Iterated wreath product of the simplex category and iterated loop spaces

Abstract

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