From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5800 Path: news.gmane.org!not-for-mail From: Andre Joyal Newsgroups: gmane.science.mathematics.categories Subject: Re: bilax_monoidal_functors Date: Fri, 14 May 2010 21:05:41 -0400 Message-ID: References: Reply-To: Andre Joyal NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1273945284 12802 80.91.229.12 (15 May 2010 17:41:24 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 15 May 2010 17:41:24 +0000 (UTC) To: "Michael Shulman" , Original-X-From: categories@mta.ca Sat May 15 19:41:23 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 1ODLMZ-0005RX-2y for gsmc-categories@m.gmane.org; Sat, 15 May 2010 19:41:23 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ODKw7-0006FF-Mh for categories-list@mta.ca; Sat, 15 May 2010 14:14:04 -0300 X-MS-TNEF-Correlator: Thread-Index: AcrzaG0u0uNIVVc8TX6JnjWcRxIW2AAX1/M+ Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5800 Archived-At: Dear Michael, > Compact means small, finite, > bounded, inaccessible by directed joins, etc. and "rigid" means = "having few > automorphisms," and I don't see that there is anything very compact or > rigid about such categories. 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? =20 I guess that in the category of R-modules over a commutative ring R,=20 a module M has a (good) dual iff it is finitely generated projective iff the endo-functor functor Hom(M,-) preserves all colimits (M is *compact* in a strong sense).=20 The rigidity terminology may have something to do with Tanaka duality. If C is a rigid monoidal category, then any monoidal natural=20 transformation u:F-->G between (strong) monoidal functors C-->E (where E is a monoidal category) is invertible. I would prefer a different terminology for monoidal categories with = duals. What about "auto-dual monoidal category"? It as a bit like "autonomous" category. Best, Andr=E9 =20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]