Lemma 16.2.11 (Left Kan extensions and Day convolution). Let \(u\colon A\to B\) be a symmetric monoidal functor, and let \(D\) be a symmetric monoidal \(\infty \)-category. Assume that the Day convolution monoidal structures on \(\Fun (A,D)\) and \(\Fun (B,D)\) exist. Suppose that for every object \(b\in B\), the \(\infty \)-category \(D\) admits colimits indexed by the relative slice category \(u_{/b}\), and that its tensor product preserves these colimits separately in each variable. Then pointwise left Kan extension along \(u\) defines a symmetric monoidal functor \[ u_!\colon \Fun (A,D)\longrightarrow \Fun (B,D). \] In particular, this applies when \(A\) and \(B\) are small, while \(D\) is cocomplete and its tensor product preserves small colimits separately in each variable.
Proof. The hypotheses on the relative slice categories ensure that pointwise left Kan extension along \(u\) exists. Its right adjoint \(u^*\) is lax symmetric monoidal by Proposition 16.2.10. By Proposition 14.3.11, it therefore suffices to show that for every finite collection of functors \(F_i\colon A\to D\), indexed by a finite set \(I\), the canonical comparison \[ u_!\bigl (\bigotimes _{i\in I}^{\Day }F_i\bigr ) \longrightarrow \bigotimes _{i\in I}^{\Day }u_!(F_i) \] is an isomorphism.
Set \[ H:=\bigotimes _D^I\circ \prod _{i\in I}F_i\colon A^I\to D. \] By the description of Day convolution in Remark 16.2.9 and the composition law for left Kan extensions, the source of the comparison is the left Kan extension of \(H\) along \[ A^I\xrightarrow {\smash {\bigotimes _A^I}}A\xrightarrow {u}B. \] On the other hand, the pointwise formula for left Kan extensions and the assumptions on the tensor product of \(D\) show that \[ \bigotimes _D^I\circ \prod _{i\in I}u_!(F_i)\colon B^I\to D \] is the left Kan extension of \(H\) along \(u^I\colon A^I\to B^I\): the relative slice of \(u^I\) over a tuple \((b_i)_{i\in I}\) is the product of the relative slices \(u_{/b_i}\), and its colimit may be computed iteratively. Applying Day convolution and composing left Kan extensions therefore identifies the target of the comparison with the left Kan extension of \(H\) along \[ A^I\xrightarrow {u^I}B^I\xrightarrow {\smash {\bigotimes _B^I}}B. \] Since \(u\) is symmetric monoidal, these two composite functors are naturally equivalent. The canonical comparison is therefore an isomorphism, as desired. □
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