From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7719 Path: news.gmane.org!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: on a subcategory of algebras for a monad Date: Sat, 11 May 2013 01:09:32 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1368398837 12106 80.91.229.3 (12 May 2013 22:47:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sun, 12 May 2013 22:47:17 +0000 (UTC) To: Emily Riehl , Original-X-From: majordomo@mlist.mta.ca Mon May 13 00:47:17 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Ubf2y-0007bf-DR for gsmc-categories@m.gmane.org; Mon, 13 May 2013 00:47:16 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:48794) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Ubf0n-0003u8-Ot; Sun, 12 May 2013 19:45:01 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Ubf0m-00075H-Mm for categories-list@mlist.mta.ca; Sun, 12 May 2013 19:45:00 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7719 Archived-At: Hi, Emily, Have you or your grad student noticed the example got by taking as your locally presentable category C the category of sets, and as your monad M = the ultrafilter or Stone-Cech monad =DF (with the Eilenberg-Moore category of= =DF-algebras being the category of compact Hausdorff spaces (and continuo= us maps))? Here, your "objects those X in C such that the unit X -> MX is an isomorp= hism" are just the finite sets, and, unless I misunderstand your limits questio= n, I fear that only finite limits will work as you desire. Does that help in any way? Cheers, -- Fred = --- ------ Original Message ------ Received: Fri, 10 May 2013 09:23:34 PM EDT From: Emily Riehl To: categories@mta.ca Subject: categories: on a subcategory of algebras for a monad > Hi, > = > I received the following question from a grad student that I was unable= to > answer, but maybe you can (shared with permission). The subcategory Com= p_M > he introduces below can equally be defined to be the inverter of the > counit of the monadic adjunction. But I don't see how this universal > property helps understand limits in the subcategory. We suspect a left > adjoint to the inclusion is unlikely. > = > Can you help? Or have you seen something like this before? > = > Best, > Emily > = > *** > ?? > Hi folks, > ?? > I'm interested in closure properties of a particular subcategory of the= > category of algebras of a monad. To be more precise, let C be a locally= > presentable category and M be a monad on C. The category of algebras Al= g_M > has all limits, and they are computed in C. Denote by Comp_M the full > subcategory of Alg_M of "M-complete objects" (does anyone have a better= > name?), with objects those X in C such that the unit X -> MX is an > isomorphism, viewed in the natural way as M-algebras (using the inverse= MX > -> X). > ?? > My question: Is Comp_M closed under (actually: sequential) limits, comp= uted > as limits in Alg_M? > ?? > For some examples that come to mind immediately, the answer is clearly = yes, > because Comp_M is either trivial (e.g., if M is the free monoid monad o= n > Sets) or all of Alg_M (i.e., if M is idempotent). A more interesting example > is Bousfield-Kan R-completion, for which I don't know the answer. > ?? > In fact, I'm interested in left exact monads; in this case, the idempot= ent > approximation is given by the equalizer of the two natural maps M -> M^= 2, > but I'm not sure if this is relevant. What I'm hoping for is a sufficie= nt > criterion or a good counterexample in the abstract situation. > ?? > Many thanks! > = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]