From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4564 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Equivalence of pseudo-limits Date: Thu, 11 Sep 2008 18:54:11 +0100 (BST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241020029 13870 80.91.229.2 (29 Apr 2009 15:47:09 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:47:09 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Fri Sep 12 11:42:09 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Sep 2008 11:42:09 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ke9kd-0005Ie-IM for categories-list@mta.ca; Fri, 12 Sep 2008 11:35:59 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 34 Original-Lines: 74 Xref: news.gmane.org gmane.science.mathematics.categories:4564 Archived-At: In the case singled out by Alex a rather direct proof can also be given. Let Psd denote the 2-category of 2-functors, pseudonatural transformations and modifications P -> Cat. The pseudolimit of a 2-functor F: P -> Cat may be identified with the hom-category Psd(1,F); and accordingly we have a pseudolimit 2-functor lim = Psd(1,-): Psd -> Cat, which sends pseudonatural equivalences F =~ G to equivalences of categories lim(F) =~ lim(G). The corresponding result for pseudolimits in other 2-categories now follows by the Cat-enriched Yoneda lemma. Richard --On 11 September 2008 12:27 Dominic Verity wrote: > 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 >> > >