{"title":"Symmetries of the cyclic nerve","authors":"David Ayala, Aaron Mazel-Gee, Nick Rozenblyum","doi":"arxiv-2405.03897","DOIUrl":null,"url":null,"abstract":"We undertake a systematic study of the Hochschild homology, i.e. (the\ngeometric realization of) the cyclic nerve, of $(\\infty,1)$-categories (and\nmore generally of category-objects in an $\\infty$-category), as a version of\nfactorization homology. In order to do this, we codify $(\\infty,1)$-categories\nin terms of quiver representations in them. By examining a universal instance\nof such Hochschild homology, we explicitly identify its natural symmetries, and\nconstruct a non-stable version of the cyclotomic trace map. Along the way we\ngive a unified account of the cyclic, paracyclic, and epicyclic categories. We\nalso prove that this gives a combinatorial description of the $n=1$ case of\nfactorization homology as presented in [AFR18], which parametrizes\n$(\\infty,1)$-categories by solidly 1-framed stratified spaces.","PeriodicalId":501143,"journal":{"name":"arXiv - MATH - K-Theory and Homology","volume":"26 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - K-Theory and Homology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2405.03897","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We undertake a systematic study of the Hochschild homology, i.e. (the
geometric realization of) the cyclic nerve, of $(\infty,1)$-categories (and
more generally of category-objects in an $\infty$-category), as a version of
factorization homology. In order to do this, we codify $(\infty,1)$-categories
in terms of quiver representations in them. By examining a universal instance
of such Hochschild homology, we explicitly identify its natural symmetries, and
construct a non-stable version of the cyclotomic trace map. Along the way we
give a unified account of the cyclic, paracyclic, and epicyclic categories. We
also prove that this gives a combinatorial description of the $n=1$ case of
factorization homology as presented in [AFR18], which parametrizes
$(\infty,1)$-categories by solidly 1-framed stratified spaces.