Completions and DK-equivalences of ${Θ_n}$-spaces

TL;DR

The paper develops Rezk-style completion for $\Theta_n$-spaces, providing a family of completions in each dimension and a total completion that localizes Segal $\Theta_n$-spaces with respect to all completeness conditions. It defines and analyzes DK-equivalences in this higher-categorical setting, linking them to levelwise weak equivalences after appropriate localizations. Using an inductive suspension framework, it proves that horizontal and higher-dimensional completions preserve the essential enriched structure while turning Segal objects into complete ones. The results yield a precise characterization: for $Se^n(S)$-fibrant objects, DK-equivalences coincide with $Cplt^n(S)$-local equivalences, and complete Segal $\Theta_n$-spaces model $(\infty,n)$-categories up to these localizations. The constructions also provide a practical, dimension-by-dimension approach to completing higher categorical structures via a sequence of functorial completions and a total completion $T$.

Abstract

We establish Rezk completion functors for $Θ_n$ spaces with respect to each and all of the completeness conditions. As a consequence, we obtain a characterization of Dwyer-Kan equivalences between Segal $Θ_n$ spaces.

Theorems & Definitions (127)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.3
• Proposition 2.5
• Definition 2.6
• Definition 2.7
• Theorem 2.8
• Definition 2.10