Higher order Toda brackets

TL;DR

This paper introduces two equivalent methods for defining higher order Toda brackets in pointed simplicial model categories, providing a diagrammatic perspective and linking it to homotopy-commutative diagram rectification.

Contribution

It presents a recursive and an enrichment-based definition of higher order Toda brackets, proving their equivalence and offering a diagrammatic interpretation as an obstruction.

Findings

The two definitions of higher order Toda brackets are shown to agree.

A diagrammatic description of Toda brackets is provided.

The work connects Toda brackets to obstructions in rectifying homotopy-commutative diagrams.

Abstract

We describe two ways to define higher order Toda brackets in a pointed simplicial model category $\mathcal{D}$: one is a recursive definition using model categorical constructions, and the second uses the associated simplicial enrichment. We show that these two definitions agree, by providing a third, diagrammatic, description of the Toda bracket, and explain how it serves as the obstruction to rectifying a certain homotopy-commutative diagram in $\mathcal{D}$.