From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/3521 Path: news.gmane.org!not-for-mail From: reinhard.boerger@FernUni-Hagen.de Newsgroups: gmane.science.mathematics.categories Subject: Re: A question about extensive categories Date: Thu, 14 Dec 2006 10:29:51 +0100 Message-ID: NNTP-Posting-Host: main.gmane.org Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7BIT X-Trace: ger.gmane.org 1241019356 9106 80.91.229.2 (29 Apr 2009 15:35:56 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:35:56 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Dec 14 20:58:45 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 14 Dec 2006 20:58:45 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Gv1J3-0003Ax-8v for categories-list@mta.ca; Thu, 14 Dec 2006 20:52:09 -0400 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 22 Original-Lines: 45 Xref: news.gmane.org gmane.science.mathematics.categories:3521 Archived-At: Dear colleagues on 29 Nov 2006 at 15:58 Jiri Adamek wrote: > Does anyone know whether every extensive and locally finitely > presentable > category fulfils the following condition: > > For every omega op-chain of coproduct injections i_n: A_n+1 -> A_n > with all A_n finitely presentable some i_n is an isomorphism. Here is the answer: Consider the category of pairs of sets X,Y together with an isomorphism s:XxY->Y, with inverse Y->XxY, Y|->(py,qy). Then X is locally finitely presentable, because it is a two-sorted variety with operations s,p,q and equations s(py,qy)=y=qs(x,y), ps(x,y)=x. One might think of the elements of Y as versions of sequences in X; an element y can be considered as a version of the sequence (pq^n(y))_n, where n runs over all natural numbers (including 0). Then p maps evey "sequence" to its initial (0-th) member, q means ommitting the initial member, an s(x,y) is obtained from y by adding the new initial member x. Then it is quite easy to see that the category is also extensive. The free algebra A=(N,S) on one element of Y can then be described as follows: N is the set of natural numbers and s is the set of all sequences (x(n))_n in N with the property that the exisit a natural number m and an integer k with x(n)=n+k for all n>m; p is the projection (x(n))_n|->x(0) to the 0-th component, and q is the left shift (x(n))_n|->(x(n+1))_n. The algebra A is freely generated by the sequence g:=(n)_n of natural numbers; every element of S can be obtained from g by first shifting left several times and then shifting right with inserting arbitary natural numbers in the initial component; every n in N can be obtained as n=pq^n(g). The free algebra on one element of X is ({0},0,...), where 0 is the empty set. Now the coproduct A+B is isomorphic to A with injections i:A->A, j:B->A, where i is the sucessor map n|->n+1 and j maps 0 to 0 in the first component; this uniquely determines the second components (because "different versions of the same sequence do not occur".) Now A_n:=n and i_n:=i for all n in N yields a counterexample to the above statement. Greetings Reinhard