There is a way to realize the (infinite) loop space which relies on the (homotopy) totalization of a cosimplicial space. Given a (nice?) topological space $X$, consider the cosimplicial space $X_{\bullet}$ having $X_k = X^{k+1}$ as objects and $$ d_i( x_0 , \ldots, x_n) = (x_0, \ldots, x_i, x_i, \ldots , x_n) \textrm{ for } 1 \le i \le n$$ $$ d_0(x_0, \ldots, x_n) = (*, x_0, \ldots, x_n) $$ $$ d_{n+1} (x_0, \ldots, x_n) = (x_0 ,\ldots, x_n, *) $$ where $*$ is the basepoint of $X$. The homotopy totalization of this cosimplicial space is **the loop space of $X$ ** (edited). This construction provides a spectral sequence for loop spaces, though I don't remember where I have seen this studied.
Consider the following alternative construction. The new cosimplicial space $\hat{X}_{\bullet}$ is the iterated join $$ X_k := \underbrace{X * \ldots * X}_{(k+1) \textrm{ times}}$$ and maps mimic the underlying maps between finite sets in $\Delta$: for $\varphi: [n] \to [m]$ we set $$ \varphi( \lambda_0 x_0 + \ldots + \lambda_n x_n) = \lambda_{\varphi(0)} x_0 + \ldots + \lambda_{\varphi(n)} x_n $$
Do you think that this construction is somehow equivalent to the previous one? Does it provide nice spectral sequences? Equivalence here can be interpreted in several ways: homotopy equivalent as cosimplicial objects; their (truncated) (homotopy) totalization are homotopy equivalent. Since it does not ask for a basepoint, it could be an un-based version. To provide some context, my $X$ is a polytope, and I was wondering if the truncated realizations of $\hat{X}$ had a nice description (iterated loop spaces would be very nice).
