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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
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Donovan Van Osdol -
Eduardo J. Dubuc -
Michael Barr -
Prof. Peter Johnstone -
Ross Street