From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5477 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 Date: Tue, 5 Jan 2010 15:04:26 -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 1262743180 31057 80.91.229.12 (6 Jan 2010 01:59:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 6 Jan 2010 01:59:40 +0000 (UTC) To: "Peter Selinger" , Original-X-From: categories@mta.ca Wed Jan 06 02:59:32 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from [138.73.1.1] (helo=mailserv.mta.ca) by lo.gmane.org with esmtp (Exim 4.50) id 1NSLBL-0007A4-28 for gsmc-categories@m.gmane.org; Wed, 06 Jan 2010 02:59:31 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NSKnt-0007hY-PW for categories-list@mta.ca; Tue, 05 Jan 2010 21:35:17 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5477 Archived-At: 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). The forgetful functor DCat ---> Cat is a right adjoint but it is not a right Quillen functor with respect to the natural model structures on these categories. In other words the forgetful functor DCat ---> Cat is wrong. This may explains why a dagger category cannot be regarded as a category equipped a homotopy invariant structure. But I claim that the notion of dagger category is perfectly reasonable from an homotopy theoretic point of view. This is because the model category DCat is combinatorial. It follows, by a general result, that the notion of of dagger category is homotopy essentially algebraic There a homotopy limit sketch whose category of models (in spaces) is Quillen equivalent to the model category DCat. This is true also for the model category Cat. There should be a notion of dagger quasi-category. A dagger simplicial set can be defined to be a simplicial set X equipped with an involutive isomorphism dag:X-->X^o =20 which is the identity on 0-cells. The category of dagger simplicial sets (and dagger preserving maps) is the category of presheaves on the category whose objects are the = ordinals [n] but where the maps [m]-->[n] are order reversing or preserving. Finally, the (homotopy) trace of a category (resp. quasi-category) has the structure of a cyclic set in the sense of Connes. I conjecture that the (homotopy) trace of a dagger category (resp. = dagger quasi-category) has the structure of a dihedral set in the sense of Fiedorowicz and Loday. Happy New Year to all! andr=E9 PS: I will be quiet during the next few weeks. -------- Message d'origine-------- De: categories@mta.ca de la part de Peter Selinger Date: dim. 03/01/2010 02:23 =C0: Categories List Objet : categories: the definition of "evil" =20 Dear all, sorry for sending yet another message on the topic of "evil" structures on categories. After some interesting private replies, as well as Dusko's latest message (which should have appeared on the list by the time you read this), I noticed that not everyone is agreeing on the technical meaning of the term "evil". I will therefore attempt to state a more precise technical definition of the term as I have used it. Perhaps 2-category theorists already have another name for this. The information definition I had used is that a structure is "evil" if it does not "transport along equivalences of categories". I thought it was reasonably obvious what was meant by "transport along", but there is actually a lot of variation in what people understand this phrase to mean. John Baez gave a pointer to a website containing a technical definition of "evil": http://ncatlab.org/nlab/show/evil. Unfortunately, this site only speaks of properties, not structures. It is easy to state what it means for a property of categories to be transported along equivalences: namely, if C has the property, and C and C' are equivalent, then C' has the property. Structures are more tricky. .... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]