How can I define $\Omega X$ when $X\in\mathsf{sSet}_{\ast}$ but it is not fibrant?. Actually, I would like to formulate the 'Group completion theorem' in simplicial setting, I mean
$H_{\ast}(M)[\pi_{0}^{-1}]\cong H_{\ast}(\Omega BM)$ , when $\pi_{0}$ is in the center of $H_{\ast}(M)$.
But it does not make sense. Please, any suggestion?.
The loop space $\Omega X$ is always a homotopy pullback of the span $* \to X \leftarrow *$.
In a model category, the homotopy pullback $D$ of a span $A \to C \leftarrow B$ can always be given by a regular pullback:
$$ \require{AMScd} \begin{CD} D @>>> PC \\ @VVV @VVV \\A \times B @>>> C \times C \end{CD} $$
where the arrow $PC \to C \times C$ is any path space fibration.
I am rusty on details, but I think in simplicial sets, $PC = C^{\Delta[1]}$ with the obvious map to $C \times C$ is such a thing.