{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,20]],"date-time":"2025-08-20T13:00:18Z","timestamp":1755694818718,"version":"3.40.5"},"reference-count":17,"publisher":"Cambridge University Press (CUP)","issue":"2","license":[{"start":{"date-parts":[[2020,2,6]],"date-time":"2020-02-06T00:00:00Z","timestamp":1580947200000},"content-version":"unspecified","delay-in-days":5,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Struct. Comp. Sci."],"published-print":{"date-parts":[[2020,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The homotopical Squier\u2019s theorem relates rewriting properties of a presentation of a monoid with homotopical invariants of this monoid. This theorem has since been extended by Guiraud and Malbos, yielding a so-called polygraphic resolution of a monoid starting from a presentation with suitable rewriting properties. In this article, we argue that cubical categories are a more natural setting in which to express and possibly extend Guiraud and Malbos construction. As a proof-of-concept, we give a new proof of Squier\u2019s homotopical theorem using cubical categories.<\/jats:p>","DOI":"10.1017\/s0960129520000018","type":"journal-article","created":{"date-parts":[[2020,2,6]],"date-time":"2020-02-06T10:25:56Z","timestamp":1580984756000},"page":"159-172","source":"Crossref","is-referenced-by-count":2,"title":["A cubical Squier\u2019s theorem"],"prefix":"10.1017","volume":"30","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9680-7129","authenticated-orcid":false,"given":"Maxime","family":"Lucas","sequence":"first","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2020,2,6]]},"reference":[{"key":"S0960129520000018_ref12","doi-asserted-by":"publisher","DOI":"10.1016\/j.jpaa.2016.09.005"},{"key":"S0960129520000018_ref5","first-page":"163","article-title":"Double categories, 2-categories, thin structures and connections","volume":"5","author":"Brown","year":"1999","journal-title":"Theory and Applications of Categories"},{"key":"S0960129520000018_ref10","doi-asserted-by":"publisher","DOI":"10.1017\/S0960129516000220"},{"key":"S0960129520000018_ref7","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2010.01.022"},{"key":"S0960129520000018_ref8","doi-asserted-by":"publisher","DOI":"10.1017\/S096012951100065X"},{"key":"S0960129520000018_ref9","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2012.05.010"},{"key":"S0960129520000018_ref17","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(87)90137-X"},{"key":"S0960129520000018_ref3","unstructured":"Brown, R. and Higgins, P. J. (1977). Sur les complexes crois\u00e9s, \u03c9-groupo\u00efdes, et T-complexes. C. R. Acad. Sci. Paris S\u00e9r. A-B 285 (16), A997\u2013A999."},{"key":"S0960129520000018_ref15","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(94)90175-9"},{"key":"S0960129520000018_ref1","doi-asserted-by":"publisher","DOI":"10.1006\/aima.2001.2069"},{"key":"S0960129520000018_ref11","first-page":"60","article-title":"Thin elements and commutative shells in cubical \u03c9-categories","volume":"14","author":"Higgins","year":"2005","journal-title":"Theory and Applications of Categories"},{"key":"S0960129520000018_ref2","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/230\/03337"},{"key":"S0960129520000018_ref14","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(87)90129-0"},{"key":"S0960129520000018_ref16","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(76)90013-X"},{"key":"S0960129520000018_ref13","doi-asserted-by":"crossref","first-page":"191","DOI":"10.21136\/HS.2018.06","article-title":"Cubical (\u03c9,p)-categories","volume":"2","author":"Lucas","year":"2018","journal-title":"Higher Structures"},{"key":"S0960129520000018_ref6","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3975(93)90054-W"},{"key":"S0960129520000018_ref4","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(81)90018-9"}],"container-title":["Mathematical Structures in Computer Science"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S0960129520000018","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,26]],"date-time":"2023-09-26T09:44:56Z","timestamp":1695721496000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S0960129520000018\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,2]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2020,2]]}},"alternative-id":["S0960129520000018"],"URL":"https:\/\/doi.org\/10.1017\/s0960129520000018","relation":{},"ISSN":["0960-1295","1469-8072"],"issn-type":[{"type":"print","value":"0960-1295"},{"type":"electronic","value":"1469-8072"}],"subject":[],"published":{"date-parts":[[2020,2]]}}}