Lemma 16.1.4. Assume \(\oDay (\Oo ,\Pp )\) exists. Then commutative algebras in \(\oDay (\Oo ,\Pp )\) correspond to operad morphisms \(\Oo \to \Pp \): \[ \CAlg (\oDay (\Oo ,\Pp )) \simeq \Fun _{\Op _{\infty }}(\Oo ,\Pp ). \]
Proof. Since \(\Comm \) is the terminal \(\infty \)-operad, we have \(\Oo \simeq \Comm \times \Oo \), and so \begin {align*} \CAlg (\oDay (\Oo ,\Pp )) &= \Fun _{\Op _{\infty }}(\Comm ,\oDay (\Oo ,\Pp )) \\ &\simeq \Fun _{\Op _{\infty }}(\Comm \times \Oo , \Pp ) \\ &\simeq \Fun _{\Op _{\infty }}(\Oo ,\Pp ). \qedhere \end {align*} □
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