Iterated wreath product of the simplex category and iterated loop spaces

Generalising Segal's approach to 1-fold loop spaces, the homotopy theory of n-fold loop spaces is shown to be equivalent to the homotopy theory of reduced Θn-spaces, where Θn is an iterated wreath product of the simplex category Δ. A sequence of functors from Θn to Γ allows for an alternative description of the Segal spectrum associated to a Γ-space. This yields a canonical reduced Θn-set model for each Eilenberg–MacLane space. The number of (n+k)-dimensional cells of the resulting CW-complex of type K(Z/2Z,n) is the kth generalised Fibonacci number of order n.