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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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..., and "Bicohomology Theory": http://www.ams.org/journals/tran/1973-183-00/S0002-9947-1973-0323873-8/S0002... With best regards, Don [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (6)
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Bisson, Terrence P -
Donovan Van Osdol -
Hans-E. Porst -
pare＠mathstat.dal.ca -
Terry Bisson
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Todd Trimble