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