Comparing localizations across adjunctions
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- by Carles Casacuberta, Oriol Raventós and Andrew Tonks PDF
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Abstract:
We show that several apparently unrelated formulas involving left or right Bousfield localizations in homotopy theory are induced by comparison maps associated with pairs of adjoint functors. Such comparison maps are used in the article to discuss the existence of functorial liftings of homotopical localizations and cellularizations to categories of algebras over monads acting on model categories, with emphasis on the cases of module spectra and algebras over simplicial operads. Some of our results hold for algebras up to homotopy as well; for example, if $T$ is the reduced monad associated with a simplicial operad and $f$ is any map of pointed simplicial sets, then $f$-localization coincides with $Tf$-localization on spaces underlying homotopy $T$-algebras, and similarly for cellularizations.References
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Additional Information
- Carles Casacuberta
- Affiliation: Facultat de Matemàtiques i Informàtica, Universitat de Barcelona (UB), Gran Via de les Corts Catalanes 585, 08007 Barcelona, Spain
- MR Author ID: 263099
- ORCID: 0000-0002-0133-7831
- Email: carles.casacuberta@ub.edu
- Oriol Raventós
- Affiliation: DLR Institute of Networked Energy Systems, Carl-von-Ossietzky-Str. 15, 26129 Oldenburg, Germany
- ORCID: 0000-0002-0512-4331
- Email: oriol.raventosmorera@dlr.de
- Andrew Tonks
- Affiliation: Department of Mathematics, University of Leicester, Leicester LE1 7RH, United Kingdom
- MR Author ID: 325533
- ORCID: 0000-0002-5456-5002
- Email: apt12@leicester.ac.uk
- Received by editor(s): September 16, 2019
- Received by editor(s) in revised form: October 25, 2020, and December 30, 2020
- Published electronically: August 18, 2021
- Additional Notes: The authors were partially supported by the Agencia Estatal de Investigación and former Spanish ministries under grants MDM-2014-0445, MTM2010-15831, MTM2013-42178-P, and AEI/FEDER grant MTM2016-76453-C2-2-P, as well as by the Agència de Gestió d’Ajuts Universitaris i de Recerca from Generalitat de Catalunya as members of the 2009 SGR 119, 2014 SGR 114, and 2017 SGR 585 research groups. The second author was supported by project CZ.1.07/2.3.00/20.0003 of the Operational Programme Education for Competitiveness of the Ministerstvo školství, mládeže a tělovýchovy of the Czech Republic and by grant SFB 1085 (Higher Invariants) of the Deutsche Forschungsgemeinschaft
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 374 (2021), 7811-7865
- MSC (2020): Primary 55P60, 18A40; Secondary 55P48
- DOI: https://doi.org/10.1090/tran/8382
- MathSciNet review: 4328684