Polyhedral coproducts

TL;DR

This paper introduces polyhedral coproducts as a dual concept to polyhedral products, exploring their properties, functorial behavior, and a loop space decomposition, enriching the understanding of these topological constructions.

Contribution

It defines polyhedral coproducts, studies their properties, and establishes a dual loop space decomposition, providing new insights into their relationship with polyhedral products.

Findings

Polyhedral coproducts interpolate between wedge and product of spaces.

A general loop space decomposition for polyhedral coproducts is established.

Polyhedral coproducts differ from polyhedral products in their functorial behavior.

Abstract

Dualising the construction of a polyhedral product, we introduce the notion of a polyhedral coproduct as a certain homotopy limit over the face poset of a simplicial complex. We begin a study of the basic properties of polyhedral coproducts, surveying the Eckmann-Hilton duals of various familiar examples and properties of polyhedral products. In particular, we show that polyhedral coproducts give a functorial interpolation between the wedge and cartesian product of spaces which differs from the one given by polyhedral products, and we establish a general loop space decomposition for these spaces which is dual to the suspension splitting of a polyhedral product due to Bahri, Bendersky, Cohen and Gitler.