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 ${\mathrm{\Theta }}_n$-spaces, where ${\mathrm{\Theta }}_n$ is an iterated wreath product of the simplex category $\mathrm{\Delta }$. A sequence of functors from ${\mathrm{\Theta }}_n$ to $\mathrm{\Gamma }$ allows for an alternative description of the Segal-spectrum associated to a $\mathrm{\Gamma }$-space. In particular, each Eilenberg-MacLane space $K(\pi ,n)$ has a canonical reduced ${\mathrm{\Theta }}_n$-set model.