From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6107 Path: news.gmane.org!not-for-mail From: Toby Bartels Newsgroups: gmane.science.mathematics.categories Subject: Re: Dual category and dual object Date: Sat, 4 Sep 2010 09:38:00 -0700 Message-ID: References: Reply-To: Toby Bartels NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1283646051 18603 80.91.229.12 (5 Sep 2010 00:20:51 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sun, 5 Sep 2010 00:20:51 +0000 (UTC) Cc: David Leduc To: categories Original-X-From: majordomo@mlist.mta.ca Sun Sep 05 02:20:50 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1Os2yX-00020T-KZ for gsmc-categories@m.gmane.org; Sun, 05 Sep 2010 02:20:49 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42865) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1Os2xP-0001Bu-NZ; Sat, 04 Sep 2010 21:19:39 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1Os2xN-0006du-CT for categories-list@mlist.mta.ca; Sat, 04 Sep 2010 21:19:37 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6107 Archived-At: David Leduc wrote: >Are the notions of dual category and dual object related? The concept of dual vector space (which may be the historical original) is an example of both, although in slightly different ways. A vector space is an object in Vect, and its dual is a dual object; taking a vector space to its dual extends to a contravariant functor, which is the same thing as a functor on a dual category. >If not, are there any good reasons to use the word "dual" for both notions? Perhaps we should say "adjoint object" instead of "dual object". Every monoidal category can be interpreted as a 2-category, and a dual object in the monoidal category generalises to an adjoint morphism in the 2-category. On the other hand, we can say "opposite category" instead of "dual category", especially since we denote the dual category of C by C^{op}. --Toby [For admin and other information see: http://www.mta.ca/~cat-dist/ ]