{"title":"A relative 2–nerve","authors":"F. Garc'ia, Tobias Dyckerhoff, Walker H. Stern","doi":"10.2140/agt.2020.20.3147","DOIUrl":null,"url":null,"abstract":"In this work, we introduce a 2-categorical variant of Lurie's relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to $\\infty$-categorical localizations, corresponds to Lurie's scaled unstraightening equivalence. In this $\\infty$-bicategorical context, the relative 2-nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie's relative nerve when restricted to 1-categories.","PeriodicalId":8433,"journal":{"name":"arXiv: Algebraic Topology","volume":"110 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2019-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv: Algebraic Topology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2140/agt.2020.20.3147","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 5
Abstract
In this work, we introduce a 2-categorical variant of Lurie's relative nerve functor. We prove that it defines a right Quillen equivalence which, upon passage to $\infty$-categorical localizations, corresponds to Lurie's scaled unstraightening equivalence. In this $\infty$-bicategorical context, the relative 2-nerve provides a computationally tractable model for the Grothendieck construction which becomes equivalent, via an explicit comparison map, to Lurie's relative nerve when restricted to 1-categories.