From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8975 Path: news.gmane.org!.POSTED!not-for-mail From: "Fred E.J. Linton" Newsgroups: gmane.science.mathematics.categories Subject: Re: Half cartesian duoical categories Date: Fri, 07 Oct 2016 15:52:21 -0400 Message-ID: Reply-To: "Fred E.J. Linton" NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable X-Trace: blaine.gmane.org 1476034924 10731 195.159.176.226 (9 Oct 2016 17:42:04 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Sun, 9 Oct 2016 17:42:04 +0000 (UTC) To: David Yetter , "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Sun Oct 09 19:41:59 2016 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.7.28]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1btI6f-0000vD-SE for gsmc-categories@m.gmane.org; Sun, 09 Oct 2016 19:41:50 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:52923) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1btI66-0006ue-6H; Sun, 09 Oct 2016 14:41:14 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1btI5u-0001Vt-7T for categories-list@mlist.mta.ca; Sun, 09 Oct 2016 14:41:02 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8975 Archived-At: Fix a duoidal category as in David's scenario, below. Write 1 and I for the monoidal unit objects for x and #, respectively. If indeed > (A x B) # (C x D) ------> (A # B) x (C # D) = is to hold, then x and # must essentially coincide. Here's why: 1) I =3D I # I =3D (1 x I) # (1 x I) =3D (1 # I) x (1 # I) =3D 1 x 1 =3D = 1 ; whence 2) A # C =3D (A x 1) # (C x 1) =3D (A # 1) x (C # 1) =3D A x C . Or perhaps David meant to posit the more usual middle 4 interchange law : (A x B) # (C x D) ------> (A # C) x (B # D) ? Wouldn't surprise me. But there I'm no help, sorry. Cheers, -- Fred --- ------ Original Message ------ Received: Fri, 07 Oct 2016 02:50:44 PM EDT From: David Yetter To: "categories@mta.ca" Subject: categories: Half cartesian duoical categories > Is there already a name in the literature for the special instance of duoidal category in which one of the monoidal structures is cartesian? I= n particular the instance in which if # denotes the non-cartesian monoidal structure and x the cartesian, the lax middle-four interchange transforma= tion has components > = > = > (A x B) # (C x D) ------> (A # B) x (C # D) ? > = > = > It has come up in my current student's dissertation work. An existing = name = and citations to papers using this specific type of duoidal category woul= d be much appreciated. > = > = > Best Thoughts, > = > David Yetter > = > Professor of Mathematics > = > Kansas State University > = > = [For admin and other information see: http://www.mta.ca/~cat-dist/ ]