Corollary 18.5.5 (Presentable operadic stabilization, [Nikolaus (2016), Proposition 4.9]). Let \(C\) be a presentably symmetric monoidal \(\infty \)-category. Then \(\Sp (C)\) admits a canonical presentably symmetric monoidal structure, and the operadic stabilization \(\oSp (\Mm _C)\) is represented by its multimorphism operad: \[ \oSp (\Mm _C)\simeq \Mm _{\Sp (C)}. \] In particular, the representability hypothesis in Corollary 18.4.7 is automatic in the presentable case.
Proof. The presentably symmetric monoidal structure on \(C\) is a commutative algebra object of \(\PrL \). Tensoring it with the commutative algebra \(\Sp \) gives another commutative algebra \[ \Sp \otimes C\in \CAlg (\PrL ). \] By Theorem 18.5.3, its underlying presentable \(\infty \)-category is naturally equivalent to \(\Sp (C)\). This equips \(\Sp (C)\) with a presentably symmetric monoidal structure. The unit map \(\An \to \Sp \) of the mode induces a strongly monoidal stabilization functor \(C\simeq \An \otimes C\to \Sp \otimes C\), whose right adjoint \(\Omega ^\infty \colon \Sp (C)\to C\) is lax symmetric monoidal.
Under the reduced-excisive model of stabilization, the resulting multimorphism operad is the full suboperad of the Day convolution operad on \(\Fun (\An _*^{\fin },C)\) spanned by the reduced excisive functors. This is exactly the construction of \(\oSp (\Mm _C)\) from Section 18.4, giving the displayed equivalence. □
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