From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6307 Path: news.gmane.org!not-for-mail From: Ross Street Newsgroups: gmane.science.mathematics.categories Subject: Re: Cat as a '2-fibration' over Set Date: Sat, 9 Oct 2010 17:12:32 +1100 Message-ID: References: Reply-To: Ross Street NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 (Apple Message framework v936) Content-Type: text/plain; charset=ISO-8859-1; format=flowed; delsp=yes Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1286664315 14839 80.91.229.12 (9 Oct 2010 22:45:15 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 9 Oct 2010 22:45:15 +0000 (UTC) Cc: categories@mta.ca To: David Roberts Original-X-From: majordomo@mlist.mta.ca Sun Oct 10 00:45:14 2010 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpy.mta.ca ([138.73.1.139]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1P4iAA-0002YF-WF for gsmc-categories@m.gmane.org; Sun, 10 Oct 2010 00:45:11 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:44248) by smtpy.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P4i9R-0003Hl-1z; Sat, 09 Oct 2010 19:44:25 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P4i9N-000051-El for categories-list@mlist.mta.ca; Sat, 09 Oct 2010 19:44:21 -0300 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6307 Archived-At: 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 =20 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/ ]