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/ ]