From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5487 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: dagger not evil (2) Date: Wed, 6 Jan 2010 10:44:08 -0500 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= 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: ger.gmane.org 1262833979 32170 80.91.229.12 (7 Jan 2010 03:12:59 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 7 Jan 2010 03:12:59 +0000 (UTC) To: "Peter Selinger" , "Categories List" Original-X-From: categories@mta.ca Thu Jan 07 04:12:49 2010 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 4.50) id 1NSino-0006LW-Pu for gsmc-categories@m.gmane.org; Thu, 07 Jan 2010 04:12:49 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NSiLD-0000aK-Db for categories-list@mta.ca; Wed, 06 Jan 2010 22:43:15 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5487 Archived-At: Dear Peter and all, In my last message, I wrote that the forgetful functor DCat ---> Cat=20 is wrong because it is not a right Quillen functor.=20 The argument is not good enough. I believe that a forgetful functor XStruc--->Cat should reflect=20 weak equivalences in addition to preserving them. =20 The forgetful functor DCat ---> Cat preserves weak equivalences but it does not reflect them.=20 Because two objects in a dagger category can be isomorphic without been unitary isomorphic. Best, aj -------- Message d'origine-------- De: Joyal, Andr=E9 Date: mar. 05/01/2010 15:04 =C0: Peter Selinger; Categories List Objet : dagger not evil =20 Dear Peter and all, I cannot resist adding my grain of salt to the ongoing=20 discussion on dagger categories. I will take the point of view of a homotopy theorist. Recall that the category of small categories Cat admits a "natural" model structure (called the "folk" model structure for the wrong reason by the folks).=20 The category of small dagger categories DCat also admits=20 a "natural" model structure. A dagger functor f:A-->B is a weak equivalence iff it is fully faithful and=20 unitary surjective (this last condition means that every object of B is unitary isomorphic to an object in the image of the functor f). The cofibrations and the trivial fibrations are as in Cat. A fibrations is a unitary isofibration (a map having the lifting property for unitary isomorphisms). .... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]