A constructive approach to higher homotopy operations

TL;DR

This paper introduces a systematic method for constructing higher homotopy operations in model categories, encompassing classical and novel unpointed operations, and presents a double induction approach for defining these obstructions.

Contribution

It provides the first explicit general construction of higher homotopy operations, including unpointed cases, using a double induction framework in model categories.

Findings

Unified construction of classical and unpointed higher homotopy operations

Double induction method for defining obstructions

Intermediate obstructions are also characterized

Abstract

In this paper we provide an explicit general construction of higher homotopy operations in model categories, which include classical examples such as (long) Toda brackets and (iterated) Massey products, but also cover unpointed operations not usually considered in this context. We show how such operations, thought of as obstructions to rectifying a homotopy-commutative diagram, can be defined in terms of a double induction, yielding intermediate obstructions as well.