Suspension spectra as spectrifications

Remark 4.3.27. In the situation of Lemma 4.3.22, the suspension spectrum \(\Sigma ^{\infty }(X)\) of a pointed object \(X\in C_*\) is the spectrification of the suspension prespectrum \[ (X,\Sigma X,\Sigma ^2X,\dots ). \] Equivalently, its \(n\)-th level is given by \[ \Sigma ^{\infty }(X)_n \simeq \colim _{k \geq 0} \Omega ^k \Sigma ^{k+n}X. \] The structure maps are the inverses of the natural isomorphisms \[ \Omega \Sigma ^{\infty }(X)_{n+1} \simeq \Omega ( \colim _{k \geq 0} \Omega ^k \Sigma ^{k+n+1} X) \simeq \colim _{k \geq 0} \Omega ^{k+1}\Sigma ^{k+n+1} X \simeq \Sigma ^{\infty }(X)_n, \] where we use that \(\Omega \colon C_* \to C_*\) preserves sequential colimits.