@inbook{2f6a0ab957da40afac4e06daf84e8037,

title = "Flagged higher categories",

abstract = "We introduce flagged (∞, n)-categories and prove that they are equivalent to Segal sheaves on Joyal{\textquoteright}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.",

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

author = "David Ayala and John Francis",

note = "Funding Information: 2010 Mathematics Subject Classification. Primary 18A05. Secondary 55U35, 55P65. Key words and phrases. (∞,n)-categories, higher categories, flagged higher categories, Segal spaces, univalence, groupoid objects, Cech nerve. The first author was supported by the National Science Foundation under award 1507704. Funding Information: The second author was supported by the National Science Foundation under award 1508040. Publisher Copyright: {\textcopyright} 2018 American Mathematical Society.",

year = "2018",

doi = "10.1090/conm/718/14489",

language = "English (US)",

series = "Contemporary Mathematics",

publisher = "American Mathematical Society",

pages = "137--173",

booktitle = "Contemporary Mathematics",

}