Proposition 4.3.29 (Associated spectrum). In the situation of Lemma 4.3.22, let \(X\) be a prespectrum in \(C\). Its associated spectrum \(X^{\mathrm {sp}}\) is naturally isomorphic to the following colimit in \(\Sp (C)\): \[ X^{\mathrm {sp}} \quad \simeq \quad \colim ( \, \Sigma ^{\infty }X_0 \to \Sigma ^{\infty -1}X_1 \to \Sigma ^{\infty -2}X_2 \to \Sigma ^{\infty -3}X_3 \to \cdots \, ). \] Here the \(n\)-th transition map is given by the composite \[ \Sigma ^{\infty - n} X_n \xrightarrow {\cong } \Sigma ^{\infty - (n+1)} \Sigma X_n \xrightarrow {\Sigma \sigma ^X_n} \Sigma ^{\infty - (n+1)} \Sigma \Omega X_{n+1} \xrightarrow {\epsilon } \Sigma ^{\infty - (n+1)}X_{n+1}, \] where the first isomorphism is from Lemma 4.3.26, and the last map is induced by the counit \(\epsilon \colon \Sigma \Omega \to \id \) of the adjunction \(\Sigma \dashv \Omega \) on \(C_*\).
Proof. Since adjunctions compose, the functor \((\Sigma ^{\infty }(-))[-n]\) is left adjoint to \(\Omega ^{\infty }((-)[n])=\Omega ^{\infty -n}\). The stated relations follow by passing to left adjoints in the previous lemma. □
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