From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9782 Path: news.gmane.org!.POSTED!not-for-mail From: Clemens Berger Newsgroups: gmane.science.mathematics.categories Subject: Re: Lawvere theories and Monads Date: Mon, 24 Dec 2018 11:22:40 +0100 Message-ID: References: Reply-To: Clemens Berger NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8"; format=flowed Content-Transfer-Encoding: 8bit X-Trace: blaine.gmane.org 1545660919 19155 195.159.176.226 (24 Dec 2018 14:15:19 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Mon, 24 Dec 2018 14:15:19 +0000 (UTC) Cc: To: Jade Master Original-X-From: majordomo@mlist.mta.ca Mon Dec 24 15:15:14 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1gbR0k-0004u1-Ki for gsmc-categories@m.gmane.org; Mon, 24 Dec 2018 15:15:14 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:35955) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1gbR2d-0004WC-Py; Mon, 24 Dec 2018 10:17:11 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1gbR1o-0007MB-I4 for categories-list@mlist.mta.ca; Mon, 24 Dec 2018 10:16:20 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9782 Archived-At: Hi, the reference you are searching comes under the title "adjoint triangle theorem" which is due to Eduardo Dubuc (cf. the nLab entry). The left adjoint f_* is uniquely determined by the fact that M_T and M_T' are "nice" monads. The explicit (known) formulas for this left adjoint imply your observation. All the best, Clemens. Le 2018-12-22 18:45, Jade Master a ??crit??: > I have a question about the relationship between Lawvere theories and > monads. Every morphism of Lawvere theories f: T ->T' induces a morphism > of > monads M_f: M_T => M_T' which can be calculated by using the universal > property of the coend formula for M_T (this can be found in Hyland's > > paper > on Lawvere theories and monads). > > On the other hand f: T->T' gives a functor f* : Mod(T') -> Mod(T) given > by > composition with f. Because everything is nice enough, f* always has a > left > adjoint f_* : Mod(T) -> Mod(T') (details of this can be found here > or in Toposes, > Triples > and Theories). > > My question is the following: What relationship is there between the > adjunction > > f_* \dashv f*: Mod(T) ->Mod(T') > > and the morphism of monads computed using coends > > M_f : M_T => M_T'? > > In the examples I can think of the components of M_f are given by the > unit > of the adjunction f_* \dashv f* but I cannot find a reference > explaining > this. It doesn't seem to be in Toposes, Triples, and Theories. > > > Thank you, > Jade Master > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]