Day convolution is characterized by an internal-hom universal property in the \(\infty \)-category of \(\infty \)-operads. This section takes that universal property as the definition of a Day convolution operad:
Definition 16.1.1. Let \(\Oo \) and \(\Pp \) be \(\infty \)-operads. An \(\infty \)-operad \(\oDay (\Oo ,\Pp )\) is called a Day convolution operad for \(\Oo \) and \(\Pp \) if it comes equipped with a morphism of \(\infty \)-operads \[ \ev \colon \oDay (\Oo ,\Pp ) \times \Oo \to \Pp \] satisfying the following universal property: if \(\Qq \) is another \(\infty \)-operad, the composite \[ \hspace {-6pt} \Fun _{\Op _{\infty }}(\Qq ,\oDay (\Oo ,\Pp )) \xrightarrow {- \times \Oo } \Fun _{\Op _{\infty }}(\Qq \times \Oo , \oDay (\Oo ,\Pp ) \times \Oo ) \xrightarrow {\ev \circ -} \Fun _{\Op _{\infty }}(\Qq \times \Oo , \Pp ) \] is an equivalence. Informally, this means we have the following classification of operad morphisms into \(\oDay (\Oo ,\Pp )\): \[ \{\text {Operad morphisms $\Qq \to \oDay (\Oo ,\Pp )$}\} \quad \leftrightsquigarrow \quad \{\text {Operad morphisms $\Qq \times \Oo \to \Pp $}\}. \] Note that this property uniquely determines \(\oDay (\Oo ,\Pp )\) if it exists.
Remark 16.1.2. Let \(\Oo \) be an \(\infty \)-operad for which \(\oDay (\Oo ,\Pp )\) exists for all \(\Pp \). Then the functor \(- \times \Oo \colon \Op _{\infty } \to \Op _{\infty }\) admits a right adjoint \[ \oDay (\Oo ,-)\colon \Op _{\infty } \to \Op _{\infty }. \]
Whenever the relevant Day convolution operads exist, this internal-hom description is functorial in both variables. An operad map \(u\colon \Oo '\to \Oo \) induces by precomposition an operad map \[ u^*\colon \oDay (\Oo ,\Pp )\longrightarrow \oDay (\Oo ',\Pp ), \] and an operad map \(v\colon \Pp \to \Pp '\) induces by postcomposition an operad map \[ v_*\colon \oDay (\Oo ,\Pp )\longrightarrow \oDay (\Oo ,\Pp '). \] The first map is adjoint to evaluation after precomposition with \(\id \times u\), while the second is adjoint to postcomposition of evaluation with \(v\). Their functoriality follows from the universal property. The same universal property gives a natural exponential law \[ \oDay \bigl (\Oo ,\oDay (\Pp ,\Qq )\bigr ) \simeq \oDay (\Oo \times \Pp ,\Qq ). \] Indeed, maps from an operad \(\Rr \) into either side are naturally equivalent to maps \(\Rr \times \Oo \times \Pp \to \Qq \).
The universal property of \(\oDay (\Oo ,\Pp )\) allows us to deduce most of its basic structure. Let us first describe its underlying \(\infty \)-category.
Lemma 16.1.3. Assume \(\oDay (\Oo ,\Pp )\) exists. Then the underlying \(\infty \)-category of \(\oDay (\Oo ,\Pp )\) is the \(\infty \)-category of functors \(\Oo _{\lra {1}} \to \Pp _{\lra {1}}\) between the underlying \(\infty \)-categories of \(\Oo \) and \(\Pp \): \[ \oDay (\Oo ,\Pp )_{\lra {1}} \quad \simeq \quad \Fun (\Oo _{\lra {1}},\Pp _{\lra {1}}). \]
Proof. By Lemma 14.1.20, the universal property of Day convolution, and Lemma 14.1.23, we have natural equivalences \begin {align*} \oDay (\Oo ,\Pp )_{\lra {1}} &\simeq \Fun _{\Op _{\infty }}(\Triv ,\oDay (\Oo ,\Pp )) \\ &\simeq \Fun _{\Op _{\infty }}(\Triv \times \Oo ,\Pp ) \\ &\simeq \Fun _{\Op _{\infty }}(\Triv _{\Oo _{\lra {1}}},\Pp ) \\ &\simeq \Fun (\Oo _{\lra {1}},\Pp _{\lra {1}}), \end {align*}
where the final equivalence is another instance of Lemma 14.1.20. □
As another consequence of the universal property, we may describe the commutative algebras in Day convolution operads:
Lemma 16.1.4. Assume \(\oDay (\Oo ,\Pp )\) exists. Then commutative algebras in \(\oDay (\Oo ,\Pp )\) correspond to operad morphisms \(\Oo \to \Pp \): \[ \CAlg (\oDay (\Oo ,\Pp )) \simeq \Fun _{\Op _{\infty }}(\Oo ,\Pp ). \]
Proof. Since \(\Comm \) is the terminal \(\infty \)-operad, we have \(\Oo \simeq \Comm \times \Oo \), and so \begin {align*} \CAlg (\oDay (\Oo ,\Pp )) &= \Fun _{\Op _{\infty }}(\Comm ,\oDay (\Oo ,\Pp )) \\ &\simeq \Fun _{\Op _{\infty }}(\Comm \times \Oo , \Pp ) \\ &\simeq \Fun _{\Op _{\infty }}(\Oo ,\Pp ). \qedhere \end {align*} □
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