From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5931 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: pullback of locally presentable categories Date: Wed, 16 Jun 2010 15:35:47 +0100 (BST) Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: dough.gmane.org 1276859275 1470 80.91.229.12 (18 Jun 2010 11:07:55 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Fri, 18 Jun 2010 11:07:55 +0000 (UTC) Cc: Gaucher Philippe , categories To: Steve Lack Original-X-From: categories@mta.ca Fri Jun 18 13:07:53 2010 connect(): No such file or directory 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 4.69) (envelope-from ) id 1OPZQP-0000xU-K4 for gsmc-categories@m.gmane.org; Fri, 18 Jun 2010 13:07:53 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OPYtm-0001mD-G1 for categories-list@mta.ca; Fri, 18 Jun 2010 07:34:10 -0300 In-Reply-To: Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5931 Archived-At: ---and I have further noticed that, although the first situation may encompass the other two, this is by the by, since the argument in this first situation is incorrect. I have only shown the vertex of the pullback to be accessible, but not necessarily locally presentable. I don't see any obvious way of rectifying this. However, the argument in the second and third situations is still valid. So to summarise:-- Given a cospan of locally presentable categories and accessible functors, - if one of the functors is an isofibration, then the pullback is accessible; - if one of the functors is monadic or comonadic, then the pullback is locally presentable. Richard --On 16 June 2010 13:57 Richard Garner wrote: > > I have just noticed that the first of the three situations I list below > actually encompasses the other two, since a monadic or comonadic functor is > necessarily an isofibration. > > > --On 16 June 2010 13:18 Richard Garner wrote: > >> >> Can I add to Steve's excellent summary of the situation the observation >> that there are circumstances under which the strict pullback of locally >> presentable categories will also be locally presentable. I know of three >> such. >> >> The first is when one of the functors being pulled back is an isofibration: >> that is, a functor admitting (necessarily cartesian) liftings of >> isomorphisms. In this case, it has been observed by Joyal and Street that >> the strict pullback also enjoys the universal property of the >> pseudopullback, and hence is locally presentable as in Steve's message. >> >> The second situation is when one of the functors in question is monadic. >> For then the vertex of the strict pullback can be constructed from >> inserters and equifiers in Acc, and hence is accessible; moreover, it is >> necessarily complete (since it projects onto a complete category via a >> limit-creating functor, namely the pullback of the monadic one), and hence >> locally presentable. >> >> The third situation is when one of the functors in question is comonadic: >> in which case the same argument pertains, but with completeness now >> substituted by cocompleteness. >> >> Note in particular that these last two circumstances include the situation >> of pulling back a reflective or coreflective subcategory. >> >> Richard >> >> --On 16 June 2010 10:51 Steve Lack wrote: >> >>> Dear Philippe, >>> >>> If you mean the literal pullback then no. But perhaps you mean the >>> pseudopullback or the iso-comma objects (where one askes for commutativity >>> of the square only up to isomorphism) in which case things look better. >>> >>> Greg Bird proved in his 1984 thesis that: >>> >>> (i) the 2-category of locally presentable categories, left adjoint >>> functors, >>> and natural transformations has all flexible limits; >>> >>> (ii) the 2-category of locally presentable categories, right adjoint >>> functors, and natural transformations has all flexible limits. >>> >>> These flexible limits include pseudopullbacks and iso-comma objects. They >>> also imply the existence of all bilimits (where everything is done up to >>> isomorphism of 1-cells, and the universal property involves just a >>> pseudonatural equivalence). In the thesis, flexible limits were called >>> "limits of retract type". >>> >>> Makkai and Pare proved in their monograph on accessible categories that: >>> >>> (iii) the 2-category of accessible categories, accessible functors, and >>> natural transformations has bilimits. >>> >>> Bilimits were there called "Limits" (capital L). >>> .... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]