Re: Derived cotriples

["Eduardo J. Dubuc" <edubuc@dm.uba.ar>]

On 21/05/12 18:23, Michael Barr wrote:
> Suppose (G,\epsilon,\delta) is a cotriple on a complete category. Let G^2
> ===> G ---> G' be a coequalizer. Then we can find canonical (perhaps
> unique) \epsilon': G' ---> Id and \delta': G' ---> G'^2 such that
> (G',\epsilon',\delta') is a new cotriple on the category and such that G
> ---> G' is a map of cotriples. It seems reasonable to call this the
> derived cotriple. This process can be repeated, apparently forever, using
> colimits at limit ordinals. If it ever stablizes, the resultant cotriple
> will be idempotent and vice versa. Does any know whether this construction
> has been studied before?
>
> Michael
>

Hi, the following is related (or the same ?):

In my thesis (SLN 145, page 135) I consider the dual case of
monads=triples in the enriched V-category case.

Considering triple T in A (with the smallness (*) condition of being the
codensity triple determined by a set of objects in A).

I construct a chain of categories

B=A_oo ---> ... ---> A_a ---> .... ---> A_b ---> ... A_1 ---> A_0=A

where A_1 is the category of algebras for the triple T in A

A_(a+1) ---> A_a ,  A_(a+1) is algebras for a triple in A_a

for a limit ordinal a,  A_a is a limit of the preceeding chain of rigth
adjoints.

B is the limit of the large tower over all the ordinals, which is shown
to exists (see (*)).

We have  for all "a"  a rigth adjoint functor  A_a ---> A determining a
triple T_a in A and also a rigth adjoint functor  B ---> A, which is
full and faithful and so the corresponding cotriple in B is the
identity, and the corresponding triple T_oo in A is idempotent. There
are maps of triples:

     T_oo ---> ... ---> T_a ---> ... ---> T_b ---> ... --->  T_1  = T

(with T_oo idempotent).

(*) The smallness condition is not needed to develop this construction,
but it is needed to prove that the process stabilizes, that is, that the
cotriple in B is the identity.

   Eduardo Dubuc


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