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

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