Simplicial resolutions and Ganea fibrations

In this work, we compare the two approximations of a path-connected space $X$, by the Ganea spaces $G_n(X)$ and by the realizations $|\Lambda_\bullet X|{n}$ of the truncated simplicial resolutions emerging from the loop-suspension cotriple $\mathrm{\Sigma }\mathrm{\Omega }$. For a simply connected space $X$, we construct maps $|\Lambda\bullet X|{n-1}\to G_n(X)\to |\Lambda\bullet X|_{n}$over$X$,uptohomotopy\mathrm{.}Inthecase$n=2$, we prove the existence of a map $G_2(X)\to \mathrm{\parallel }{\mathrm{\Lambda }}_{\bullet }X{\mathrm{\parallel }}_1$ over $X$ (up to homotopy) and conjecture that this map exists for any $n$.