The envelope allows us to construct Day convolution operads with arbitrary target operads. The key formal point is that every \(\infty \)-operad embeds as a full suboperad of the multimorphism operad of its envelope, and that Day convolution restricts along full suboperads.
Lemma 18.3.1. Let \(i\colon \Oo \hookrightarrow \Qq \) be a full suboperad. For every \(\infty \)-operad \(\Rr \), the induced functor \[ \Fun _{\Op _{\infty }}(\Rr ,\Oo ) \to \Fun _{\Op _{\infty }}(\Rr ,\Qq ) \] is fully faithful. Its essential image consists of those operad maps \(\Rr \to \Qq \) whose colors land in \(\Oo _{\lra {1}} \subseteq \Qq _{\lra {1}}\).
Proof. This is clear from the fact that the total category \(\Oo ^{\otimes }\) is the full subcategory of \(\Qq ^{\otimes }\) spanned by those finite tuples whose entries are colors of \(\Oo \). □
Lemma 18.3.2. Let \(i\colon \Oo \hookrightarrow \Qq \) be a full suboperad, let \(C\) be a symmetric monoidal \(\infty \)-category, and assume that \(\oDay (\Mm _C,\Qq )\) exists. Let \[ \Ww \subseteq \oDay (\Mm _C,\Qq ) \] be the full suboperad spanned by those functors \(C \to \Qq _{\lra {1}}\) whose values lie in \(\Oo _{\lra {1}}\). Then \(\Ww \) is a Day convolution operad \(\oDay (\Mm _C,\Oo )\).
Proof. The evaluation map for \(\oDay (\Mm _C,\Qq )\) restricts to a map \[ \Ww \times \Mm _C \to \Oo , \] because on colors it sends a pair \((F,c)\) to \(F(c)\). We verify the defining universal property. Let \(\Rr \) be an arbitrary \(\infty \)-operad. By Lemma 18.3.1, maps \(\Rr \to \Ww \) are precisely maps \(\Rr \to \oDay (\Mm _C,\Qq )\) whose color functors land in \(\Oo _{\lra {1}}\). Transposing along the Day convolution adjunction for \(\Qq \), these are precisely maps \[ \Rr \times \Mm _C \to \Qq \] whose colors land in \(\Oo _{\lra {1}}\). Applying Lemma 18.3.1 again, this is the same as giving a map \(\Rr \times \Mm _C \to \Oo \). This identification is natural in \(\Rr \), hence \(\Ww \) satisfies the universal property of \(\oDay (\Mm _C,\Oo )\). □
Theorem 18.3.3. Let \(C\) be a symmetric monoidal \(\infty \)-category. For every \(\infty \)-operad \(\Oo \), the Day convolution operad \(\oDay (\Mm _C,\Oo )\) exists.
Proof. Put \(D := \Env (\Oo )\). By Lemma 17.3.17, the unit \[ \eta _{\Oo }\colon \Oo \to \Mm _D \] is fully faithful: its essential image is the full suboperad spanned by the singleton tuples of colors in \(D\). We may therefore identify \(\Oo \) with this full suboperad by Lemma 14.1.8. The Day convolution operad \(\oDay (\Mm _C,\Mm _D)\) exists by Theorem 16.2.4, so Lemma 18.3.2 gives the result. □
Remark 18.3.4. A relative version of this result was proved by Lurie (2017), Theorem 2.2.6.2: if \(f\colon \Oo ' \to \Pp \) is a cocartesian fibration of \(\infty \)-operads, then the pullback functor \(f^*\colon (\Op _{\infty })_{/\Pp } \to (\Op _{\infty })_{/\Oo '}\) admits a right adjoint \[ \Nm _{\Oo '/\Pp }\colon (\Op _{\infty })_{/\Oo '} \to (\Op _{\infty })_{/\Pp }. \] By precomposing this functor with \(f^*\), it then follows that the product functor \(\Oo ' \times _{\Pp } - \colon (\Op _{\infty })_{/\Pp } \to (\Op _{\infty })_{/\Pp }\) admits a right adjoint \[ \oDay _{/\Pp }(\Oo ', -) \colon (\Op _{\infty })_{/\Pp } \to (\Op _{\infty })_{/\Pp }. \] If \(\Pp = \Comm \) is the terminal \(\infty \)-operad and \(\Oo ' \simeq \Mm _C\) for some \(C\), this specializes to Theorem 18.3.3.
Theorem 18.3.5 (Multimorphism formula for Day convolution). Let \(C\) be a small symmetric monoidal \(\infty \)-category and let \(\Oo \) be an \(\infty \)-operad. For every finite set \(I\) and functors \(F_i,G\colon C \to \Oo _{\lra {1}}\), there is a natural equivalence \[ \oDay (\Mm _C,\Oo )(\{F_i\}_{i \in I};G) \simeq \int _{\{c_i\}_{i \in I}\in C^I} \Oo (\{F_i(c_i)\}_{i \in I};G(\bigotimes \nolimits ^I_C c_i)). \]
Proof. Put \(D := \Env (\Oo )\) and identify \(\Oo \) with the full suboperad of \(\Mm _D\) spanned by the singleton tuples, as in the proof of Theorem 18.3.3. By Lemma 18.3.2, the operad \(\oDay (\Mm _C,\Oo )\) is the full suboperad of \(\oDay (\Mm _C,\Mm _D)\) spanned by the \(\Oo _{\lra {1}}\)-valued functors. Therefore its multimorphism animae agree with the corresponding multimorphism animae in \(\oDay (\Mm _C,\Mm _D)\). By Proposition 16.2.6, we get \[ \oDay (\Mm _C,\Oo )(\{F_i\}_{i \in I};G) \simeq \Nat \left (\bigotimes \nolimits ^I_D\circ \prod _{i \in I}F_i,G\circ \bigotimes \nolimits ^I_C\right ). \] Using the end formula for natural transformations from Proposition 23.6.6, the right-hand side is equivalent to \[ \int _{\{c_i\}_{i \in I}\in C^I} \Hom _D\left (\bigotimes \nolimits ^I_D F_i(c_i),G(\bigotimes \nolimits ^I_C c_i)\right ). \] By Lemma 17.3.17, the integrand is naturally equivalent to \[ \Oo (\{F_i(c_i)\}_{i \in I};G(\bigotimes \nolimits ^I_C c_i)). \] Taking ends preserves this equivalence of integrands, giving the desired formula. □
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