Relative Day convolution

Remark 16.2.5. The above description of \(\oDay (\Mm _C,\Mm _D)\) immediately generalizes to a description of the \(\Oo \)-monoidal Day convolution operad \(\oDay _{/\Oo }(\Mm _{C/\Oo }, \Mm _{D/\Oo }) \in (\Op _{\infty })_{/\Oo }\) for \(\Oo \)-monoidal \(\infty \)-categories \(C\) and \(D\). The proof is essentially identical, up to replacing \(\Op _{\infty }\) with \((\Op _{\infty })_{/\Oo }\) everywhere. It is this generality in which Hinich and Winges state and prove the result. Note that Theorem 16.2.2, the principal external input about Day convolution, is already stated for an arbitrary \(\infty \)-operad \(\Oo \).

Concretely, the universal property reads \[ \Fun _{(\Op _{\infty })_{/\Oo }}\bigl (\Qq , \oDay _{/\Oo }(\Mm _{C/\Oo },\Mm _{D/\Oo })\bigr ) \quad \simeq \quad \Fun _{(\Op _{\infty })_{/\Oo }}\bigl (\Qq \times _{\Oo } \Mm _{C/\Oo }, \Mm _{D/\Oo }\bigr ) \] for every \(\infty \)-operad \(\Qq \) over \(\Oo \). Moreover, since both sides are described by the same pullback square, base change along an operad map \(\Oo \to \Comm \) identifies \[ \Oo \times _{\Comm } \oDay (\Mm _C,\Mm _D) \quad \simeq \quad \oDay _{/\Oo }(\Mm _{C/\Oo },\Mm _{D/\Oo }). \] In other words, the underlying \(\Oo \)-monoidal \(\infty \)-category of the Day convolution is the \(\Oo \)-monoidal Day convolution. For \(\Oo =\Assoc \), this is the relative version underlying the final assertion of Observation 16.6.7, where the relevant functor is lax monoidal but not lax symmetric monoidal.