From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2702 Path: news.gmane.org!not-for-mail From: Oswald Wyler Newsgroups: gmane.science.mathematics.categories Subject: Re: comparing cotriples via an adjoint pair Date: Sat, 22 May 2004 11:17:59 -0400 (EDT) Message-ID: References: <5.1.0.14.2.20040520232118.0235f070@mailbox.syr.edu> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018838 5388 80.91.229.2 (29 Apr 2009 15:27:18 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:27:18 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Sun May 23 18:00:11 2004 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 23 May 2004 18:00:11 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 1BS025-0005Mj-00 for categories-list@mta.ca; Sun, 23 May 2004 17:57:21 -0300 In-Reply-To: <5.1.0.14.2.20040520232118.0235f070@mailbox.syr.edu> Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 24 Original-Lines: 32 Xref: news.gmane.org gmane.science.mathematics.categories:2702 Archived-At: On Thu, 20 May 2004, Gaunce Lewis wrote: > I have encountered a situation in which I have two categories C, D which > are related by a pair of adjoint functors L from C to D and R from D to > C. Also, there is a cotriple S on C and a cotriple T on D. Finally, there > is a natural isomorphism f from RT to SR. It seems that if a couple of > diagrams relating f to the structure maps of the cotriples commute, then > there is an induced adjoint pair relating the two coalgebra categories. Is > this, or something similar to it, in the literature in some easily > referenced place? > > Thanks, > Gaunce This situation has been encountered since at least 1970 by various categorists, including myself. A relevant paper is: D. Pumpl\"un, Eine Bemerkung \"uber Monaden und adjungierte Funktoren, Math. Annalen 185, 329-337 (1970). If Gaunce's two commuting diagrams are the usual ones, then his conjecture is correct. Observe that in this situation, we have not just a pair but a quadruple of dual categories, replacing C and D by their duals, or inverting the direction of arrows, or both. This may just be a "folk theorem", but it should have been published by someone, somewhere, and I would also like to have an easily accessible reference, or references. Oswald Wyler