From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6264 Path: news.gmane.org!not-for-mail From: Thomas Streicher Newsgroups: gmane.science.mathematics.categories Subject: Re: Not invariant but good Date: Thu, 30 Sep 2010 13:11:21 +0200 Message-ID: <20100930111121.GA25969@mathematik.tu-darmstadt.de> References: Reply-To: Thomas Streicher NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii X-Trace: dough.gmane.org 1285898773 8408 80.91.229.12 (1 Oct 2010 02:06:13 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 1 Oct 2010 02:06:13 +0000 (UTC) Cc: categories@mta.ca To: Michael Shulman Original-X-From: majordomo@mlist.mta.ca Fri Oct 01 04:06:09 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.138]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P1V0i-0008Ca-0x for gsmc-categories@m.gmane.org; Fri, 01 Oct 2010 04:06:08 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:36469) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P1Uzt-00068H-Sc; Thu, 30 Sep 2010 23:05:17 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P1Uzq-0001ly-S6 for categories-list@mlist.mta.ca; Thu, 30 Sep 2010 23:05:15 -0300 Content-Disposition: inline In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6264 Archived-At: Dear Mike, > I'm not sure what you mean here. The notion of fibration in a > 2-category can be defined as a property if you like: a morphism E --> > B in a 2-category K is a fibration if all the induced functors K(X,E) > --> K(X,B) are fibrations and all commutative squares induced by > morphisms X --> X' are morphisms of fibrations (preserve cartesian > arrows). This is equivalent to giving some structure on E --> B, but > that structure is unique up to unique isomorphism when it exists. The definition you give entails that a "generalised" fibration is actually a Grothendieck fibration (since Cat(1,E) is isomorphic to E). This way you don't get closure under precomposition by equivalences. I also don't see why cartesiannness of the functors induced by X --> X' should amount to a choice of structure (cartesiannness of a functor is a property and not additional structure). Moreover, this requirement is a property of Grothendieck fibrations which can be established when having strong choice available. I was rather alluding to the notion of fibration in 2-cats as can be found in part B of the Elephant where a fibration is defined as a 1-arrow together with additional structure. The definition you gave above (which is not more general) is the obvious thing to do in case K is not wellpointed enough (as Cat is). Moreover, your definition of fibration in a 2-category is based on Grothendieck fibrations and thus employs equality of 1-cells. Since 1-cells are objects of a category it should be "evil" to speak about their equality. Not that it were a problem to me... Thomas PS Your definition of fibration in a 2-cat looks much simpler than what I could find in the papers by Street and Johnstone. That's nothing to complain about but where is it from? It seems to me the appropriate one when generalising from Cat to more general 2-cats. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]