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

Abstract

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

43 pages, v4: corrected the definition for higher DK-equivalences. Comments welcome