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

TL;DR

This paper extends the homotopy coherent nerve construction to $( abla,n)$-categories, establishing a Quillen equivalence between models of enriched categories and Segal category objects for higher categories.

Contribution

It constructs a homotopy coherent nerve for $( abla,n)$-categories and proves a Quillen equivalence, generalizing the $( abla,1)$ case to higher categories.

Findings

Establishes a right Quillen equivalence for $( abla,n)$-categories.

Provides an equivalence between homotopy coherent diagrams and Segal category functors.

Enables defining $( abla,n)$-categories via different models.

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.