n -category \n$\\mathcal{A}(x,y)$\n . We also show that our construction is functorial with respect to weak \n$\\omega$\n -functors introduced by Garner.","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":"32 1","pages":"420 - 441"},"PeriodicalIF":0.4000,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Structures in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/S0960129522000111","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 1
Abstract
Abstract Classical definitions of weak higher-dimensional categories are given inductively, for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak n-categories for all natural numbers n, or of weak
$\omega$
-categories, take more sophisticated approaches, and the nature of the ‘hom is often not immediate from the definitions’. In this paper, we focus on Leinster’s definition of weak
$\omega$
-category based on an earlier definition by Batanin and construct, for each weak
$\omega$
-category
$\mathcal{A}$
, an underlying (weak
$\omega$
-category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak
$\omega$
-category
$\mathcal{A}(x,y)$
. We also show that our construction is functorial with respect to weak
$\omega$
-functors introduced by Garner.
期刊介绍:
Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.