{"title":"Lie algebra models for unstable homotopy theory","authors":"Gijs Heuts","doi":"10.1201/9781351251624-16","DOIUrl":"https://doi.org/10.1201/9781351251624-16","url":null,"abstract":"Quillen showed how to describe the homotopy theory of simply-connected rational spaces in terms of differential graded Lie algebras. Here we survey a generalization of Quillen's results that describes the $v_n$-periodic localizations of homotopy theory (where rational corresponds to $n=0$) in terms of spectral Lie algebras. The latter form an extension of the theory of Lie algebras to the setting of stable homotopy theory. This is a chapter written for the Handbook of Homotopy Theory edited by Haynes Miller.","PeriodicalId":378948,"journal":{"name":"Handbook of Homotopy Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125979704","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An introduction to higher categorical algebra","authors":"David Gepner","doi":"10.1201/9781351251624-13","DOIUrl":"https://doi.org/10.1201/9781351251624-13","url":null,"abstract":"This article is a survey of algebra in the $infty$-categorical context, as developed by Lurie in \"Higher Algebra\", and is a chapter in the \"Handbook of Homotopy Theory\". We begin by introducing symmetric monoidal stable $infty$-categories, such as the derived $infty$-category of a commutative ring, before turning to our main example, the $infty$-category of spectra. We then go on to consider ring spectra and their $infty$-categories of modules, as well as basic constructions such as localization, completion, and dualizability. We conclude with a brief account of the cotangent complex and deformation theory.","PeriodicalId":378948,"journal":{"name":"Handbook of Homotopy Theory","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116489463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Topological cyclic homology","authors":"L. Hesselholt, T. Nikolaus","doi":"10.1201/9781351251624-15","DOIUrl":"https://doi.org/10.1201/9781351251624-15","url":null,"abstract":"This survey of topological cyclic homology is a chapter in the Handbook on Homotopy Theory. We give a brief introduction to topological cyclic homology and the cyclotomic trace map following Nikolaus-Scholze, followed by a proof of B\"okstedt periodicity that closely resembles B\"okstedt's original unpublished proof. We explain the extension of B\"{o}kstedt periodicity by Bhatt-Morrow-Scholze from perfect fields to perfectoid rings and use this to give a purely p-adic proof of Bott periodicity. Finally, we evaluate the cofiber of the assembly map in p-adic topological cyclic homology for the cyclic group of order p and a perfectoid ring of coefficients.","PeriodicalId":378948,"journal":{"name":"Handbook of Homotopy Theory","volume":"7 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"120937078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Polyhedral products and features of their homotopy theory","authors":"A. Bahri, M. Bendersky, F. Cohen","doi":"10.1201/9781351251624-3","DOIUrl":"https://doi.org/10.1201/9781351251624-3","url":null,"abstract":"A polyhedral product is a natural subspace of a Cartesian product that is specified by a simplicial complex. The modern formalism arose as a generalization of the spaces known as moment-angle complexes which were developed within the nascent subject of toric topology. This field, which began as a topological approach to toric geometry and aspects of symplectic geometry, has expanded rapidly in recent years. The investigation of polyhedral products and their homotopy theoretic properties has developed to the point where they are studied in various fields of mathematics far removed from their origin. In this survey, we provide a brief historical overview of the development of this subject, summarize many of the main results and describe applications.","PeriodicalId":378948,"journal":{"name":"Handbook of Homotopy Theory","volume":"76 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124926910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Unstable motivic homotopy theory","authors":"K. Wickelgren, B. Williams","doi":"10.1201/9781351251624-22","DOIUrl":"https://doi.org/10.1201/9781351251624-22","url":null,"abstract":"We give an introduction to unstable motivic homotopy theory of Morel and Voevodsky, and survey some results.","PeriodicalId":378948,"journal":{"name":"Handbook of Homotopy Theory","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126083515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Goodwillie calculus","authors":"G. Arone, Michael Ching","doi":"10.1201/9781351251624-1","DOIUrl":"https://doi.org/10.1201/9781351251624-1","url":null,"abstract":"We survey the theory and applications of Goodwillie's calculus of homotopy functors and related topics.","PeriodicalId":378948,"journal":{"name":"Handbook of Homotopy Theory","volume":"83 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114458178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Topological modular and automorphic forms","authors":"M. Behrens","doi":"10.1201/9781351251624-6","DOIUrl":"https://doi.org/10.1201/9781351251624-6","url":null,"abstract":"This article is a brief survey of the theory of topological modular forms (TMF) and the theory of topological automorphic forms (TAF). It will be a chapter in forthcoming \"Handbook of Homotopy Theory\" edited by Haynes Miller.","PeriodicalId":378948,"journal":{"name":"Handbook of Homotopy Theory","volume":"os-18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127857742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Little discs operads, graph complexes and Grothendieck-Teichmüller groups","authors":"B. Fresse","doi":"10.1201/9781351251624-11","DOIUrl":"https://doi.org/10.1201/9781351251624-11","url":null,"abstract":"This paper is a survey on the homotopy theory of $E_n$-operads written for the new handbook of homotopy theory.","PeriodicalId":378948,"journal":{"name":"Handbook of Homotopy Theory","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125479912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Moduli spaces of manifolds: a user's guide","authors":"Søren Galatius, O. Randal-Williams","doi":"10.1201/9781351251624-12","DOIUrl":"https://doi.org/10.1201/9781351251624-12","url":null,"abstract":"We survey recent work on moduli spaces of manifolds with an emphasis on the role played by (stable and unstable) homotopy theory. The theory is illustrated with several worked examples.","PeriodicalId":378948,"journal":{"name":"Handbook of Homotopy Theory","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129681314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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