Show that for any sequence \(Z_1 \to Z_2 \to \cdots\), the natural map \(\operatorname{hocolim} \Omega Z_n \to \Omega \operatorname{hocolim} Z_n\) is a weak homotopy equivalence, where the direct limits mean mapping telescopes.
Solution
Let \[ T=\operatorname{hocolim}(Z_1\to Z_2\to\cdots) \] be the mapping telescope, and let \[ T_\Omega=\operatorname{hocolim}(\Omega Z_1\to\Omega Z_2\to\cdots). \] There is a natural map \[ \kappa:T_\Omega\longrightarrow\Omega T \] obtained by regarding a loop in a stage \(Z_n\) as a loop in the corresponding slice of the telescope.
For every \(j\ge0\), compactness of \(S^j\) implies \[ \pi_j(T_\Omega) \cong\varinjlim_n\pi_j(\Omega Z_n). \tag{1} \] Indeed, a map from a compact sphere into a telescope has image in a finite subtelescope, and a finite subtelescope deformation retracts onto its last stage. The same argument gives \[ \pi_{j+1}(T) \cong\varinjlim_n\pi_{j+1}(Z_n). \tag{2} \] Using \(\pi_j(\Omega Z_n)=\pi_{j+1}(Z_n)\), equations (1) and (2) identify both sides with the same direct limit: \[ \pi_j(T_\Omega) \cong \varinjlim_n\pi_{j+1}(Z_n) \cong \pi_j(\Omega T). \] Under these identifications, \(\kappa_*\) is the identity on the direct limit. This also gives the required bijection on path components when \(j=0\).
Hence \(\kappa\) induces isomorphisms on every homotopy group and a bijection on components. Therefore \[ \boxed{\operatorname{hocolim}\Omega Z_n\longrightarrow \Omega\operatorname{hocolim}Z_n\text{ is a weak homotopy equivalence}.} \]