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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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On the unicity of the theory of higher categories
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by Clark Barwick and Christopher Schommer-Pries HTML | PDF
J. Amer. Math. Soc. 34 (2021), 1011-1058


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.
  • © Copyright 2021 Clark Barwick; Christopher Schommer-Pries
  • Journal: J. Amer. Math. Soc. 34 (2021), 1011-1058
  • MSC (2020): Primary 18N60, 18N65
  • DOI:
  • MathSciNet review: 4301559