From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4375 Path: news.gmane.org!not-for-mail From: Steve Vickers Newsgroups: gmane.science.mathematics.categories Subject: Re: Finite categories and filtered colimits, Date: Wed, 23 Apr 2008 14:28:16 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 (Apple Message framework v753) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1241019905 13009 80.91.229.2 (29 Apr 2009 15:45:05 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:45:05 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Wed Apr 23 18:15:02 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 23 Apr 2008 18:15:02 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JomCn-0004Su-BD for categories-list@mta.ca; Wed, 23 Apr 2008 18:08:41 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 24 Original-Lines: 59 Xref: news.gmane.org gmane.science.mathematics.categories:4375 Archived-At: Dear Francois, I don't know how relevant this is to your thinking, but I thought I'd =20= mention the result is constructively false. For an example, let C be the poset with two elements b <=3D t. The only =20= idempotents are the identities on b and t, so of course they split. Let p be a truth value, and let I =3D {b} u {t | p}, an ideal (hence =20 directed) in C. If this has a colimit, then it must be either b or t. =20= The colimit is b iff not p (so I =3D {b}), and it follows that if the =20= colimit is t we have not not p. Hence existence of the colimit gives =20 (not p or not not p) for every p, which is not intuitionistically valid. In fact one way to regard the set of truth values (i.e. the subobject =20= classifier) is as the ideal completion of C. Regards, Steve. On 21 Apr 2008, at 20:04, lamarche wrote: > Fellow category theorists, > > I'm looking for a ref for the following result: > > Let C be a finite category. Then TFAE > > -- C has colimits for all small filtered (well, directed is =20 > probably better) diagrams. > > -- Idempotents split in C. > > > This doesn't seem to be in Makkai-Par=E9 or Adamek-Rosicky, but =20 > surely somebody must have observed this. Actually the result is a =20 > bit more general since idempotents split in *any* category that has =20= > filtered colimits. > > > Thanks in advance, > > Fran=E7ois Lamarche > > > > > >