Completions and DK-equivalences of ${\mathrm{\Theta }}_n$-spaces

We establish Rezk completion functors for ${\mathrm{\Theta }}_n$-spaces with respect to each and all of the completeness conditions. As a consequence, we obtain a characterization completeness of Segal ${\mathrm{\Theta }}_n$-spaces as locality with respect to higher-dimensional Dwyer-Kan equivalences.