Lemma 18.4.10 (Day convolution on pointed objects). Let \(C\) be an \(\infty \)-category with finite products and finite colimits such that its cartesian product preserves finite colimits separately in both variables. Then there is a natural equivalence \[ (\Mm _{(C,\times )})_* \simeq \Mm _{(C_*,\wedge )}. \]
Proof. The operad \((\Mm _{(C,\times )})_*\) is the full suboperad of the Day convolution operad on \(\Ar (C)\) spanned by the arrows with source the terminal object. The monoidal localization \[ \cofib \colon \Ar (C)^{\square }\longrightarrow (C_*,\wedge ) \] of Lemma 16.3.4 identifies this full suboperad with the multimorphism operad of its local objects. □
Generated from the authoritative LaTeX source.