Corollary 8.3.1. The Eilenberg–MacLane functor \(H\colon \Ab \longrightarrow \Sp _{\geq 0}\) is canonically lax symmetric monoidal, hence induces functors \[ H\colon \Alg (\Ab )\longrightarrow \Alg (\Sp ) \qquadtext {and} H\colon \CAlg (\Ab )\longrightarrow \CAlg (\Sp ). \]
Proof. Since \(H\) is right adjoint to the symmetric monoidal functor \(\pi _0\colon \Sp _{\geq 0} \to \Ab \), it follows from Proposition 14.3.6 that \(H\) is canonically lax symmetric monoidal. Every lax symmetric monoidal functor preserves associative and commutative algebra objects, as does the symmetric monoidal inclusion \(\Sp _{\geq 0}\hookrightarrow \Sp \). □
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