Derived cotriples

Derived cotriples

[Michael Barr]

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

-- 
The United States has the best congress money can buy.


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Re: Derived cotriples

[Prof. Peter Johnstone]

The dual construction was studied by Sabah Fakir in "Monade idempotente
associee a une monade", C.R. Acad. Sci. Paris 270 (1970), A99-101.

Peter Johnstone

On Mon, 21 May 2012, 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
>
>


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Re: Derived cotriples

[Ross Street]

On 23/05/2012, at 2:34 AM, Prof. Peter Johnstone wrote:

> The dual construction was studied by Sabah Fakir in "Monade idempotente
> associee a une monade", C.R. Acad. Sci. Paris 270 (1970), A99-101.

Brian Day made good use of that construction to revisit the Applegate-Tierney tower.
That might be relevant to Mike Barr too.
See 
Day, Brian. On adjoint-functor factorisation. 
Category Seminar (Proc. Sem., Sydney, 1972/1973), pp. 1--19. 
Lecture Notes in Math., Vol. 420, Springer, Berlin, 1974.

Ross

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Re: Derived cotriples

[Eduardo J. Dubuc]

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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Re: Derived cotriples

[Donovan Van Osdol]

In my 1969 Ph.D. thesis I showed that, given a "good category for
sheaf theory" and a topological space, the associated sheaf functor
arises as the dual of your construction.  Basically, I needed the
"goodness" hypothesis so that the equalizer itself would construct the
associated sheaf and thus I would not need to iterate your
construction.  The triple used, in this case, was the original
Godement standard construction.  Details can be found in Springer
Lecture Notes in Mathematics, volume 236.

Don


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