Lemma 4.3.22 (Suspension spectra). Let \(C\) be an \(\infty \)-category admitting finite limits, finite colimits, and sequential colimits, and assume that \(\Omega \colon C_* \to C_*\) preserves sequential colimits. Then the functor \(\Omega ^{\infty }\colon \Sp (C) \to C_*\) admits a left adjoint \[ \Sigma ^{\infty }\colon C_* \to \Sp (C). \]
Proof. Filtered colimits in \(\An _*\) are created in \(\An \), and the loop functor \(\Omega \colon \An _* \to \An _*\) preserves them. Hence the levelwise colimit in \(\PSp (\An )\) of a filtered diagram of spectra again has isomorphisms as its structure maps, so it is already a spectrum. The final claim follows by evaluating at level zero. □
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