A homotopy coherent nerve for (∞,n)-categories

Abstract

In the case of (∞,1)-categories, the homotopy coherent nerve gives a right Quillen equivalence between the models of simplicially enriched categories and of quasi-categories. This shows that homotopy coherent diagrams of (∞,1)-categories can equivalently be defined as functors of quasi-categories or as simplicially enriched functors out of the homotopy coherent categorifications.In this paper, we construct a homotopy coherent nerve for (∞,n)-categories. We show that it realizes a right Quillen equivalence between the models of categories strictly enriched in (∞,n−1)-categories and of Segal category objects in (∞,n−1)-categories. This similarly enables us to define homotopy coherent diagrams of (∞,n)-categories equivalently as functors of Segal category objects or as strictly enriched functors out of the homotopy coherent categorifications.