From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/256 Path: news.gmane.org!not-for-mail From: Urs Schreiber Newsgroups: gmane.science.mathematics.categories Subject: Re: pasting along an adjunction Date: Sat, 18 Apr 2009 15:42:54 +0200 Message-ID: Reply-To: Urs Schreiber NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1240081195 15790 80.91.229.12 (18 Apr 2009 18:59:55 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Sat, 18 Apr 2009 18:59:55 +0000 (UTC) To: Andrew Salch , categories@mta.ca Original-X-From: categories@mta.ca Sat Apr 18 21:01:14 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1LvFms-00052s-Jx for gsmc-categories@m.gmane.org; Sat, 18 Apr 2009 21:01:14 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1LvFGD-0006KW-9b for categories-list@mta.ca; Sat, 18 Apr 2009 15:27:29 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:256 Archived-At: On Fri, Apr 17, 2009 at 8:06 PM, Andrew Salch wrote: > In a recent paper of Connes and Consani, > they consider the following "pasting along > an adjunction": [...] > I would like to know if this "pasting along > an adjunction" is a special case of some > more general construction already known > to category theory, and if basic properties > of pasting along an adjunction have already > been worked out and written down somewhere. In section 2.3.1 of "Higher Topos Theory" http://arxiv.org/abs/math.CT/0608040 Jacob Lurie motivates the notion of "inner fibrations" and of Cartesian fibrations of (oo,1)-categories as a generalization of this "pasting" construction. "Pasting" along any bifunctor C^op x D --> Set is the same as having an inner fibration over the interval (which is an arbitrary functor for 1-categories), and the particular "pasting" that you mention, over hom_D(F(-),-) coming from a functor F : C \to D, gives a Cartesian fibration over the interval (top of p. 88, leading over to section 2.4). Best, Urs