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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?

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    $\begingroup$ I don't think the ordering is playing a role here: $S^p\wedge \Omega^pX$ is contravariantly functorial for arbitrary injections of the sets $\{1,\dotsc,p\}$. I've not seen this variant but it is dual to the fact that $\Omega^p(S^p\wedge X)$ is covariantly functorial for the same maps. This comes up a lot (implicitly or explicitly) when setting up the model category of symmetric spectra. $\endgroup$
    – Neil Strickland
    May 29, 2020 at 19:35

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