On the graph products of simplicial groups and connected Hopf algebras

TL;DR

This paper explores the structure of graph products of simplicial groups and connected Hopf algebras, showing they model loop spaces of polyhedral products and establishing their homology and morphisms.

Contribution

It introduces a unified framework for analyzing graph products of simplicial groups and Hopf algebras using polyhedral products, including structure theorems and morphism analysis.

Findings

Graph products model loop spaces of polyhedral products over flag complexes

Homology of these graph products is characterized

Structure theorems for graph products are established

Abstract

In this paper we study the classifying spaces of graph products of simplicial groups and connected Hopf algebras over a field, and show that they can be uniformly treated under the framework of polyhedral products. It turns out that these graph products are models of the loop spaces of polyhedral products over a flag complex and their homology, respectively. Certain morphisms between graph products are also considered. In the end we prove the structure theorems of such graph products in the form we need.