A closed model structure for $n$-categories, internal $Hom$, $n$-stacks and generalized Seifert-Van Kampen

TL;DR

This paper develops a new model category framework for $n$-categories, enabling internal homs and the construction of higher categories, and proves a generalized Seifert-Van Kampen theorem for Tamsamani's Poincaré $n$-groupoid.

Contribution

It introduces a closed model structure for $n$-categories with internal homs, facilitating the construction of $n+1$-categories and extending classical theorems to higher categorical contexts.

Findings

Established a closed model category for $n$-categories with internal $Hom$.

Constructed the $n+1$-category $nCAT$ using internal $Hom$.

Proved a generalized Seifert-Van Kampen theorem for Tamsamani's Poincaré $n$-groupoid.

Abstract

We define a closed model category containing the $n$-nerves defined by Tamsamani, and admitting internal $Hom$. This allows us to construct the $n+1$-category $nCAT$ by taking the internal $Hom$ for fibrant objects. We prove a generalized Seifert-Van Kampen theorem for Tamsamani's Poincar\'e $n$-groupoid of a topological space. We give a still-speculative discussion of $n$-stacks, and similarly of comparison with other possible definitions of $n$-category.