From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7370 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: Alternative closed structure on Cat Date: Fri, 6 Jul 2012 08:12:23 +1000 Message-ID: References: Reply-To: Ross Street NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 (Apple Message framework v1084) Content-Type: text/plain; charset=windows-1252 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1341578411 10255 80.91.229.3 (6 Jul 2012 12:40:11 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 6 Jul 2012 12:40:11 +0000 (UTC) To: Categories list Original-X-From: majordomo@mlist.mta.ca Fri Jul 06 14:40:10 2012 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.80]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Sn7pS-0005nj-4c for gsmc-categories@m.gmane.org; Fri, 06 Jul 2012 14:40:10 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:54322) by smtpx.mta.ca with esmtp (Exim 4.77) (envelope-from ) id 1Sn7oW-0006Tm-Lh; Fri, 06 Jul 2012 09:39:12 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Sn7oW-0004qd-FR for categories-list@mlist.mta.ca; Fri, 06 Jul 2012 09:39:12 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7370 Archived-At: On 05/07/2012, at 11:38 PM, Peter Selinger wrote: > Power and Robinson state in [1, Section 2] that the tensor you > describe is indeed part of a monoidal closed structure: the function > category has as objects all functors, and as morphisms the (not > necessarily natural) transformations. Moreover, Power and Robinson > state that this is the unique other symmetric monoidal closed > structure on Cat, i.e., there are no others besides this one and the > "usual" one. I have never seen a proof of this last fact. >=20 > [1] J. Power and E. Robinson. "Premonoidal categories and notions of > computation." Mathematical Structures in Computer Science 7(5): > 445-452, 1997. (www.eecs.qmul.ac.uk/~edmundr/pubs/mscs97/premoncat.ps) Yes, this is what I call the "funny" tensor product on Cat. Categories enriched in Cat with the funny tensor product are called "sesquicategories": they are less than 2-categories as they have whiskering but only ambiguous horizontal composition of 2-cells. There is a bit of literature on all this. For example, it is mentioned = in Categorical structures, Handbook of Algebra Volume 1 (editor M. = Hazewinkel; Elsevier Science, Amsterdam 1996; ISBN 0 444 82212 7) = 529-577. and/or Higher categories, strings, cubes and simplex equations, Applied = Categorical Structures 3 (1995) 29- 77 & 303; MR96b:18009. As to finding all the symmetric monoidal closed structures on a locally = finitely presentable category,=20 the object-in-two-categories technique is provided by F. Foltz, GM. Kelly and C. Lair, Algebraic categories with few monoidal = biclosed structures or none,=20 J. Pure and Applied Algebra 17 (1980) 171=96177. Perhaps they even give the Cat example. Best wishes, Ross= [For admin and other information see: http://www.mta.ca/~cat-dist/ ]