On the unicity of the theory of higher categories
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- by Clark Barwick and Christopher Schommer-Pries;
- J. Amer. Math. Soc. 34 (2021), 1011-1058
- DOI: https://doi.org/10.1090/jams/972
- Published electronically: April 20, 2021
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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 $(\mathbb {Z}/2)^n$.References
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Bibliographic 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: https://doi.org/10.1090/jams/972
- MathSciNet review: 4301559