### Abstract

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 language | English (US) |
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Title of host publication | Contemporary Mathematics |

Publisher | American Mathematical Society |

Pages | 137-173 |

Number of pages | 37 |

DOIs | |

State | Published - Jan 1 2018 |

### Publication series

Name | Contemporary Mathematics |
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Volume | 718 |

ISSN (Print) | 0271-4132 |

ISSN (Electronic) | 1098-3627 |

### Keywords

- (∞, 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

*Contemporary Mathematics*(pp. 137-173). (Contemporary Mathematics; Vol. 718). American Mathematical Society. https://doi.org/10.1090/conm/718/14489