Re: question on pseudomorphisms

[burroni@math.jussieu.fr]

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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