From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/6298 Path: news.gmane.org!not-for-mail From: David Roberts Newsgroups: gmane.science.mathematics.categories Subject: Cat as a '2-fibration' over Set Date: Thu, 7 Oct 2010 19:48:09 +1030 Message-ID: Reply-To: David Roberts NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 X-Trace: dough.gmane.org 1286564197 9355 80.91.229.12 (8 Oct 2010 18:56:37 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 8 Oct 2010 18:56:37 +0000 (UTC) To: categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Fri Oct 08 20:56:33 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 1P4I7J-0001NW-9j for gsmc-categories@m.gmane.org; Fri, 08 Oct 2010 20:56:29 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:51374) by smtpx.mta.ca with esmtp (Exim 4.71) (envelope-from ) id 1P4I6X-00049e-61; Fri, 08 Oct 2010 15:55:41 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1P4I6U-0007dJ-1d for categories-list@mlist.mta.ca; Fri, 08 Oct 2010 15:55:38 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:6298 Archived-At: Hi all, 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. This can be easily seen by constructing, given a category C = (C_1 \rightrightarrows C_0) and a function f:A \to C_0, the set of arrows A^2 \times_{f,C_0^2}C_1 (the pullback of (s,t):C_1 \to C_0^2) of the category C[f]. The cartesian lift of f is then the canonical functor F:C[f]\to C. Now given another function g:A\to C_0 -- giving rise to G:C[g]\to C -- and a natural transformation F \Rightarrow G there is a canonical isomorphism C[f]\simeq C[g] over C. Thus if we think of Cat as a 2-category, there is something extra going on. For example, one gets a pseudofunctor Set \to 2Cat on choosing specified pullbacks to define C[f]. Has this phenomenon been studied before? (I would think so) Does this make Obj a fibration of 2-categories (see e.g. Hermida, or Bakovic)? Or is this a more 'classical' concept? More basically, where was this fact first pointed out? David [For admin and other information see: http://www.mta.ca/~cat-dist/ ]