Is there a generalization of Segal's theorem that the inclusion of $X_1$ into $\Omega|X_*|$ is a weak equivalence for a $\Gamma$-space $X_*$ if $X_1$ is group like? Specifically, I am looking for a result something like $\hat{X} \rightarrow \Omega ^n |X_{*,\dots,*}|$ is a weak equivalence where $X_{*,\dots,*}$ is a multi-simplicial space subject to some conditions like a $\Gamma$-space and $\hat{X}$ is a space derived from the spaces of $X_{*,\dots,*}$, subject to some condition on $\pi_0,\dots,\pi_{n-1}$
1 Answer
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The paper "Iterated monoidal categories" by Balteanu, Fiedorowicz, Schwanzl, Vogt gives an analogous result for functors $(\Delta ^{op})^n \rightarrow Top$ and n-fold loop spaces.