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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Homotopy theory of algebras of substitudes and their localisation
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by Michael Batanin and David White PDF
Trans. Amer. Math. Soc. 375 (2022), 3569-3640 Request permission


We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler and operads coloured by a category) equipped with a (semi)model structure lifted from the model structure on the underlying presheaves. We are especially interested in the case when the model structure on presheaves is a Cisinski style localisation with respect to a proper Grothendieck fundamental localiser. For example, for $\mathtt {W}= \mathtt {W}_{\infty }$ the minimal fundamental localiser, the local objects in such a localisation are locally constant presheaves, and local algebras of substitudes are exactly algebras whose underlying presheaves are locally constant.

We investigate when this localisation has nice properties. We single out a class of such substitudes which we call left localisable and show that the substitudes for $n$-operads, symmetric, and braided operads are in this class. As an application we develop a homotopy theory of higher braided operads and prove a stabilisation theorem for their $\mathtt {W}_k$-localisations. This theorem implies, in particular, a generalisation of the Baez-Dolan stabilisation hypothesis for higher categories.

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Additional Information
  • Michael Batanin
  • Affiliation: Institute of Mathematics of Czech Academy of Sciences, Zitna 25, Prague 1, Czech Republic
  • MR Author ID: 249592
  • Email:
  • David White
  • Affiliation: Department of Mathematics and Computer Science, Denison University, Granville, Ohio 43023
  • Email:
  • Received by editor(s): February 5, 2020
  • Received by editor(s) in revised form: November 6, 2021
  • Published electronically: February 17, 2022
  • Additional Notes: The first author was financially supported by Max Plank Institut für Mathematik and Institut des Hautes Étude Scientifiques. The first author was also financially supported by Praemium Academiæ of M. Markl, RVO: 67985840 and the grant GAČR EXPRO 19-28628X. The second author was supported by the National Science Foundation under Grant No. IIA-1414942, the Australian Academy of Science, and the Australian Category Theory Seminar.
  • © Copyright 2022 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 375 (2022), 3569-3640
  • MSC (2020): Primary 18D20, 55P48
  • DOI:
  • MathSciNet review: 4402670