From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6835 Path: news.gmane.org!not-for-mail From: Todd Trimble Newsgroups: gmane.science.mathematics.categories Subject: Re: a coalgebras over fields question. Date: Wed, 24 Aug 2011 12:46:55 -0400 Message-ID: References: Reply-To: Todd Trimble NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; format=flowed; charset=iso-8859-1; reply-type=original Content-Transfer-Encoding: 7BIT X-Trace: dough.gmane.org 1314227862 23587 80.91.229.12 (24 Aug 2011 23:17:42 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 24 Aug 2011 23:17:42 +0000 (UTC) Cc: Categories list To: "Bisson, Terrence P" Original-X-From: majordomo@mlist.mta.ca Thu Aug 25 01:17:37 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.128]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1QwMhT-00081S-Q5 for gsmc-categories@m.gmane.org; Thu, 25 Aug 2011 01:17:36 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:50891) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1QwMfT-0004T5-LJ; Wed, 24 Aug 2011 20:15:31 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QwMfS-0006Ro-Vm for categories-list@mlist.mta.ca; Wed, 24 Aug 2011 20:15:30 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6835 Archived-At: 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" To: 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/ ]