From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/9735 Path: news.gmane.org!.POSTED!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: characterization of flp endofucntors on Set? Date: Tue, 23 Oct 2018 11:01:58 +1100 Message-ID: References: <8064398f1cab41cca1233e5714fbf6dc@ME2PR01MB2756.ausprd01.prod.outlook.com> <1540252689.3296447.1551069960.4BC0C607@webmail.messagingengine.com> Reply-To: Richard Garner NNTP-Posting-Host: blaine.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: 7bit X-Trace: blaine.gmane.org 1540307963 18549 195.159.176.226 (23 Oct 2018 15:19:23 GMT) X-Complaints-To: usenet@blaine.gmane.org NNTP-Posting-Date: Tue, 23 Oct 2018 15:19:23 +0000 (UTC) To: Thomas Streicher , categories@mta.ca Original-X-From: majordomo@mlist.mta.ca Tue Oct 23 17:19:19 2018 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp2.mta.ca ([198.164.44.40]) by blaine.gmane.org with esmtp (Exim 4.84_2) (envelope-from ) id 1gEySk-0004iv-PH for gsmc-categories@m.gmane.org; Tue, 23 Oct 2018 17:19:18 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:57497) by smtp2.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1gEyUo-0004MC-6z; Tue, 23 Oct 2018 12:21:26 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1gEyTi-00048w-Tn for categories-list@mlist.mta.ca; Tue, 23 Oct 2018 12:20:18 -0300 In-Reply-To: <1540252689.3296447.1551069960.4BC0C607@webmail.messagingengine.com> Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:9735 Archived-At: To be clear: by "subdirect product" I mean "reduced product by a filter" and in this case all the sets are the same so I suppose I should really say "reduced power". On Tue, Oct 23, 2018, at 10:58 AM, Richard Garner wrote: > Dear Thomas, > > I don't know a complete answer to your question - I think there is > probably not an entirely elementary characterisation. The partial > answers I am aware of (which probably you are aware of too) have a lot > to do with filters.> > - Trnkova classifies in "On descriptive classification of set-functors > I" different kinds of limit-preserving endofunctor of Set. In > particular, she shows that an endofunctor of set preserves finite > limits if and only if it preserves finite products and is not the > reflector of Set into 0<=1.> > - Blass in "Exact functors and measurable cardinals" shows that any > finite-limit preserving endofunctor of Set is a directed union of > subdirect products. Of course, this is not so far away from the "for- > free" characterisation of the flp endofunctors of Set as ind-objects > in Set^op.> > - Rather trivially, an flp and finitary endofunctor of Set must be of > the form FA = { continuous maps X ---> disc(A) of finite support } > for some Stone space X, because flp finitary endofunctors of Set are > the same as flp functors Set_f-->Set, and these are all of the form > Stone(X,-):Set_f-->Set. I don't how to extend this even to > endofunctors of rank aleph_1.> > My knowledge of the literature in this area is patchy, so I am also > interested to see what other answers you might receive!> > Richard > > [For admin and other information see: http://www.mta.ca/~cat-dist/ ]