Long Box Bracket Operations in Homotopy Theory

In any 2-category C with zeros (for example the topological 2-category T op∗ of based spaces, based maps and track classes of homotopies) box bracket operations have been defined and studied by Hardie-Marcum-Oda [4]. These are secondary homotopy operations and if (as we will suppose) C admits a suspension functor Σ (see [3]) they take values in morphism groups HC(ΣW,X), here denoted π(ΣW,X), of the homotopy category HC of C. In T op∗ the classical (3-fold) Toda bracket [9] {a, f, w}, defined for nullhomotopic composites a ◦ f and f ◦ w, is a particular example of a box bracket. In T op∗ higher order (or long n-fold) Toda brackets may also be considered. Different authors (e.g. see Spanier [8], Gray [2], Cohen [1], Walker [10] for general n; Ôguchi [6] for n = 4) have employed different methods to define long Toda brackets, but all definitions are quite complicated because coherence conditions for higher homotopies arise. In fact no definition of higher order Toda brackets in general categorical terms seems to be known although Sagave [7] has given a definition for triangulated categories. Essentially the notion of mapping cone seems to be required. Not unsurprisingly the formulation of long box brackets has been lacking. This is true even in the case of an appropriate triple (3-fold) box bracket (cf [3]). Recently a categorical 4-fold (or quaternary) box bracket has been treated [5]; it takes its values in π(ΣW,X) and is useful for elucidating Toda’s result [9, Proposition 1.5]. In the present work we present for T op∗ an inductive construction of long box brackets so that an n-fold box bracket (n ≥ 2) takes values in π(ΣW,X) and has an (n+1)-fold Toda bracket as a particular example. Thus (in relation to [3]) we construct a 3-fold (or triple) box bracket with values in π(ΣW,X); this triple box bracket has the 4-fold (or quaternary) Toda bracket as a particular example. For n = 4 we arrive at a 4-fold (or quaternary) box bracket (different from that of [5]) with values in π(ΣW,X) and with the 5-fold Toda bracket as a particular example.