From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2302 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: Can we construct free semi-lattice from free dist. lattice? Date: Tue, 27 May 2003 21:52:21 +0100 (BST) Message-ID: References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018560 3451 80.91.229.2 (29 Apr 2009 15:22:40 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:22:40 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Tue May 27 20:05:46 2003 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 27 May 2003 20:05:46 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 19KnUY-0000fi-00 for categories-list@mta.ca; Tue, 27 May 2003 20:04:26 -0300 In-Reply-To: X-Scanner: exiscan for exim4 (http://duncanthrax.net/exiscan/) *19KlQj-0003n7-Ib*t1T2DscuJf6* Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 52 Original-Lines: 26 Xref: news.gmane.org gmane.science.mathematics.categories:2302 Archived-At: On Tue, 27 May 2003, Christopher Townsend wrote: > Dear All, > I have a problem which I had thought very specialised, but actually can be > stated very generally: - > > I have a category C with finite limits, and so I also have a category > DLat(C) of distributive lattices which, lets say, has coequalizers. If free > distibrutive lattices can be constructed (i.e. if there exists F:C->DLat(C) > left adjoint to the forgetful functor) then do free semilattices exist? > I suspect the answer is no, for irritatingly trivial reasons. If C is a pointed category (i.e. has a zero object), then any internal distributive lattice in C has its top and bottom elements equal, and so is degenerate (i.e. isomorphic to the terminal object 1). Hence the free-distributive-lattice functor exists, and is the constant functor with value 1. But I'm sure there must be examples of pointed, finitely complete categories which don't have a free-semilattice functor (though I have to admit I don't have one at my fingertips). Peter Johnstone