question on pseudomorphisms

question on pseudomorphisms

[burroni]

Dear all,

Has the following question been already studied and, if it is the case  
(it is my opinion), where ?
Let T be a monad on Cat (not necessarily a 2-monad), C the category of  
T-algebras and C' the catégorie with the same objets but with  
pseudomorphisms (morphisms up to natural isomorphisms --- eventually  
with coherences).
The inclusion i : C --> C' has a left adjoint j : C' --> C.

The question is : for all T-algebra A, is the canonical morphisms m :  
A --> i(j(A)) an equivalence (of the underlying categories) ?

Regards,
Albert



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Re: question on pseudomorphisms

[burroni]

Dear Steve,
Thank you for your precious informations (I have never read the  
article "two-dimensional ...").
But my interest is really in the case of non necessarily 2-monad over  
Cat and I am convinced that the adjonction j -| i exists, with or  
without coherences (for the pseudomorphisms).
Best regards,
Albert

Steve Lack <s.lack@uws.edu.au> a écrit :

> Dear Albert,
>
> In the case where T is a 2-monad with rank, and the pseudomorphisms are (as
> usual) assumed to be coherent, then this was proved by Blackwell-Kelly-Power
> in the paper 2-dimensional monad theory. (Here "rank" means that the
> 2-functor T preserves alpha-filtered colimit for some alpha - without some
> such assumption, I don't know how you can prove that the left adjoint j
> exists, and I suspect it does not.)
>
> If T has rank but pseudomorphisms are not required to be coherent, then the
> adjoint j will exist, but the morphism A-->ijA need not be an equivalence
> (take the identity monad for example).
>
> If T is not even a 2-monad then I'm not sure what coherence of the 2-cells
> would mean, but in any case there will be problems.
>
> Regards,
>
> Steve Lack.
>
>

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Re: question on pseudomorphisms

[Steve Lack]

Dear Albert,

In the case where T is a 2-monad with rank, and the pseudomorphisms are (as
usual) assumed to be coherent, then this was proved by Blackwell-Kelly-Power
in the paper 2-dimensional monad theory. (Here "rank" means that the
2-functor T preserves alpha-filtered colimit for some alpha - without some
such assumption, I don't know how you can prove that the left adjoint j
exists, and I suspect it does not.)

If T has rank but pseudomorphisms are not required to be coherent, then the
adjoint j will exist, but the morphism A-->ijA need not be an equivalence
(take the identity monad for example).

If T is not even a 2-monad then I'm not sure what coherence of the 2-cells
would mean, but in any case there will be problems.

Regards,

Steve Lack.


On 23/11/09 11:55 PM, "burroni@math.jussieu.fr" <burroni@math.jussieu.fr>
wrote:

> Dear all,
> 
> Has the following question been already studied and, if it is the case
> (it is my opinion), where ?
> Let T be a monad on Cat (not necessarily a 2-monad), C the category of
> T-algebras and C' the catégorie with the same objets but with
> pseudomorphisms (morphisms up to natural isomorphisms --- eventually
> with coherences).
> The inclusion i : C --> C' has a left adjoint j : C' --> C.
> 
> The question is : for all T-algebra A, is the canonical morphisms m :
> A --> i(j(A)) an equivalence (of the underlying categories) ?
> 
> Regards,
> Albert
> 
> 
> 
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]