a coalgebras over fields question.

a coalgebras over fields question.

[Bisson, Terrence P]

Hi,  I have heard that naive questions are allowed at the cat list, so here  goes:
   
The diagonal map in spaces often gives a co-commutative diagonal map in homology,
so I want to understand the special properties of the category of co-commutative coalgebras.

It seems to be a "well-known fact" that 
  the category of co-commutative coalgebras over a field is a cartesian closed category,
but I can't seem to find much discussion of the internal hom.  
Can you suggest any reference?   

Thanks, Terry Bisson



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Re: a coalgebras over fields question.

[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

----- Original Message -----
From: "Bisson, Terrence P" <bisson@canisius.edu>
To: <categories@mta.ca>
Sent: Wednesday, August 24, 2011 12:58 AM
Subject: categories: a coalgebras over fields question.


> Hi,  I have heard that naive questions are allowed at the cat list, so
> here  goes:
>
> The diagonal map in spaces often gives a co-commutative diagonal map in
> homology,
> so I want to understand the special properties of the category of
> co-commutative coalgebras.
>
> It seems to be a "well-known fact" that
>  the category of co-commutative coalgebras over a field is a cartesian
> closed category,
> but I can't seem to find much discussion of the internal hom.
> Can you suggest any reference?
>
> Thanks, Terry Bisson
>



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Re: a coalgebras over fields question.

[Terry Bisson]

Todd Trimble <trimble1 <at> optonline.net> writes:

  >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.


  Hans-E. Porst <porst@math.uni-bremen.de>   pointed me to 
similar results in his recent paper
"On subcategories of the category of Hopf algebras",  Arabian
Journal of Science and Enginering, (2011), DOI 10.1007/s13369-011-0090-4 

Others recommended his article  "On corings and comodules", 
Archivum Mathematicum 42 (2006), no. 4, 419-425. 

Preprints of these are available at
http://www.math.uni-bremen.de/~porst/

I am finding these comments very helpful.  Thanks greatly, Terry Bisson


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Re: a coalgebras over fields question.

[Hans-E. Porst]

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
FB 3: Mathematics                               Phone: +49 421 21863701
University of Bremen                            Secr.: +49 421 21863700
D-28334 Bremen                                  Fax:   +49 421 2184856


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Re: a coalgebras over fields question.

[Donovan Van Osdol]

Dear Terry,
     Since the impetus for your question seems to come from
algebraic topology, you might find two papers (each about
forty years old!) of at least passing interest:

"Coalgebras, Sheaves, and Cohomology":
http://www.ams.org/journals/proc/1972-033-02/S0002-9939-1972-0294447-7/S0002-9939-1972-0294447-7.pdf,  and

"Bicohomology Theory":
http://www.ams.org/journals/tran/1973-183-00/S0002-9947-1973-0323873-8/S0002-9947-1973-0323873-8.pdf

With best regards,
Don


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Re: a coalgebras over fields question.

[pare]

Hi Terry (and others),

You might have a look at my paper with Grunenfelder "Families Parametrized
by Coalgebras", J. Alg. 107 (1987), 316-375, which gives a different
perspective on the topic.

Bob

> Hi,  I have heard that naive questions are allowed at the cat list, so
> here  goes:
>
> The diagonal map in spaces often gives a co-commutative diagonal map in
> homology,
> so I want to understand the special properties of the category of
> co-commutative coalgebras.
>
> It seems to be a "well-known fact" that
>   the category of co-commutative coalgebras over a field is a cartesian
> closed category,
> but I can't seem to find much discussion of the internal hom.
> Can you suggest any reference?
>
> Thanks, Terry Bisson
>

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Re: a coalgebras over fields question.

[Todd Trimble]

Dear All,

I had posted the message below yesterday, but the message had
some html in it (from previous messages in the exchange, I believe).
It is somewhat redundant today because Professor Porst's paper
has now been mentioned in a different posting, but it can still serve
as a note of appreciation to him. :-)

Best regards,

Todd

----- Original Message -----
From: Todd Trimble
To: Hans-E. Porst
Cc: Bisson, Terrence P ; Categories list
Sent: Thursday, August 25, 2011 11:49 AM
Subject: Re: categories: Re: a coalgebras over fields question.


Indeed, Hans-E. Porst has done a lot of valuable work in the area
of coalgebras and comodules. I would also recommend his article
"On corings and comodules", Archivum Mathematicum 42 (2006),
no. 4, 419-425.

Todd


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