A homotopy coherent nerve for $(\infty ,n)$-categories

Abstract

In the case of $(\infty ,1)$-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 $(\infty ,1)$-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 $(\infty ,n)$-categories. We show that it realizes a right Quillen equivalence between the models of categories strictly enriched in $(\infty ,n-1)$-categories and of Segal category objects in $(\infty ,n-1)$-categories. This similarly enables us to define homotopy coherent diagrams of $(\infty ,n)$-categories equivalently as functors of Segal category objects or as strictly enriched functors out of the homotopy coherent categorifications.

55 pages; v3: final version to appear in JPAA. v2 supersedes v1, which contains some errors. v2 is a complete rewrite that proves a similar main result to v1 but in the model of Segal category objects in (oo,n-1)-categories instead of complete Segal objects in (oo,n-1)-categories