Lemma 4.3.21. Filtered colimits in \(\Sp \) are computed levelwise. In particular, the functor \(\Omega ^{\infty }\colon \Sp \to \An \) preserves filtered colimits.
Proof. Let \(X\colon I \to \Sp (C)\) be a diagram. Compute its colimit in \(\PSp (C)\), where colimits are pointwise by Lemma 4.3.8, and then apply spectrification. Since spectrification is left adjoint to the inclusion \(\Sp (C) \hookrightarrow \PSp (C)\), the resulting spectrum has the universal property of the colimit in \(\Sp (C)\). □
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