Definition 18.4.9. If \(\Oo \) is an \(\infty \)-operad with an operadic terminal object, equip \([1]\) with the symmetric monoidal structure given by the minimum, as used in the proof of Lemma 16.3.3, and denote by \[ \Oo _* \quad \subseteq \quad \oDay (\Mm _{[1]}, \Oo ) \] the full suboperad whose colors are those functors \(F\colon [1] \to \Oo _{\lra {1}}\) satisfying \(F(0) \simeq *\). If in addition \(\Oo \) has finite operadic products, equip \(\Span (\Fin )\) with the symmetric monoidal structure induced by the cartesian product of finite sets, and denote by \[ \oCGrp (\Oo ) \quad \subseteq \quad \oCMon (\Oo ) \quad \subseteq \quad \oDay (\Mm _{\Span (\Fin )}, \Oo ) \] the full suboperads whose colors are those functors \(F\colon \Span (\Fin ) \to \Oo _{\lra {1}}\) that are commutative groups or commutative monoids in \(\Oo _{\lra {1}}\), respectively.
Notice that this is not the cartesian monoidal structure on \(\Span (\Fin )\): its categorical products are disjoint unions by Lemma 13.3.8. The cartesian product of finite sets is the monoidal structure used in the Day convolution description of Proposition 16.4.1.
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