From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7921 Path: news.gmane.org!not-for-mail From: Michael Shulman Newsgroups: gmane.science.mathematics.categories Subject: Re: adjoints to lax-idempotent algebra structures Date: Mon, 18 Nov 2013 12:40:38 -0800 Message-ID: References: Reply-To: Michael Shulman NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1384821887 20789 80.91.229.3 (19 Nov 2013 00:44:47 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 19 Nov 2013 00:44:47 +0000 (UTC) Cc: categories To: "Prof. Peter Johnstone" Original-X-From: majordomo@mlist.mta.ca Tue Nov 19 01:44:53 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 1ViZQz-00066c-CA for gsmc-categories@m.gmane.org; Tue, 19 Nov 2013 01:44:53 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:40636) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ViZQ7-0004EN-VZ; Mon, 18 Nov 2013 20:43:59 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ViZQ8-0001FH-Ia for categories-list@mlist.mta.ca; Mon, 18 Nov 2013 20:44:00 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7921 Archived-At: Thanks again to everyone who replied with examples and comments. I've created an nLab page which hopefully includes everything I learned: http://ncatlab.org/nlab/show/continuous+algebra On Sun, Nov 17, 2013 at 9:14 AM, Michael Shulman wrote: > On Sun, Nov 17, 2013 at 6:53 AM, Prof. Peter Johnstone > wrote: >> What you can say about them in general >> is contained in Corollary B1.1.15 of the Elephant (page 254): they >> are exactly the retracts of free algebras (provided idempotent 2-cells >> split in the underlying 2-category), and they all occur as coadjoint >> retracts of free algebras. > > That's exactly the sort of thing I was looking for; thanks! > > Continuous categories are one of the examples I had in mind. Another > interesting almost-example is totally distributive categories. And > when T is a monad for coproducts, such a left adjoint seems to > decompose every object into a coproduct of connected ones (although I > have not analyzed this case carefully). > > T-continuous is a reasonable name, but it would also be nice for a > name to suggest B1.1.15. Is there a general name for algebras that > are retracts of free ones? In particular cases they are "projective" > or "cofibrant", but it seems doubtful that either of those terms > applies literally here. > > Mike [For admin and other information see: http://www.mta.ca/~cat-dist/ ]