{"title":"A combinatorial model for the path fibration","authors":"Manuel Rivera, Samson Saneblidze","doi":"10.1007/s40062-018-0216-4","DOIUrl":null,"url":null,"abstract":"<p>We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set <i>X</i> we associate a necklical set <span>\\({\\widehat{{\\varvec{\\Omega }}}}X\\)</span> such that its geometric realization <span>\\(|{\\widehat{{\\varvec{\\Omega }}}}X|\\)</span>, a space built out of gluing cubical cells, is homotopy equivalent to the based loop space on |<i>X</i>| and the differential graded module of chains <span>\\(C_*({\\widehat{{\\varvec{\\Omega }}}}X)\\)</span> is a differential graded associative algebra generalizing Adams’ cobar construction.</p>","PeriodicalId":49034,"journal":{"name":"Journal of Homotopy and Related Structures","volume":"14 2","pages":"393 - 410"},"PeriodicalIF":0.5000,"publicationDate":"2018-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s40062-018-0216-4","citationCount":"6","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Homotopy and Related Structures","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s40062-018-0216-4","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 6
Abstract
We introduce the abstract notion of a necklical set in order to describe a functorial combinatorial model of the path fibration over the geometric realization of a path connected simplicial set. In particular, to any path connected simplicial set X we associate a necklical set \({\widehat{{\varvec{\Omega }}}}X\) such that its geometric realization \(|{\widehat{{\varvec{\Omega }}}}X|\), a space built out of gluing cubical cells, is homotopy equivalent to the based loop space on |X| and the differential graded module of chains \(C_*({\widehat{{\varvec{\Omega }}}}X)\) is a differential graded associative algebra generalizing Adams’ cobar construction.
期刊介绍：
Journal of Homotopy and Related Structures (JHRS) is a fully refereed international journal dealing with homotopy and related structures of mathematical and physical sciences.
Journal of Homotopy and Related Structures is intended to publish papers on
Homotopy in the broad sense and its related areas like Homological and homotopical algebra, K-theory, topology of manifolds, geometric and categorical structures, homology theories, topological groups and algebras, stable homotopy theory, group actions, algebraic varieties, category theory, cobordism theory, controlled topology, noncommutative geometry, motivic cohomology, differential topology, algebraic geometry.
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