A Coherent Homotopy Category of 2-track Commutative Cubes

We consider a category H ⊗ (the homotopy category of homotopy squares) whose objects are homotopy commutative squares of spaces and whose morphisms are cubical diagrams subject to a coherent homotopy relation. The main result characterises the isomorphisms of H ⊗ to be the cube morphisms whose forward arrows are homotopy equivalences. As a first application of the new category we give a direct 2-track theoretic definition of the quaternary Toda bracket operation. Subject classifications : [2000] 18D05, 18B30, 55P10.