I would like to understand whether the following construction has been studied. Let me model the pointed sphere as $S^n = I^n / \partial I^n$, and let $\Omega^n (-) := map((I^n, \partial I^n), (-, *))$.
Let $X$ be a based space. Write $X_p := S^{p+1} \wedge \Omega^{p+1} X$. There are maps $$d_i : S^{p+1} \wedge \Omega^{p+1} X \longrightarrow S^{p} \wedge \Omega^{p} X$$ given by the formula $d_i([t_0, ..., t_p], f) = ([t_0, ..., \hat{t_i}, .., t_p], f\vert_{I^i \times \{t_i\} \times I^{p-i}})$. These form the face maps for a semi-simplicial space $X_\bullet$. Furthermore, evaluation yields an aumentation $\epsilon: X_\bullet \to X$. On geometric realisation we therefore obtain a map $$|X_\bullet| \to X.$$
Is this construction familiar to anyone?