Flagged higher categories

David Ayala, John Francis

Research output: Chapter in Book/Report/Conference proceedingChapter


We introduce flagged (∞, n)-categories and prove that they are equivalent to Segal sheaves on Joyal’s category Θ n. As such, flagged (∞, n)-categories provide a model-independent formulation of Segal sheaves. This result generalizes the statement that n-groupoid objects in spaces are effective, as we explain and contextualize as an instance of Koszul duality. Along the way, we establish a useful expression for the univalent-completion of such a Segal sheaf. Finally, we conjecture a characterization of flagged (∞, n)-categories as stacks on (∞, n)-categories that satisfy descent with respect to colimit diagrams that do not generate invertible i-morphisms for any i.

Original languageEnglish (US)
Title of host publicationContemporary Mathematics
PublisherAmerican Mathematical Society
Number of pages37
StatePublished - Jan 1 2018

Publication series

NameContemporary Mathematics
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627


  • (∞, n)-categories
  • Cech nerve
  • Flagged higher categories
  • Groupoid objects
  • Higher categories
  • Segal spaces
  • Univalence

ASJC Scopus subject areas

  • Mathematics(all)

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  • Cite this

    Ayala, D., & Francis, J. (2018). Flagged higher categories. In Contemporary Mathematics (pp. 137-173). (Contemporary Mathematics; Vol. 718). American Mathematical Society. https://doi.org/10.1090/conm/718/14489