Re: Reference for lifting an adjunction to a monoidal one

["F. Lucatelli Nunes" <flucatellinunes@gmail.com>]

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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