Discreteness and completeness for ${\mathrm{\Theta }}_n$-models of $(\mathrm{\infty },n)$-categories

We establish cartesian model structures for variants of ${\mathrm{\Theta }}_n$-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 ${\mathrm{\Theta }}_n$-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 ${\mathrm{\Theta }}_n$-diagrams. In the process, we give a characterization of the Dwyer-Kan equivalences in the ${\mathrm{\Theta }}_n$-space model, generalizing the one given by Rezk for complete Segal spaces.