From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6316 Path: news.gmane.org!not-for-mail From: Ronnie Newsgroups: gmane.science.mathematics.categories Subject: Re: Cat as a '2-fibration' over Set Date: Wed, 13 Oct 2010 10:01:24 +0100 Message-ID: Reply-To: Ronnie NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1286965301 30708 80.91.229.12 (13 Oct 2010 10:21:41 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Wed, 13 Oct 2010 10:21:41 +0000 (UTC) To: "categories@mta.ca" Original-X-From: majordomo@mlist.mta.ca Wed Oct 13 12:21:38 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 1P5ySk-000461-0x for gsmc-categories@m.gmane.org; Wed, 13 Oct 2010 12:21:34 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:36972) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P5yRj-00005Q-7P; Wed, 13 Oct 2010 07:20:31 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P5yRg-00032G-39 for categories-list@mlist.mta.ca; Wed, 13 Oct 2010 07:20:28 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6316 Archived-At: Dear All, There is a strong emphasis on cocartesian morphisms of groupoids in the=20 various editions of `Topology and Groupoids' (but called there=20 `universal morphisms'), following Philip Higgins' paper 1964 paper on=20 groupoids. Philip's work on this idea was nicely jacked up by him to=20 induced morphisms of crossed modules, i.e. using the cofibration XMod=20 \to Group(oid)s giving the base group (or groupoid!), with a good=20 application to 2nd relative homotopy groups, and this appeared in our=20 papers. In higher dimensions, you get the Relative Hurewicz Theorem this=20 way. One of the applications to groupoids I like is that the cocartesian=20 morphism from the groupoid I (indiscrete on 0,1) over the identification {0,1} \to {0} gives I \to Z=3D integers. This seems a enough good reason=20 why the fundamental group of the circle is the integers! Doing this for categories instead of groupoids gives of course `2' \to N=20 is cocartesian over the same identification, which thus gives another=20 formulation of induction! In the new book `Nonabelian algebraic topology' (in final stages,=20 downloadable from my web page, and final comments welcome) we emphasise=20 the fibrations and cofibrations of categories approach. Ross asks about the Beck-B=E9nabou-Roubaud-Chevalley condition: I would=20 like to know of applications to the matters considered in these two books= ! Ronnie On 09/10/2010 07:12, Ross Street wrote: > On 07/10/2010, at 8:18 PM, David Roberts wrote: > >> To start with think of Cat as a 1-category. The functor Obj:Cat \to >> Set sending a small category to its set of objects is a fibration. > > > Dear David > > In a daring version of an undergraduate algebra unit on groups, > I taught the notions of cartesian and opcartesian morphism > for a functor and looked at them for the functor ob : Cat --> Set. > The goal was to give a groupoid proof of the Nielsen-Schreier > theorem using fibrations in the small (between groupoids) and in the > large. I achieved the goal to my own satisfaction; I think most of > the students thought otherwise. A core of them liked it. This is > the most explicit category theory I have tried to teach pre fourth > year honours. > > My inspiration very definitely came from Ronnie Brown's topology book(s= ). > > I'm not at work today (Saturday, and a grandson's birthday party) > so I can't check whether these constructions of direct and inverse > images for ob : Cat --> Set are in that book, whether it is the > ob : Gpd --> Set case that is there, or what. Ronnie can tell us > perhaps. Anyway, it is essentially there. It may not be phrased in > terms of cartesian morphisms. > >> Has this phenomenon been studied before? (I would think so) >> Does this make Obj a fibration of 2-categories (see e.g. Hermida, or=20 >> Bakovic)? >> Or is this a more 'classical' concept? More basically, where was this >> fact first pointed out? > > I too would like to know of other references. > > I am ashamed to say I hadn't thought about the 2-fibrational aspects > of ob : Cat --> Set. > > Also, how about the Beck-B=E9nabou-Roubaud-Chevalley condition? > > Ross > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]