{"title":"自由分解空间","authors":"Philip Hackney, Joachim Kock","doi":"10.1007/s13348-024-00446-8","DOIUrl":null,"url":null,"abstract":"<p>We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by inert maps. We show that left Kan extension along the inclusion takes general objects to Möbius decomposition spaces and general maps to CULF maps. We establish an equivalence of <span>\\(\\infty \\)</span>-categories <img alt=\"\" src=\"//media.springernature.com/lw177/springer-static/image/art%3A10.1007%2Fs13348-024-00446-8/MediaObjects/13348_2024_446_IEq3_HTML.gif\" style=\"width:177px;max-width:none;\"/>. Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions.</p>","PeriodicalId":50993,"journal":{"name":"Collectanea Mathematica","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Free decomposition spaces\",\"authors\":\"Philip Hackney, Joachim Kock\",\"doi\":\"10.1007/s13348-024-00446-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by inert maps. We show that left Kan extension along the inclusion takes general objects to Möbius decomposition spaces and general maps to CULF maps. We establish an equivalence of <span>\\\\(\\\\infty \\\\)</span>-categories <img alt=\\\"\\\" src=\\\"//media.springernature.com/lw177/springer-static/image/art%3A10.1007%2Fs13348-024-00446-8/MediaObjects/13348_2024_446_IEq3_HTML.gif\\\" style=\\\"width:177px;max-width:none;\\\"/>. Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions.</p>\",\"PeriodicalId\":50993,\"journal\":{\"name\":\"Collectanea Mathematica\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2024-09-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Collectanea Mathematica\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s13348-024-00446-8\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Collectanea Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s13348-024-00446-8","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by inert maps. We show that left Kan extension along the inclusion takes general objects to Möbius decomposition spaces and general maps to CULF maps. We establish an equivalence of \(\infty \)-categories . Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions.
期刊介绍：
Collectanea Mathematica publishes original research peer reviewed papers of high quality in all fields of pure and applied mathematics. It is an international journal of the University of Barcelona and the oldest mathematical journal in Spain. It was founded in 1948 by José M. Orts. Previously self-published by the Institut de Matemàtica (IMUB) of the Universitat de Barcelona, as of 2011 it is published by Springer.
. Although free decomposition spaces are rather simple objects, they abound in combinatorics: it seems that all comultiplications of deconcatenation type arise from free decomposition spaces. We give an extensive list of examples, including quasi-symmetric functions.
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