From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4561 Path: news.gmane.org!not-for-mail From: "Dominic Verity" Newsgroups: gmane.science.mathematics.categories Subject: Re: Equivalence of pseudo-limits Date: Thu, 11 Sep 2008 12:27:33 +1000 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241020027 13859 80.91.229.2 (29 Apr 2009 15:47:07 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:07 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Sep 11 14:33:08 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Sep 2008 14:33:08 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Kdpw0-0002YJ-Gp for categories-list@mta.ca; Thu, 11 Sep 2008 14:26:24 -0300 Content-Disposition: inline Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 31 Original-Lines: 55 Xref: news.gmane.org gmane.science.mathematics.categories:4561 Archived-At: Hi Alex, This property is certainly known to hold for a much larger class of 2-categorical limits - the flexible limits, which class includes the classes of pseudo, lax and op-lax limits. I believe you will find a proof of this result in the Bird, Kelly, Power and Street paper entitled "Flexible Limits for 2-Categories". Failing that the Power and Robinson paper "A characterisation of PIE limits" probably contains this result is some form. You can also find explications of the pseudo and lax results in earlier works by Street, although obvious candidates for the best one to consult elude me at the moment. The most elementary proof of the flexible limit result starts by observing that all flexible limits can be constructed using products, splittings of idempotents and a couple of less familiar, exclusively 2-categorical, limits called inserters and equifiers. It is then straight forward to verify the result you mention for each of these particular limits and then to infer that it must therefore hold for all flexible limits. In the early 1990's Robert Pare introduced a class of limits called the persistent limits. These were defined for 2-categories, but made use of his double categorical approach to 2-limits. Persistent limits are precisely those limits which have the stability property you seek, but with regard to a slightly more general class of double categorical diagram transformations whose 1-cellular components are all equivalences. In my thesis (1992), I prove that the class of flexible limits introduced by Bird, Kelly, Power and Street is identical to Pare's class of persistent limits - thus closing the circle and demonstrating that the flexible limits are in a natural sense the largest class of 2-limits which have this property. Regards Dominic Verity 2008/9/10 Alex Hoffnung > Hi all, > > Given an indexing 2-category J, a pair of parallel functors > F,G : J ----> CAT, and a natural equivalence f : F ==> G, > the pseudo-limits of F and G should be equivalent. > > I am trying to find out what paper, if any, I can cite for this theorem. > Or > maybe this is just the type of thing that nobody has bothered to write > down. > Any help would be appreciated. > > best, > Alex Hoffnung >