From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9097 Path: news.gmane.org!.POSTED!not-for-mail From: "F. Lucatelli Nunes" Newsgroups: gmane.science.mathematics.categories Subject: Re: Reference for lifting an adjunction to a monoidal one Date: Sun, 5 Feb 2017 01:31:58 +0000 Message-ID: References: Reply-To: "F. Lucatelli Nunes" NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: blaine.gmane.org 1486317850 23341 195.159.176.226 (5 Feb 2017 18:04:10 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 5 Feb 2017 18:04:10 +0000 (UTC) To: David Roberts , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Sun Feb 05 19:04:05 2017 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 1caRAR-0005lj-AB for gsmc-categories@m.gmane.org; Sun, 05 Feb 2017 19:04:03 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:33015) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1caRAE-0003au-Ur; Sun, 05 Feb 2017 14:03:50 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1caR9l-0003ct-6Z for categories-list@mlist.mta.ca; Sun, 05 Feb 2017 14:03:21 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9097 Archived-At: Of course, there was a mistake in "Consider the forgetful 2-functor U: Lax-Alg\to X. Let U(f) be a *left adjoint* 1-cell. f is left adjoint if and only if f is a pseudomorphism (and not just a lax morphism)." Sorry. Best wishes 2017-02-04 0:36 GMT+00:00 F. Lucatelli Nunes : > Dear David Roberts, > > Sorry. I did not read the details of your statement. As Richard Garner > observed, it is incorrect. > > > Another way of stating the relevant result of Kelly is the following: > > "Consider the forgetful 2-functor U: Lax-Alg\to X. Let U(f) be a right > adjoint 1-cell. > f is left adjoint if and only if f is a pseudomorphism (and not just a lax > morphism)." > This means that there is a right adjoint g to f (if (U(f) is left adjoint > and f is a pseudomorphism) in Lax-Alg. > > To get an adjunction in Ps-Alg, you should, now, ask whether this lifted g > is also a pseudomorphism (which means to verify if the mate of the > structure of f is an isomorphism). > > In other words, in the context of strong monoidal functors, considering the > forgetful functor F: StrongMonoidal\to Cat, assume that f\dashv F(g) is an > adjunction in Cat. > g is right adjoint if and only if its mate is an isomorphism (that is to > say, the induced oplax structure in f is a strong structure: Beck Chevalley > Condition) > > > Anyways, "Doctrinal Adjunction" (Kelly) is what you are looking for. You > will probably find what you want about lifting of adjoints there. > I would also recommend "Two-Dimensional Monadicity" of John Bourke > (Advances in Mathematics) 2014. > > > Best Regards > > 2017-01-30 2:37 GMT+00:00 Richard Garner : > >> >> >> Dear David, >> >> I am sure you will get a few responses telling you that the result, as >> you state it, is not quite correct. What is correct is that, given an >> adjunction L -| R: UC <---> UD: >> >> a) endowments of L with oplax monoidal structure are in bijection, under >> the mates correspondence, with endowments of R with lax monoidal >> structure >> >> b) given endowments of L and R with lax monoidal structure, the unit and >> counit of the adjunction satisfy the conditions to be monoidal >> transformations if and only if the given lax constraint cells on L are >> inverse to the oplax constraint cells induced from R via a) >> >> whence: >> >> c) liftings of the adjunction L -| R to an adjunction in the 2-category >> of monoidal categories, lax monoidal functors and monoidal >> transformations are in bijective correspondence with endowments of L >> with strong monoidal structure >> >> There is a dual b') of b) giving the dual >> >> c') liftings of the adjunction L -| R to an adjunction in the 2-category >> of monoidal categories, oplax monoidal functors and monoidal >> transformations are in bijective correspondence with endowments of R >> with strong monoidal structure >> >> of c). All of this follows from the general considerations in Kelly >> "Doctrinal adjunction" SLNM 420, though it would be more perspicuous to >> prove it directly following Kelly's schema. >> >> Richard >> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]