From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7924 Path: news.gmane.org!not-for-mail From: Peterson Clayton Newsgroups: gmane.science.mathematics.categories,gmane.spam.detected Subject: Questions on compact closed categories Date: Tue, 19 Nov 2013 13:17:24 +0000 Message-ID: Reply-To: Peterson Clayton NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1384889654 27595 80.91.229.3 (19 Nov 2013 19:34:14 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Tue, 19 Nov 2013 19:34:14 +0000 (UTC) To: "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Tue Nov 19 20:34:19 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.186]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Vir3z-0005Gn-L2 for gsmc-categories@m.gmane.org; Tue, 19 Nov 2013 20:34:19 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:41035) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1Vir3F-00014M-Nn; Tue, 19 Nov 2013 15:33:33 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Vir3E-00059n-Ar for categories-list@mlist.mta.ca; Tue, 19 Nov 2013 15:33:32 -0400 Accept-Language: fr-CA, en-US Content-Language: fr-FR Precedence: bulk X-Spam-Report: 6.2 points; * 1.9 DATE_IN_PAST_06_12 Date: is 6 to 12 hours before Received: date * 2.5 LOCALPART_IN_SUBJECT Local part of To: address appears in Subject * 1.8 MIME_QP_LONG_LINE RAW: Quoted-printable line longer than 76 chars Xref: news.gmane.org gmane.science.mathematics.categories:7924 gmane.spam.detected:5116287 Archived-At: Dear list members, I am currently working in categorical logic with something that might be ca= lled a "compact closed deductive system", that is, a deductive system (in t= he sense of Lambek) defined as a compact closed category (i.e., a *-autonom= ous category where the tensor unit is a dualizing object). I have two questions. First, it appeared to me that we can show in a compact closed deductive sys= tem that every arrow is an isomorphism. Hence, if there is a deduction arro= w from A to B, then A is isomorphic (logically equivalent) to B. Is this re= sult accurate? Does this generalize to any compact closed category? Secondly, I wonder what happens if we add an arbitrary arrow A --> B to the= category. Put differently, what happens if we add A --> B as an axiom to a= compact closed deductive system? Does this also yield an isomorphism betwe= en A and B (assuming that the first result is adequate)? Or is it possible = to add some axioms that are not necessarily isomorphisms? I hope my question is clear, and if not I would be happy to clarify myself,= so do not hesitate to contact me.=20 Any lead will be appreciated. Thanks in advance for those who will respond. Yours, Clayton Peterson Ph. D. candidate Universit=E9 de Montr=E9al clayton.peterson@umontreal.ca [For admin and other information see: http://www.mta.ca/~cat-dist/ ]