Segal's multisimplicial spaces

TL;DR

This paper generalizes Segal's conditions for simplicial spaces to multisimplicial spaces, enabling an $n$-fold delooping of classifying spaces of categories, with applications to the reduced bar construction.

Contribution

It extends Segal's results from simplicial to multisimplicial spaces, facilitating higher-dimensional delooping in algebraic topology.

Findings

Provides sufficient conditions for multisimplicial spaces to model loop spaces

Enables $n$-fold delooping of classifying spaces

Applicable to the reduced bar construction for categories

Abstract

Some sufficient conditions on a simplicial space $X$ guaranteeing that $X_1\simeq \Omega|X|$ were given by Segal. We give a generalization of this result for multisimplicial spaces. This generalization is appropriate for the reduced bar construction, providing an $n$-fold delooping of the classifying space of a category.