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

TL;DR

This paper develops and compares various $ heta_n$-space models for $( abla,n)$-categories, establishing their equivalences and providing criteria for different combinations of discreteness and completeness conditions.

Contribution

It introduces new $ heta_n$-space models with discreteness conditions, proves their equivalence to existing models, and generalizes Dwyer-Kan equivalences for these frameworks.

Findings

All models are equivalent to the $ heta_n$-space model.

A criterion for non-overlapping models based on discreteness and completeness.

Generalization of Dwyer-Kan equivalences to $ heta_n$-space models.

Abstract

We establish cartesian model structures for variants of $\Theta_n$-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 $\Theta_n$-space model, and we give a criterion for which combinations of discreteness and completeness give non-overlapping models. These models can be thought of as generalizations of Segal categories in the framework of $\Theta_n$-diagrams. In the process, we give a characterization of the Dwyer-Kan equivalences in the $\Theta_n$-space model, generalizing the one given by Rezk for complete Segal spaces.