  We provide a general definition of higher homotopy operations, encompassing
most known cases, including higher Massey and Whitehead products, and long Toda
brackets. These operations are defined in terms of the W-construction of
Boardman and Vogt, applied to the appropriate diagram category; we also show
how some classical families of polyhedra (including simplices, cubes,
associahedra, and permutahedra) arise in this way.
