From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3817 Path: news.gmane.org!not-for-mail From: Newsgroups: gmane.science.mathematics.categories Subject: Re: Maps of monads - references Date: Tue, 10 Jul 2007 17:31:21 PDT Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019541 10427 80.91.229.2 (29 Apr 2009 15:39:01 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:39:01 +0000 (UTC) To: Original-X-From: rrosebru@mta.ca Tue Jul 10 22:28:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 10 Jul 2007 22:28:16 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1I8QuU-0000gI-TN for categories-list@mta.ca; Tue, 10 Jul 2007 22:22:30 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 11 Original-Lines: 72 Xref: news.gmane.org gmane.science.mathematics.categories:3817 Archived-At: There's also: The formal theory of monads II, R. Street and S. Lack,J. Pure Appl. Algebra 175 (1-3) (2002) 243-265; MR2003m:18007 (preprint available from the homepages of the authors). Street's work is about 2-categories of monads and many people needed the simply categorical level, because of the Linear Logic connection. So there are several printed versions of the restricted result, including one of our group, I believe, in Relating Categorical Semantics for Intuitionistic Linear Logic, (M. Maietti, P. Maneggia, V. de Paiva and E. Ritter) in Applied Categorical Structures, volume 13(1):1--36, 2005.=20 But given our application we prove it for comonads, lifting both to Eilenberg-Moore coalgebras and to co-Kleisli categories. Dr Valeria de Paiva PARC 3333 Coyote Hill Road Palo Alto, CA 94304 USA -----Original Message----- From: Steve Vickers [mailto:s.j.vickers@cs.bham.ac.uk]=20 Sent: Tuesday, July 10, 2007 12:39 PM To: categories@mta.ca Subject: categories: Re: Maps of monads - references 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=20 > 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=20 > 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=20 > for references to where this has appeared in the literature. I'm=20 > 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=20 > correspondence with extensions of F to a functor between the=20 > respective Kleisli categories of S and T. > > Thanks in advance. > > --Steve Costenoble > > >