From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/275 Path: news.gmane.org!not-for-mail From: categories Newsgroups: gmane.science.mathematics.categories Subject: Re: question on finiteness in toposes Date: Sat, 11 Jan 1997 13:14:47 -0400 (AST) Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241016864 24998 80.91.229.2 (29 Apr 2009 14:54:24 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 14:54:24 +0000 (UTC) To: categories Original-X-From: cat-dist Sat Jan 11 13:15:57 1997 Original-Received: by mailserv.mta.ca; id AA05824; Sat, 11 Jan 1997 13:14:47 -0400 Original-Lines: 35 Xref: news.gmane.org gmane.science.mathematics.categories:275 Archived-At: Date: Sat, 11 Jan 1997 08:38:43 -0500 (EST) From: Peter Freyd The answer that first occures me for Thomas Streicher's question is that in Set^2_kf the terminator generates, hence if it were a topos it would have to be a boolean topos. Which it clearly isn't. Thomas wrote: One knows that for any topos E that the full subcategory of decidable K-finite objects forms a topos itself with 2 = 1+1 as subobject classifier. It is also said that E_kf, the full subcat of E on K-finite objects need not form a topos. That's what I could find out from PTJ's Topos Theory. The counterexample given there is E = Set^2 (where 2 = 0 -> 1). The K-finite objects in Set^2 are the surjective maps between finite sets. It is clear that E_kf is not closed under equalisers taken in E (!). Nevertheless, I think that E_kf itself does have equalisers : if f,g : X -> Y then take the equaliser e_0 of f_0 and g_0 and take the epi-mono-factorisation of x o e_0 : E_0 >---> X_0 | | | epi | x where X = X_0 -> X_1 V V E_1 >---> X_1 this clearly demonstrates that the inclusion E_kf >--> E does not preserve equalisers BUT it does not show that E_kf is not a topos. I would be interested in a reference or example where E_kf really is not a topos. Maybe, E = Set^2 alraedy works but it must have another defect than not being clossed under subobjects w.r.t. E because the decidable K-finite objects have this "defect" as well. Thomas Streicher