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Discreteness and completeness for $\Theta_n$-models of $(\infty,n)$-categories

Julia E. Bergner

Vol. 6 (2024), No. 1, 49–96

We establish cartesian model structures for variants of Θn-spaces in which we replace some or all of the completeness conditions by discreteness conditions. We prove that they are all equivalent to each other and to the Θn-space model, and we give a criterion for which combinations of discreteness and completeness give nonoverlapping models. These models can be thought of as generalizations of Segal categories in the framework of Θn-diagrams. In the process, we give a characterization of the Dwyer–Kan equivalences in the Θn-space model, generalizing the one given by Rezk for complete Segal spaces.

$(\infty,n)$-categories, model categories, $\Theta_n$-spaces, Segal categories, Dwyer–Kan equivalences
Mathematical Subject Classification
Primary: 18N65, 18N50, 18N40
Secondary: 55U35, 55U40, 18D15, 18D20, 18G30, 18G55
Received: 21 October 2022
Revised: 12 May 2023
Accepted: 29 May 2023
Published: 20 January 2024
Julia E. Bergner
Department of Mathematics
University of Virginia
Charlottesville, VA
United States