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Let $L: \mathcal C \leftrightarrows \mathcal D : R$ be an adjoint pair and $\mathcal C = (\mathcal C,\otimes)$ be a monoidal category. I wonder under what conditions on the pair $L\dashv R$ we can transfer the monoidal structure to $\mathcal D$, so the adjunction becomes monoidal (not neccessary in the strong sense). What are the relevant references here that I can read?

So for $x,\ y, \ z \in \mathcal D$ we can consider something like $x\square y = R(Lx\otimes Ly)$, but then we need to agree all the triples like $(x \square y)\square z$, $x \square (y \square z)$ etc. If the adjoint pair is an equivalence then it is ok, but perhaps there are other weaker conditions.

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    $\begingroup$ if $R$ is fully faithful equivalently, $L$ is a localization, or the co-unit $LR\to id$ is an isomorphism), and you have the following condition: "if $L(x\to y)$ is an isomorphism, then so is $L(x\otimes z\to y\otimes z)$ for all $z$", then you can do it. Is that the type of condition you're looking for ? Or are you also trying to get away from the fully faithful case $\endgroup$
    – Maxime Ramzi
    Commented Apr 30, 2022 at 7:48
  • $\begingroup$ @MaximeRamzi , for my applications fully faithful is too much, but faithful should be ok. I have an impression that there should be a general theory for answering such questions, which I'm just not aware about. $\endgroup$
    – fyo
    Commented Apr 30, 2022 at 8:43
  • $\begingroup$ I'm not sure, but if $R$ is faithful then maybe your adjunction is monadic, or close to being so. For monadic adjunctions, there are some things you can say if the monad is strong, which should translate to some conditions on the adjunction. You could imagine a situation similar to Sets and abelian groups $\endgroup$
    – Maxime Ramzi
    Commented May 1, 2022 at 10:49
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    $\begingroup$ @W.Rether : I'm not sure, it seems like there is Brian Day's "Note on monoidal localizations", and there is also something in Lurie's Higher Algebra in the context of $\infty$-categories $\endgroup$
    – Maxime Ramzi
    Commented Aug 13, 2022 at 15:27
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    $\begingroup$ @W.Rether I guess that would be 4.1.7.4. for localizations + the fact that an adjunction in which the right adjoint is fully faithful is a localization, and maybe you would also benefit from 7.3.2.7. after having 4.1.7.4 $\endgroup$
    – Maxime Ramzi
    Commented Aug 15, 2022 at 9:59

1 Answer 1

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A relevant reference is §3 of Kelly's Doctrinal adjunction: in particular, see Theorem 3.1, which states that, if the adjunction is reflective, then the subcategory inherits monoidal structure if and only if $$L(\eta_X \otimes \eta_Y) : L(X \otimes Y) \to L(RLX \otimes RLY)$$ is invertible for all $X, Y \in \mathcal C$ (see (3.6) ibid.).

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