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I would like to prove that a monoidal functor $$\Phi\colon \mathbf{V}\to \mathbf{V'}$$ induces a functor $$\Phi^\#\colon \mathbf{V}\text{-Cat}\to \mathbf{V'}\text{-Cat}$$

and in particular I would like to check that if $\Phi$ admits a left/right adjoint, then $\Phi^\#$ admits one too. The problem is that I'm stuck in finding the "right" definitions involved (especially that of monoidal functor); I have to silently suppose that $\Phi$ is strong (or "non-lax") monoidal, i.e. $\Phi(A\otimes B)\cong \Phi(A)\otimes '\Phi(B)$, $\Phi(I)\cong I'$ for all $A,B\in \mathbf{V}$ and the "initial" objects $I\in \mathbf V$, $I'\in \mathbf V'$. Such a restrictive assumption leaves me unsatisfied, but I'm not really keen on non-strict monoidal functors...

As a side question, it seems to me this is a well-established result in enriched category theory, but I'm not able to find a precise reference proving the result from the beginning: Kelly treats the result as a well known folklore, saying in the first pages of Basic concepts of ECT

[we do not] discuss the change of base-category given by a symmetric monoidal functor $\mathbf{V}\to \mathbf{V'}$ and the induced 2-functor $\mathbf{V}\text{-Cat}\to \mathbf{V'}\text{-Cat}, [...]$

and John Gray, in his article Closed Categories, Lax Limits and Homotopy Limits just gives a statement of the claim I would like to prove. Again, can you help me?

Thanks a lot.

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  • $\begingroup$ The change of base is a 2-functor $\mathbf{CBase}:\mathbf{MonCat}\rightarrow\mathbf{2\text{-}Cat}$, so if you have an adjunction in $\mathbf{MonCat}$, you must have an adjunction in $\mathbf{2\text{-}Cat}$... $\endgroup$ Aug 28, 2018 at 18:28

3 Answers 3

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I believe you'll find the following article helpful:

Change of Base, Cauchy Completeness and Reversibility (pdf)

(In A. Labella & V. Schmitt, Theory and Applications of Categories, Vol. 10: 10, 2002, pp. 187–219.)

This linked article addresses much of what you're looking for, and includes references which may be of help to you. (E.g. G.M. Kelly, Basic Concepts of Enriched Category Theory, London Mathematical Society Lecture Notes Series 64, Cambridge University Press, 1982.)

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The first part of your first question (the existence of the $2$-functor $\Phi^\sharp$) can be found in Borceux's "Handbook of categorical algebra", vol 2, proposition 6.4.3. Notice that Borceux's monoidal functors are lax ones.

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  • $\begingroup$ Thanks, I already found it but I think it's nothing difficult to define $\Phi^\#$... the real problem by now seems to be "What is an adjunction between two 2-categories?" $\endgroup$
    – fosco
    Jul 9, 2011 at 13:47
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Another really good (and very readable) reference on this is Geoff Cruttwell's thesis:

G. S. H. Cruttwell, ‘Normed Spaces and the Change of Base for Enriched Categories’

Chapter 4 explicitly gives these constructions in detail. In particular, you need a monoidal adjunction to induce an adjunction between $\mathbf{V}$-$\mathrm{Cat}$ and $\mathbf{V}^\prime$-$\mathrm{Cat}$. However, a functor has a monoidal right adjoint if and only if it is strong monoidal and has a right adjoint functor in the ordinary sense. This will subsume all Cartesian (i.e. product-preserving) functors which admit right adjoints for example.

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