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

Abstract

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

43 pages; title changed from original version. Corrected versions of Proposition 5.4 and Example 7.9 included in this version, which appear in an erratum to the published paper

References in corpus (2)

• Model structures for $(\infty ,n)$-categories on (pre)stratified simplicial sets and prestratified simplicial spaces
• On the equivalence between ${\Theta }_n$-spaces and iterated Segal spaces