Composition properties of box brackets

Abstract

In the homotopy theory of a 2-category with zeros and having a suspension functor we establish various composition properties of box brackets, including new formulae involving 2-sided matrix Toda brackets and classical Toda brackets. We are lead to define and study a new secondary homotopy operation called the box quartet operation. In the topological category this operation satisfies two triviality properties, one of which may be viewed as the foundation upon which an important classical mod zero result on Toda brackets rests. New insights and computations in the homotopy groups of spheres are obtained.