From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7286 Path: news.gmane.org!not-for-mail From: "Eduardo J. Dubuc" Newsgroups: gmane.science.mathematics.categories Subject: Re: Derived cotriples Date: Tue, 22 May 2012 14:41:08 -0300 Message-ID: References: Reply-To: "Eduardo J. Dubuc" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit X-Trace: dough.gmane.org 1337812190 23945 80.91.229.3 (23 May 2012 22:29:50 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 23 May 2012 22:29:50 +0000 (UTC) Cc: Categories list To: Michael Barr Original-X-From: majordomo@mlist.mta.ca Thu May 24 00:29:50 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.80]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1SXK3w-0004BJ-Ka for gsmc-categories@m.gmane.org; Thu, 24 May 2012 00:29:48 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:43620) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1SXK2y-0004jx-Nr; Wed, 23 May 2012 19:28:48 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1SXK30-0003e4-88 for categories-list@mlist.mta.ca; Wed, 23 May 2012 19:28:50 -0300 User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.7; rv:12.0) Gecko/20120428 Thunderbird/12.0.1 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7286 Archived-At: On 21/05/12 18:23, Michael Barr wrote: > Suppose (G,\epsilon,\delta) is a cotriple on a complete category. Let G^2 > ===> G ---> G' be a coequalizer. Then we can find canonical (perhaps > unique) \epsilon': G' ---> Id and \delta': G' ---> G'^2 such that > (G',\epsilon',\delta') is a new cotriple on the category and such that G > ---> G' is a map of cotriples. It seems reasonable to call this the > derived cotriple. This process can be repeated, apparently forever, using > colimits at limit ordinals. If it ever stablizes, the resultant cotriple > will be idempotent and vice versa. Does any know whether this construction > has been studied before? > > Michael > Hi, the following is related (or the same ?): In my thesis (SLN 145, page 135) I consider the dual case of monads=triples in the enriched V-category case. Considering triple T in A (with the smallness (*) condition of being the codensity triple determined by a set of objects in A). I construct a chain of categories B=A_oo ---> ... ---> A_a ---> .... ---> A_b ---> ... A_1 ---> A_0=A where A_1 is the category of algebras for the triple T in A A_(a+1) ---> A_a , A_(a+1) is algebras for a triple in A_a for a limit ordinal a, A_a is a limit of the preceeding chain of rigth adjoints. B is the limit of the large tower over all the ordinals, which is shown to exists (see (*)). We have for all "a" a rigth adjoint functor A_a ---> A determining a triple T_a in A and also a rigth adjoint functor B ---> A, which is full and faithful and so the corresponding cotriple in B is the identity, and the corresponding triple T_oo in A is idempotent. There are maps of triples: T_oo ---> ... ---> T_a ---> ... ---> T_b ---> ... ---> T_1 = T (with T_oo idempotent). (*) The smallness condition is not needed to develop this construction, but it is needed to prove that the process stabilizes, that is, that the cotriple in B is the identity. Eduardo Dubuc [For admin and other information see: http://www.mta.ca/~cat-dist/ ]