From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6843 Path: news.gmane.org!not-for-mail From: "Hans-E. Porst" Newsgroups: gmane.science.mathematics.categories Subject: Re: a coalgebras over fields question. Date: Sat, 27 Aug 2011 10:12:23 +0200 Message-ID: References: Reply-To: "Hans-E. Porst" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v1084) Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1314455486 5746 80.91.229.12 (27 Aug 2011 14:31:26 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 27 Aug 2011 14:31:26 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Sat Aug 27 16:31:22 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 1QxJur-0008BO-4j for gsmc-categories@m.gmane.org; Sat, 27 Aug 2011 16:31:21 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:58925) by smtpy.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1QxJtd-00052r-Hq; Sat, 27 Aug 2011 11:30:05 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1QxJtc-0002LH-Ry for categories-list@mlist.mta.ca; Sat, 27 Aug 2011 11:30:04 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6843 Archived-At: 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=20 Regards, Hans Am 24.08.2011 um 18:46 schrieb Todd Trimble: > Dear Terry, >=20 > 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). >=20 > If you want a quick proof of cartesian closure based on the > adjoint functor theorem, try this paper by Michael Barr: >=20 > ftp://ftp.math.mcgill.ca/pub/barr/coalgebra.pdf >=20 > which gives a proof that applies more generally to coalgebras > over a general commutative ring. >=20 > 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 >=20 > CocommCoalg ~ Lex(CommAlg_{fd}, Set), >=20 > where the right side is the category of left exact functors on the > category of finite-dim commutative algebras/k. >=20 >> =46rom 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 >=20 > A* \otimes_k C --> D >=20 > (NB: if C and C' are cocommutative coalgebras over k, then > C' \otimes_k C is their cartesian product in CocommCoalg.) >=20 > Best regards, >=20 > Todd Trimble >=20 --=20 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/ ]