From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5797 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: Re: "compact", "rigid", or "autonomous"? Date: Thu, 13 May 2010 10:37:58 -0700 Message-ID: Reply-To: John Baez NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 X-Trace: dough.gmane.org 1273843422 3446 80.91.229.12 (14 May 2010 13:23:42 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 14 May 2010 13:23:42 +0000 (UTC) Original-X-From: categories@mta.ca Fri May 14 15:23:40 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OCura-0004Sr-8Q for gsmc-categories@m.gmane.org; Fri, 14 May 2010 15:23:38 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OCuLU-0002QV-Qd for categories-list@mta.ca; Fri, 14 May 2010 09:50:28 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5797 Archived-At: Mike wrote: One reason I like "autonomous" to mean a symmetric monoidal category in > which all objects have duals is that the only alternative names I have heard > for such a thing convey misleading intuition to me. They are sometimes > called "compact closed" or (I think) "rigid" monoidal categories... Yes, I think "rigid" is traditional in algebraic geometry. Perhaps some wiser head could explain how it originated! Personally I often use "compact', since "compact closed" seems redudant. And if I were king of the world, I'd use "with duals for objects". Regarding the relation of this use of "compact" to the use in topology: The only relationship I can think of is that a > compact subset of a Hausdorff space is closed, and a symmetric monoidal > category with duals for objects is also automatically closed, but of course > these two meanings of "closed" are totally different. Perhaps someone > can enlighten me? > I don't know if this is what people were thinking when they first applied "compact" to categories, or just my own rationalization, but: A compact subset is closed, but it has a very nice property: its image under any continuous map is again closed. Similarly a compact category is a closed, but it has a very nice property: its essential image under any symmetric monoidal functor is again compact. I don't claim this justifies the terminology, but it helped me learn to live with it. Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]