The Toda bracket in the homotopy category of a track bicategory

We show that the tertiary homotopy composition operation (quaternary Toda bracket) can, with the advantage of reduced indeterminacy, be replaced by a triple Toda bracket in a bicategory with zeros and invertible 2-morphisms. We define the bracket via a pasting operation in such a category, recovering standard properties in the new setting, and apply it to a certain bicategory whose objects are pointed maps, 1-morphisms are squares with commuting homotopies, and whose 2-morphisms are 2-tracks (Appl. Cat. Str. 8 (2000) 209). We indicate that Hopf invariant detection formulae involving the new triple bracket yield a natural extension of the classical Hopf–Toda technique for constructing non-trivial homotopy classes.