From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9091 Path: news.gmane.org!.POSTED!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Reference for lifting an adjunction to a monoidal one Date: Mon, 30 Jan 2017 13:37:03 +1100 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: 7bit X-Trace: blaine.gmane.org 1486155538 23721 195.159.176.226 (3 Feb 2017 20:58:58 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Fri, 3 Feb 2017 20:58:58 +0000 (UTC) To: David Roberts , "categories@mta.ca, list" Original-X-From: majordomo@mlist.mta.ca Fri Feb 03 21:58:54 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 1cZkwY-0005yF-OP for gsmc-categories@m.gmane.org; Fri, 03 Feb 2017 21:58:54 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:59683) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1cZkwC-00039f-MV; Fri, 03 Feb 2017 16:58:32 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1cZkvj-0000No-IS for categories-list@mlist.mta.ca; Fri, 03 Feb 2017 16:58:03 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9091 Archived-At: {Note from moderator: Apologies to Richard and Steve whose posts were inadvertently placed in the wrong folder...} 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 On Sat, Jan 28, 2017, at 12:08 PM, David Roberts wrote: > Hi all, > > I need a textbook or otherwise standard reference for the fact that if > one > has a pair of monoidal categories C, D, and an adjunction L -| R: UC <--> > UD between their underlying categories, then if one of L or R lift to a > (strong) monoidal functor, then the adjunction lifts to an adjunction in > the 2-category of monoidal categories, strong monoidal functors and > monoidal natural transformations. > > (Mac Lane of course only treats the case of strict monoidal functors, at > least in my, older, edition of his book) > > Thanks, > David [For admin and other information see: http://www.mta.ca/~cat-dist/ ]