From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/10075 Path: news.gmane.org!.POSTED.blaine.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Re: Monoidal product functor is strong monoidal, when? Date: Mon, 16 Dec 2019 10:49:16 +1030 Message-ID: Reply-To: David Roberts Mime-Version: 1.0 Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Injection-Info: blaine.gmane.org; posting-host="blaine.gmane.org:195.159.176.226"; logging-data="120010"; mail-complaints-to="usenet@blaine.gmane.org" To: "categories@mta.ca list" Original-X-From: majordomo@rr.mta.ca Mon Dec 16 21:35:17 2019 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.55]) by blaine.gmane.org with esmtps (TLS1.2:ECDHE_RSA_AES_256_GCM_SHA384:256) (Exim 4.89) (envelope-from ) id 1igx5I-000Uzy-4J for gsmc-categories@m.gmane.org; Mon, 16 Dec 2019 21:35:16 +0100 Original-Received: from rr.mta.ca ([198.164.44.159]:56150) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1igx4O-0008EW-QX; Mon, 16 Dec 2019 16:34:20 -0400 Original-Received: from majordomo by rr.mta.ca with local (Exim 4.92.1) (envelope-from ) id 1igx3a-0002bJ-VE for categories-list@rr.mta.ca; Mon, 16 Dec 2019 16:33:30 -0400 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:10075 Archived-At: Hi all again, thanks to those who replied off-list. The canonical reference is Joyal=E2=80=93Street's Braided monoidal categories. (Someone else also poin= ted out that algebras for the E_2 operad are equivalent to E_1 algebras in the category of E_1 algebras.) However, my *actual* desired result is that the multiplication of a _symmetric_ monoidal category is a braided functor (i.e. commutes with the braiding=3Dsymmetry in this case). I proved this to my own satisfaction, but my proof is not very nice, and I'm searching for a cleaner verification of the required commuting diagram. Surely this was also known! And if so, what's a good reference (I expect it to be even earlier than Joyal=E2=80=93Street). Thanks, David PS this question relating to Lawvere's 2015 invited CT address might be of interest to people here: https://mathoverflow.net/questions/348436/the-barr-boole-galois-topos-a-mod= ification-of-sets-to-play-well-with-schemes David Roberts Webpage: https://ncatlab.org/nlab/show/David+Roberts Blog: https://thehighergeometer.wordpress.com On Wed, 11 Dec 2019 at 16:59, David Roberts wrot= e: > > Hi all, > > I have half convinced myself (without checking details) that if I have > a braided monoidal category (C,@), then the monoidal product @: C x C > --> C is strong monoidal. Is this true? What's a reference for this I > could point to? > > Thanks, > David > > David Roberts > Webpage: https://ncatlab.org/nlab/show/David+Roberts > Blog: https://thehighergeometer.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]