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Model ∞-categories III: the fundamental theorem

Mazel-Gee, Aaron (2021) Model ∞-categories III: the fundamental theorem. New York Journal of Mathematics, 27 . pp. 551-599. ISSN 1076-9803. https://resolver.caltech.edu/CaltechAUTHORS:20210517-103120472

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Abstract

We prove that a model structure on a relative ∞-category (M,W) gives an efficient and computable way of accessing the hom-spaces hom_([M[W-1])(x,y) in the localization. More precisely, we show that when the source x ∈ M is cofibrant and the target y ∈ M is fibrant, then this hom-space is a "quotient" of the hom-space hom_M(x,y) by either of a left homotopy relation or a right homotopy relation.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://nyjm.albany.edu/j/2021/27-22.htmlPublisherArticle
https://arxiv.org/abs/1510.04777arXivDiscussion Paper
Alternate Title:Model infinity-categories III: the fundamental theorem
Additional Information:© 2021 The Author(s). Received December 19, 2017. Published: April 12, 2021. We wish to thank Omar Antolín-Camarena, Tobi Barthel, Clark Barwick, Rune Haugseng, Gijs Heuts, Zhen Lin Low, Mike Mandell, Justin Noel, and Aaron Royer for many very helpful conversations. Additionally, we gratefully acknowledge the financial support provided both by the NSF graduate research fellowship program (grant DGE-1106400) and by UC Berkeley's geometry and topology RTG (grant DMS-0838703) during the time that this work was carried out. As this paper is the culmination of its series, we would also like to take this opportunity to extend our thanks to the people who have most influenced the entire project, without whom it certainly could never have come into existence: Bill Dwyer and Dans Kan and Quillen, for the model-categorical foundations; André Joyal and Jacob Lurie, for the ∞-categorical foundations; Zhen Lin Low, for countless exceedingly helpful conversations; Eric Peterson, for his tireless, dedicated, and impressive TeX support, and for listening patiently to far too many all-too-elaborate "here's where I'm stuck" monologues; David Ayala, for somehow making it through every last one of these papers and providing numerous insightful comments and suggestions, and for providing much-needed encouragement through the tail end of the writing process; Clark Barwick, for many fruitful conversations (including the one that led to the realization that there should exist a notion of "model ∞-categories" in the first place!), and for his consistent enthusiasm for this project; Peter Teichner, for his trust in allowing free rein to explore, for his generous support and provision for so many extended visits to so many institutions around the world, and for his constant advocacy on our behalf; and, lastly and absolutely essentially, Katherine de Kleer, for her boundless love, for her ample patience, and for bringing beauty and excitement into each and every day.
Funders:
Funding AgencyGrant Number
NSF Graduate Research FellowshipDGE-1106400
NSFDMS-0838703
Subject Keywords:∞-category, model structure, hom-space, localization
Classification Code:2010 Mathematics Subject Classifiication: 55U35, 18G55, 55P60, 55U40
Record Number:CaltechAUTHORS:20210517-103120472
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210517-103120472
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:109148
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:17 May 2021 17:54
Last Modified:17 May 2021 17:54

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