From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/258 Path: news.gmane.org!not-for-mail From: F William Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Re: Category of categories with pullbacks is cartesian closed Date: Wed, 22 Apr 2009 10:29:43 -0400 Message-ID: Reply-To: F William Lawvere NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1240449832 2480 80.91.229.12 (23 Apr 2009 01:23:52 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 23 Apr 2009 01:23:52 +0000 (UTC) To: , "John Bourke" Original-X-From: categories@mta.ca Thu Apr 23 03:25:11 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 1Lwngd-0004ZP-2z for gsmc-categories@m.gmane.org; Thu, 23 Apr 2009 03:25:11 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Lwn4o-0005uA-Gf for categories-list@mta.ca; Wed, 22 Apr 2009 21:46:06 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:258 Archived-At: Hi John Interesting result. Especially the way in which the correct 2-structure follows from cartesian closure. An analogous result may be the (exension from=20 posetal domains to categories of) closure under=20 filtered colimits of various shapes. Possibly there is a common generalization to connected limits=20 or colimits ? Bill =20 On Thu 04/16/09 9:22 PM , John Bourke johnb@maths.usyd.edu.au sent: > Dear category theorists, > I noticed recently that the category whose objects are categories with > pullbacks and whose morphisms are pullback preserving functors is cartesi= an > closed. Given a pair of categories with pullbacks A and B, the internal > hom [A,B] has objects: pullback preserving functors from A to B, and > morphisms: cartesian natural transformations.I have posted a short paper = on the arxiv proving this fact: > http://arxiv.org/abs/0904.2486It seems like a fairly natural fact but is = not to my knowledge in the > literature. I am wondering whether anyone was previously aware of this > result, and if so whether it might be mentioned somewhere in the > literature?Thanks, > John Bourke, > University of Sydney. >=20 >=20 >=20 >=20 >=20 >=20