A categorical approach to matrix Toda brackets

Abstract

In this paper we give a categorical treatment of matrix Toda brackets, both in the pre- and post-compositional versions. Explicitly the setting in which we work is, à la Gabriel-Zisman, a 2 2 -category with zeros. The development parallels that in the topological setting but with homotopy groups replaced by nullity groups of invertible 2 2 -morphisms. A central notion is that of conjugation of 2 2 -morphisms. Our treatment of matrix Toda brackets is carried forward to the point of establishing appropriate indeterminacies.