Flagged higher categories

TL;DR

This paper introduces flagged $( abla,n)$-categories, proving their equivalence to Segal sheaves on Joyal's ${f heta}_n$, providing a model-independent framework and generalizing effectiveness of $n$-groupoid objects in spaces.

Contribution

It establishes a new model-independent formulation of Segal sheaves via flagged $( abla,n)$-categories and generalizes effectiveness results for $n$-groupoid objects.

Findings

Flagged $( abla,n)$-categories are equivalent to Segal sheaves on ${f heta}_n$.

Provides an expression for univalent-completion of Segal sheaves.

Conjectures a characterization of flagged $( abla,n)$-categories as stacks satisfying certain descent conditions.

Abstract

We introduce \emph{flagged $(\infty,n)$-categories} and prove that they are equivalent to Segal sheaves on Joyal's category ${\mathbf\Theta}_n$. As such, flagged $(\infty,n)$-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. Along the way, we establish a useful expression for the univalent-completion of such a Segal sheaf. Finally, we conjecture a characterization of flagged $(\infty,n)$-categories as stacks on $(\infty,n)$-categories that satisfy descent with respect to colimit diagrams that do not generate invertible $i$-morphisms for any $i$.