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On the unicity of the theory of higher categories


Authors: Clark Barwick and Christopher Schommer-Pries
Journal: J. Amer. Math. Soc. 34 (2021), 1011-1058
MSC (2020): Primary 18N60, 18N65
DOI: https://doi.org/10.1090/jams/972
Published electronically: April 20, 2021
MathSciNet review: 4301559
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Abstract: We axiomatise the theory of $(\infty ,n)$-categories. We prove that the space of theories of $(\infty ,n)$-categories is a $B(\mathbb {Z}/2)^n$. We prove that Rezk’s complete Segal $\Theta _n$ spaces, Simpson and Tamsamani’s Segal $n$-categories, the first author’s $n$-fold complete Segal spaces, Kan and the first author’s $n$-relative categories, and complete Segal space objects in any model of $(\infty , n-1)$-categories all satisfy our axioms. Consequently, these theories are all equivalent in a manner that is unique up to the action of $(\mathbb {Z}/2)^n$.


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Additional Information

Clark Barwick
Affiliation: School of Mathematics, University of Edinburgh, Edinburgh EH9 3FD, United Kingdom
MR Author ID: 780183
ORCID: 0000-0002-2362-3441

Christopher Schommer-Pries
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556
MR Author ID: 733206

Received by editor(s): December 28, 2018
Received by editor(s) in revised form: August 4, 2020, and December 23, 2020
Published electronically: April 20, 2021
Additional Notes: The second author was supported by NSF fellowship DMS-0902808.
Article copyright: © Copyright 2021 Clark Barwick; Christopher Schommer-Pries