Reference for lifting an adjunction to a monoidal one

Reference for lifting an adjunction to a monoidal one

[David Roberts]

Hi all,

I need a textbook or otherwise standard reference for the fact that if one
has a pair of monoidal categories C, D, and an adjunction L -| R: UC <-->
UD between their underlying categories, then if one of L or R lift to a
(strong) monoidal functor, then the adjunction lifts to an adjunction in
the 2-category of monoidal categories, strong monoidal functors and
monoidal natural transformations.

(Mac Lane of course only treats the case of strict monoidal functors, at
least in my, older, edition of his book)

Thanks,
David


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Re: Reference for lifting an adjunction to a monoidal one

[Richard Garner]

{Note from moderator: Apologies to Richard and Steve whose posts were 
inadvertently placed in the wrong folder...}

Dear David,

I am sure you will get a few responses telling you that the result, as
you state it, is not quite correct. What is correct is that, given an
adjunction L -| R: UC <---> UD:

a) endowments of L with oplax monoidal structure are in bijection, under
the mates correspondence, with endowments of R with lax monoidal
structure

b) given endowments of L and R with lax monoidal structure, the unit and
counit of the adjunction satisfy the conditions to be monoidal
transformations if and only if the given lax constraint cells on L are
inverse to the oplax constraint cells induced from R via a)

whence:

c) liftings of the adjunction L -| R to an adjunction in the 2-category
of monoidal categories, lax monoidal functors and monoidal
transformations are in bijective correspondence with endowments of L
with strong monoidal structure

There is a dual b') of b) giving the dual

c') liftings of the adjunction L -| R to an adjunction in the 2-category
of monoidal categories, oplax monoidal functors and monoidal
transformations are in bijective correspondence with endowments of R
with strong monoidal structure

of c). All of this follows from the general considerations in Kelly
"Doctrinal adjunction" SLNM 420, though it would be more perspicuous to
prove it directly following Kelly's schema.

Richard


On Sat, Jan 28, 2017, at 12:08 PM, David Roberts wrote:
> Hi all,
>
> I need a textbook or otherwise standard reference for the fact that if
> one
> has a pair of monoidal categories C, D, and an adjunction L -| R: UC <-->
> UD between their underlying categories, then if one of L or R lift to a
> (strong) monoidal functor, then the adjunction lifts to an adjunction in
> the 2-category of monoidal categories, strong monoidal functors and
> monoidal natural transformations.
>
> (Mac Lane of course only treats the case of strict monoidal functors, at
> least in my, older, edition of his book)
>
> Thanks,
> David

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Re: Reference for lifting an adjunction to a monoidal one

[F. Lucatelli Nunes]

Of course, there was a mistake in
"Consider the forgetful 2-functor U: Lax-Alg\to X. Let U(f) be a
*left adjoint* 1-cell.
f is left adjoint if and only if f is a pseudomorphism (and not just a lax
morphism)."

Sorry.
Best wishes

2017-02-04 0:36 GMT+00:00 F. Lucatelli Nunes <flucatellinunes@gmail.com>:

> Dear David Roberts,
>
> Sorry. I did not read the details of your statement. As Richard Garner
> observed, it is incorrect.
>
>
> Another way of stating the relevant result of Kelly is the following:
>
> "Consider the forgetful 2-functor U: Lax-Alg\to X. Let U(f) be a right
> adjoint 1-cell.
> f is left adjoint if and only if f is a pseudomorphism (and not just a lax
> morphism)."
> This means that there is a right adjoint g to f (if (U(f) is left adjoint
> and f is a pseudomorphism) in Lax-Alg.
>
> To get an adjunction in Ps-Alg, you should, now, ask whether this lifted g
> is also a pseudomorphism (which means to verify if the mate of the
> structure of f is an isomorphism).
>
> In other words, in the context of strong monoidal functors, considering the
> forgetful functor F: StrongMonoidal\to Cat, assume that f\dashv F(g) is an
> adjunction in Cat.
> g is right adjoint if and only if its mate is an isomorphism (that is to
> say, the induced oplax structure in f is a strong structure: Beck Chevalley
> Condition)
>
>
> Anyways, "Doctrinal Adjunction" (Kelly) is what you are looking for. You
> will probably find what you want about lifting of adjoints there.
> I would also recommend "Two-Dimensional Monadicity" of John Bourke
> (Advances in Mathematics) 2014.
>
>
> Best Regards
>
> 2017-01-30 2:37 GMT+00:00 Richard Garner <richard.garner@mq.edu.au>:
>
>>
>>
>> Dear David,
>>
>> I am sure you will get a few responses telling you that the result, as
>> you state it, is not quite correct. What is correct is that, given an
>> adjunction L -| R: UC <---> UD:
>>
>> a) endowments of L with oplax monoidal structure are in bijection, under
>> the mates correspondence, with endowments of R with lax monoidal
>> structure
>>
>> b) given endowments of L and R with lax monoidal structure, the unit and
>> counit of the adjunction satisfy the conditions to be monoidal
>> transformations if and only if the given lax constraint cells on L are
>> inverse to the oplax constraint cells induced from R via a)
>>
>> whence:
>>
>> c) liftings of the adjunction L -| R to an adjunction in the 2-category
>> of monoidal categories, lax monoidal functors and monoidal
>> transformations are in bijective correspondence with endowments of L
>> with strong monoidal structure
>>
>> There is a dual b') of b) giving the dual
>>
>> c') liftings of the adjunction L -| R to an adjunction in the 2-category
>> of monoidal categories, oplax monoidal functors and monoidal
>> transformations are in bijective correspondence with endowments of R
>> with strong monoidal structure
>>
>> of c). All of this follows from the general considerations in Kelly
>> "Doctrinal adjunction" SLNM 420, though it would be more perspicuous to
>> prove it directly following Kelly's schema.
>>
>> Richard
>>

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