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Consider the 2-category $\mathsf{MonCat}$ where objects are monoidal categories, 1-cells are strong monoidal functors, and 2-cells are monoidal natural transformations.

Given arrows $f: \mathsf{C} \to \mathsf{D}, g: \mathsf{C} \to \mathsf{E}$ in this category, we can obviously ask for a left Kan extension $Lan_fg : \mathsf{D} \to \mathsf{E}$.

If we instead work in $\mathsf{MonCat}^\mathrm{lax}$, the 2-category of monoidal categories, lax monoidal functors and monoidal natural transformations, this left Kan extension is known to exist under some reasonable assumptions.

Questions:

  1. Under reasonable assumptions on the categories involved, does the "strong monoidal Kan extension" exist?
  2. Does the "lax monoidal Kan extension" recover the strong version in the case where the two input functors are strong monoidal? In other words, are the subcategories of strong monoidal functors stable under left Kan extension along strong monoidal functors?

At the highest level of generality, we can consider lax or strong $\mathcal{O}$-monoidal functors between $\mathcal{O}$-monoidal $\infty$-categories, in the sense of Lurie. In that context, the lax left kan extensions seem to correspond to Lurie's "operadic Kan extensions", but again there does not seem to be a strong version of this in the literature.

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    $\begingroup$ Related, but not related: on the problem of when a product preserving functor is still product preserving when left Kan-extended, and on the fact that the previos fact is a "miracle" $\endgroup$
    – fosco
    Commented Jul 1, 2020 at 15:51
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    $\begingroup$ Also, see here! arxiv.org/pdf/1809.10481.pdf I wasn't aware of the existence of this note. $\endgroup$
    – fosco
    Commented Jul 2, 2020 at 7:55
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    $\begingroup$ @fosco: Yes, I saw that note - it seems to say something about when the ordinary kan extension is also a "lax monoidal kan extension" - but I'm not having much luck figuring out if it's also strong when you extend a strong functor $\endgroup$
    – Eigil Fjeldgren Rischel
    Commented Jul 2, 2020 at 11:15
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    $\begingroup$ This paper studies the lifting of Kan extensions that are "algebraic" over a monad on double categories; section 2 of this paper by Weber has an analogous 2-categorical approach. Roughly, to lift lax monoidal structure it suffices that the tensor of $E$ preserves the left Kan cell in both variables; to lift the colax structure it suffices that the left Kan cell is preserved by composition with the colax structure cell of $f$. In the pseudo case having both assumptions is sufficient. $\endgroup$
    – Roald Koudenburg
    Commented Jul 2, 2020 at 15:23

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