Re: a coalgebras over fields question.

["Hans-E. Porst" <porst@math.uni-bremen.de>]

Those interested in similar results might also look at my recent paper

On subcategories of the category of Hopf algebras, Arabian Journal of Science and Enginering, (2011), DOI 10.1007/s13369-011-0090-4

and the references therein.

A preprint of which as available at

http://www.math.uni-bremen.de/~porst/dvis/PORST_Hopf_fin.pdf 


Regards,
Hans


Am 24.08.2011 um 18:46 schrieb Todd Trimble:

> Dear Terry,
> 
> The category of cocommutative coalgebras over a field k has a
> lot of nice properties: it is not only cartesian closed, it is also
> extensive and locally finitely presentable (hence also complete
> and cocomplete, and even total).
> 
> If you want a quick proof of cartesian closure based on the
> adjoint functor theorem, try this paper by Michael Barr:
> 
> ftp://ftp.math.mcgill.ca/pub/barr/coalgebra.pdf
> 
> which gives a proof that applies more generally to coalgebras
> over a general commutative ring.
> 
> One absolutely crucial fact in this whole business is sometimes
> called the fundamental theorem for (cocommutative) coalgebras:
> each is the filtered colimit of its finite-dimensional subcoalgebras.
> (Here we are working over a field k. The situation is more subtle
> over a general commutative ring R.) The finite-dim cocommutative
> coalgebras over k coincide with the finitely presentable objects in
> CocommCoalg_k, and the category of finite-dim cocommutative
> coalgebras is dual to the category of finite-dim commutative
> algebras/k. One concludes (a la Gabriel-Ulmer duality) that there
> is an equivalence
> 
> CocommCoalg ~ Lex(CommAlg_{fd}, Set),
> 
> where the right side is the category of left exact functors on the
> category of finite-dim commutative algebras/k.
> 
>> From there, one can derive how exponentials should work: if
> C and D are cocommutative coalgebras, then their exponential
> D^C is the coalgebra which represents the left exact functor
> which takes a finite-dim algebra A to the set of coalgebra
> homomorphisms
> 
> A* \otimes_k C --> D
> 
> (NB: if C and C' are cocommutative coalgebras over k, then
> C' \otimes_k C  is their cartesian product in CocommCoalg.)
> 
> Best regards,
> 
> Todd Trimble
> 

-- 
Hans-E. Porst                                 porst@math.uni-bremen.de
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