From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8688 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: coherence for symmetric monoidal and (co)affine categories Date: Mon, 24 Aug 2015 11:11:19 +0800 Message-ID: References: Reply-To: John Baez NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: ger.gmane.org 1440616897 353 80.91.229.3 (26 Aug 2015 19:21:37 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 26 Aug 2015 19:21:37 +0000 (UTC) Cc: categories To: Paul Blain Levy Original-X-From: majordomo@mlist.mta.ca Wed Aug 26 21:21:25 2015 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.22]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1ZUgG8-0001uh-3s for gsmc-categories@m.gmane.org; Wed, 26 Aug 2015 21:21:20 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44738) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1ZUgFE-0001oc-8K; Wed, 26 Aug 2015 16:20:24 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1ZUgFF-0008V7-KU for categories-list@mlist.mta.ca; Wed, 26 Aug 2015 16:20:25 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8688 Archived-At: Hi - Paul Blain Levy writes: It seems to be folklore that > > (1) symm(C) is the free symmetric monoidal category on C > (2) coaff(C) is the free coaffine category (symmetric monoidal category > with initial unit) on C. > > [....] But for general C, where are these statements proved? I'll admit to having used the first statement without proof on page 14 here: http://arxiv.org/abs/q-alg/9702014 As I get older I feel worse about doing such things. Perhaps an efficient proof of both statements could be developed from here: Symmetric monoidal sketches Martin Hyland and John Power https://www.dpmms.cam.ac.uk/~martin/Research/Oldpapers/hp00.pdf Theorem 2.5 gives the free symmetric monoidal category on a "symmetric monoidal sketch". Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]