Lemma 4.3.26. In the situation of Lemma 4.3.22, the functor \(\Omega ^{\infty - n} \colon \Sp (C) \to C_*\) admits a left adjoint \[ \Sigma ^{\infty - n}\colon C_* \to \Sp (C) \] given by \(\Sigma ^{\infty - n}(X) := (\Sigma ^{\infty }(X))[-n]\). It satisfies the relations \[ \Sigma ^{\infty - n} \Sigma ^n \cong \Sigma ^{\infty } \qquadtext { and } \Sigma ^{\infty - (n+1)} \Sigma \simeq \Sigma ^{\infty - n}. \]
Proof. This follows from stability of \(\Sp (C)\) and the fact that \(\Omega ^{\infty }\colon \Sp (C)\to C_*\) preserves finite limits. □
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