From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5495 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: Re: dagger not evil Date: Thu, 7 Jan 2010 19:45:59 -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 1262917951 930 80.91.229.12 (8 Jan 2010 02:32:31 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 8 Jan 2010 02:32:31 +0000 (UTC) To: "Toby Bartels" , Original-X-From: categories@mta.ca Fri Jan 08 03:32:23 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 1NT4eD-0002PI-8h for gsmc-categories@m.gmane.org; Fri, 08 Jan 2010 03:32:21 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NT4Ib-0003vZ-Pz for categories-list@mta.ca; Thu, 07 Jan 2010 22:10:01 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5495 Archived-At: Dear Toby You wrote: >While the obvious forgetful functor DCat -> Cat is wrong, >is there a right one? In particular, we have a functor >Cat -> Grpd that takes the lluf subcategory (LS) of invertible = morphisms >and the functor DCat -> Grpd that takes the LS of unitary morphisms; >is there a functor DCat -> Cat that completes a commutative triangle? I will try answer your question, but my answer is wonkish. First, a category can be regarded as a (simplicial) diagram of = groupoids. More precisely, every category C has a "Rezk nerve" RN(C) which is a simplicial object in the category of groupoids. By definition, we have RN(C)_n=3DIsoNat([n],C) for every non-negative integer n, where IsoNat([n],C) denotes the groupoid of natural isomorphisms in the category of functors [n]-->C. The nerve RN(C) was first introduced by Charles Rezk in http://arxiv.org/abs/math/9811037 The functor RN has very nice properties.=20 It embeds the category Cat in the category of simplicial groupoids Simp(Grpd).=20 The embedding respects (ie preserves and reflects) the equivalences defined on both sides, where a map of simplicial groupoids f:X-->Y =20 is defined to be an equivalence if it is an equivalence levelwise. It can be proved that RN is a right Quillen functor with respect to the natural model structure on Cat and with respect to the Reedy model structure on Simp(Grpd). A dagger category can also be regarded as a (dagger simplicial) diagram = of groupoids. More precisely, every dagger category C has a "unitary nerve" UN(C) which is a dagger simplicial object in the category of groupoids. By definition, we have UN(C)_n=3DUIsoNat([n],C) for every non-negative integer n, where UIsoNat([n],C) denotes the groupoid of unitary natural = isomorphisms in the category of functors [n]-->C. The functor UN embeds the category DCat in the category of dagger simplicial groupoids DSimp(Grpd).=20 The embedding respects the equivalences defined on both sides, where a map of dagger simplicial groupoids f:X-->Y =20 is defined to be an equivalence if it is an equivalence levelwise. You wrote: >Less rigorously but more concretely, can we start with Hilb+ >(the dagger category with all bounded linear maps as morphisms) >and systematically derive the class of short linear maps, >much as we can systematically derive the class of unitary maps? >Offhand, I don't see how to do this. I am afraid I dont have an answer to this question.=20 But I will think about the problem. Best, Andr=E9 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]