  It is well known since Stasheff's work that 1-fold loop spaces can be
described in terms of the existence of higher homotopies for associativity
(coherence conditions) or equivalently as algebras of contractible
non-symmetric operads. The combinatorics of these higher homotopies is well
understood and is extremely useful.
  For $n \ge 2$ the theory of symmetric operads encapsulated the corresponding
higher homotopies, yet hid the combinatorics and it has remain a mystery for
almost 40 years. However, the recent developments in many fields ranging from
algebraic topology and algebraic geometry to mathematical physics and category
theory show that this combinatorics in higher dimensions will be even more
important than the one dimensional case.
  In this paper we are going to show that there exists a conceptual way to make
these combinatorics explicit using the so called higher nonsymmetric
$n$-operads.
