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$F : \mathbf{MonCat} \to \mathbf{2Cat}$ is the 2-functor for change of enrichment. What is the maximal subcategory of $\mathbf{MonCat}$ whose arrows $b : V \to W$ each induce an equivalence of categories $F(b) : V\mathbf{Cat} \cong W\mathbf{Cat}$?

My current guess is that we can take some restricted portion of the poset of embeddings of categories in $\mathbf{MonCat}$, perhaps using some sort of adjointness requirement. I convinced myself that this works with a diagram chase, but I think I'm wrong.

This question was split from a more general question on MSE about implications of identifying such subcategories.

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    $\begingroup$ Do you actually have any example of such a functor $b$ between monoidal categories such that $F(b)$ is an equivalence ? I don't think it exists (unless of course $b$ is an equivalence). $\endgroup$
    – Simon Henry
    Commented Jul 14, 2021 at 15:28
  • $\begingroup$ I had suspected the full inclusions of subcategories. They should be full & faithful. The induced change-of-homs arrows in 2Cat would then also be full & faithful, and they're already essentially surjective, so they would be equivalences. $\endgroup$
    – Corbin
    Commented Jul 14, 2021 at 22:35
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    $\begingroup$ But they are not essentially surjective in general. $\endgroup$
    – Simon Henry
    Commented Jul 14, 2021 at 22:48
  • $\begingroup$ Maybe I'm confused over this fragment in nLab; F is defined s.t. it "canonically induces a 2-functor F* : VCat -> WCat ... s.t. F*(C) has the same objects as C", and that seems like a special case of being eso. $\endgroup$
    – Corbin
    Commented Jul 15, 2021 at 17:16
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    $\begingroup$ I think you are confused over what "$F^*$ essentially surjective" mean: it would mean that every $W$-enriched category if equivalent to one of the form $F^*C$ for $C$ a $V$-enriched category. There is simply no relation with the nLab claim. $\endgroup$
    – Simon Henry
    Commented Jul 15, 2021 at 17:22

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Here's a partial confirmation of Simon Henry's hunch:

Observation: Let $F: V \to W$ be a strong monoidal functor. Suppose that $V$ and $W$ have initial objects $\emptyset$ preserved by $\otimes$ in each variable separately, and preserved by $F$. Suppose that $F$ induces an equivalence $VCat \to WCat$. Then $F$ is an equivalence.

Proof: The assumptions allow us to form, for each $v \in V$, the category $\Sigma(v)$, which has two objects $0,1$, with $Hom(0,1) = v$, $Hom(0,0) = Hom(1,1) = I$, $Hom(1,0) = \emptyset$, and to observe that $\Sigma_V : V \to VCat$, $v \mapsto \Sigma v$ is a fully faithful functor. The same also holds in $WCat$. Moreover, if $F$ induces an equivalence it must carry $V$-functors which induce bijections on isomorphism classes of objects to $W$-functors with the same property. From this it soon follows that the restriction of $F$ to $V$-categories of the form $\Sigma(v)$ is inverse to the restriction of $F^{-1}$ to the $\Sigma(w)$'s, and so because $\Sigma_V$ and $\Sigma_W$ are fully faithful, it follows that $F$ is an equivalence.

Possibly the general case can be reduced to this one via Day convolution or something?

I think it's a bit more interesting to look a the the 2-category $V-Mod$ rather than $VCat$. I believe that Ross Street may have written about the analogous question here, but I'm not sure of the reference.

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    $\begingroup$ The reference you are referring to is "Cauchy Characterization of enriched categories". Also "Enriched categories as a free cocompletion" by Garner and Shulman seems on point to me. I believe a combination of the two should deliver the correct answer to the question. $\endgroup$
    – Ivan Di Liberti
    Commented Jul 14, 2021 at 16:31
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    $\begingroup$ @Tim, I don't see how your $\Sigma_V$ is fully faithful, even in case $V = Set$. $\endgroup$
    – Michal R. Przybylek
    Commented Jul 16, 2021 at 14:13
  • $\begingroup$ @MichalR.Przybylek for V=Set, it says that given sets A,B a functor from the category with two objects x,y with no automorphisms but the identity, and an arrow from x to y for each element of A and no arrows in the opposite direction, to the category with B instead, is given exactly by a function A to B $\endgroup$
    – Alex Mine
    Commented Jul 17, 2021 at 5:15
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    $\begingroup$ @AlexMine, what you have said is simply not true --- you are missing two obvious constant functors. $\endgroup$
    – Michal R. Przybylek
    Commented Jul 18, 2021 at 11:18
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    $\begingroup$ @MichalR.Przybylek Sorry, I meant to say that $\Sigma_V$ is fully faithful if you take its codomain to be the category of $V$-categories and identity-on-objects $V$-functors. I should probably make explicit the argument that any equivalence $V-Cat \simeq W-Cat$ will carry identity-on-objects $V$-functors in the image of $\Sigma_V$ to identity-on-objects $W$-functors in the image of $\Sigma_W$. $\endgroup$
    – Tim Campion
    Commented Jul 20, 2021 at 13:07

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