From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/2194 Path: news.gmane.org!not-for-mail From: "Prof. Peter Johnstone" Newsgroups: gmane.science.mathematics.categories Subject: Re: preservation of exponentials Date: Thu, 20 Feb 2003 16:59:59 +0000 (GMT) Message-ID: References: <200302181332.OAA06673@fb04209.mathematik.tu-darmstadt.de> NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII X-Trace: ger.gmane.org 1241018480 2950 80.91.229.2 (29 Apr 2009 15:21:20 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:21:20 +0000 (UTC) To: categories@mta.ca Original-X-From: rrosebru@mta.ca Thu Feb 20 13:07:36 2003 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 20 Feb 2003 13:07:36 -0400 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.10) id 18lu84-0003Fx-00 for categories-list@mta.ca; Thu, 20 Feb 2003 13:05:00 -0400 In-Reply-To: <200302181332.OAA06673@fb04209.mathematik.tu-darmstadt.de> X-Scanner: exiscan for exim4 (http://duncanthrax.net/exiscan/) *18lu3D-0007Dt-00*zX9VwTnLJ9s* Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 50 Original-Lines: 27 Xref: news.gmane.org gmane.science.mathematics.categories:2194 Archived-At: On Tue, 18 Feb 2003, Thomas Streicher wrote: > Recently when rereading an old paper I came across a passage insinuating > that every finite limit preserving full and faithful functor between toposes > does also preserve exponentials. > I am sceptical because I don't see any obvious reason for it. It is certainly > wrong for ccc's (a counterexample is the inclusion of open sets of reals into > powersets of reals). On the other hand Yoneda functors and direct image parts > of injective geom morphs do preserve exponentials. > So I was thinking of inverse image parts of connected geom.morph.'s. > Of course, \Delta : Set -> Psh(C) for a connected C does preserve exponentials. > What about Delta : Set -> Sh(X) for X connected but not locally connected, > e.g. take for X Cantor space with a focal point added? > If a full and faithful functor between ccc's has a left adjoint which preserves binary products, then it preserves exponentials (Elephant, A1.5.9(ii)). In the absence of a left adjoint, the result is not true in general: Set --> Sh(X) for X connected but not locally connected gives a counterexample, as you suggest, and so does the inclusion (continuous G-sets) --> (arbitrary G-sets) for a topological group G. Peter Johnstone