From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3814 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Maps of monads - references Date: Tue, 10 Jul 2007 20:39:24 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019539 10403 80.91.229.2 (29 Apr 2009 15:38:59 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:38:59 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue Jul 10 20:05:02 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 10 Jul 2007 20:05:02 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1I8OfF-0001K5-36 for categories-list@mta.ca; Tue, 10 Jul 2007 19:58:37 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 8 Original-Lines: 35 Xref: news.gmane.org gmane.science.mathematics.categories:3814 Archived-At: See Ross Street "The formal theory of monads", JPAA 2 (1972) 149-168 for the general definition, in the abstract setting of a 2-category instead of Cat. Actually, there are two obvious generalizations of the TTT definition ("monad functors" and "monad opfunctors"), for the two possible directions of F. Steve Vickers. On 10 Jul 2007, at 14:25, Steven R. Costenoble wrote: > In Toposes, Triples, and Theories, Barr and Wells define a morphism > of triples (which, being a student of Peter May, I will call a map of > monads) in the context of two monads on a given category C. I have a > situation where I have two categories C and D, a monad S on C, a > monad T on D, and a functor F: C -> D. There is a fairly obvious > generalization of the TTT definition, to say that a map from S to T > is a natural transformation FS -> TF making certain diagrams commute. > My guess is that someone else noticed this long ago, so I'm looking > for references to where this has appeared in the literature. I'm > particularly interested in references that include the fact (at > least, I'm pretty sure it's a fact) that such maps are in one-to-one > correspondence with extensions of F to a functor between the > respective Kleisli categories of S and T. > > Thanks in advance. > > --Steve Costenoble > > >