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 $\pi\_0,\dots,\pi\_{n-1}$



Multi-simplicial generalization of $\Gamma$-spaces
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.


