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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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An $(\infty ,2)$-categorical pasting theorem
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by Philip Hackney, Viktoriya Ozornova, Emily Riehl and Martina Rovelli HTML | PDF
Trans. Amer. Math. Soc. 376 (2023), 555-597


We show that any pasting diagram in any $(\infty ,2)$-category has a homotopically unique composite. This is achieved by showing that the free 2-category generated by a pasting scheme is the homotopy colimit of its cells as an $(\infty ,2)$-category. We prove this explicitly in the simplicial categories model and then explain how to deduce the model-independent statement from that calculation.
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Additional Information
  • Philip Hackney
  • Affiliation: Department of Mathematics, University of Louisiana at Lafayette, Louisiana
  • MR Author ID: 758568
  • ORCID: 0000-0001-8050-7431
  • Email:
  • Viktoriya Ozornova
  • Affiliation: Max Planck Institute for Mathematics, Bonn, Germany
  • MR Author ID: 1124392
  • Email:
  • Emily Riehl
  • Affiliation: Department of Mathematics, Johns Hopkins University, Baltimore, Maryland
  • MR Author ID: 718246
  • Email:
  • Martina Rovelli
  • Affiliation: Department of Mathematics and Statistics, University of Massachusetts Amherst, Massachusetts
  • MR Author ID: 1204481
  • Email:
  • Received by editor(s): July 1, 2021
  • Received by editor(s) in revised form: May 6, 2022, and July 5, 2022
  • Published electronically: October 3, 2022
  • Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140, which began while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2020 semester. The authors learned a lot from fruitful discussions with the other members of the MSRI-based working group on $(\infty ,2)$-categories. This work was supported by a grant from the Simons Foundation (#850849, PH). The second author thankfully acknowledges the financial support by the DFG grant OZ 91/2-1 with the project nr. 442418934. The third author was supported from the NSF via DMS-1652600, from the ARO under MURI Grant W911NF-20-1-0082, from the AFOSR under award number FA9550-21-1-0009, and by the Johns Hopkins President’s Frontier Award program. The fourth author is deeply appreciative of the Mathematical Sciences Institute at the Australian National University for their support during the pandemic year.
  • © Copyright 2022 by the authors
  • Journal: Trans. Amer. Math. Soc. 376 (2023), 555-597
  • MSC (2020): Primary 18N65; Secondary 55U35, 18N10
  • DOI:
  • MathSciNet review: 4510118