From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4389 Path: news.gmane.org!not-for-mail From: Michael Barr Newsgroups: gmane.science.mathematics.categories Subject: Re: Further to my question on adjoints Date: Mon, 12 May 2008 11:51:53 -0400 (EDT) Message-ID: References: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed X-Trace: ger.gmane.org 1241019915 13080 80.91.229.2 (29 Apr 2009 15:45:15 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:45:15 +0000 (UTC) To: Categories list Original-X-From: rrosebru@mta.ca Tue May 13 14:15:07 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 13 May 2008 14:15:07 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JvxqY-0005s1-1p for categories-list@mta.ca; Tue, 13 May 2008 13:59:26 -0300 In-Reply-To: X-PMX-Version: 5.4.0.320885, Antispam-Engine: 2.5.2.313940, Antispam-Data: 2008.4.7.44415 X-McGill-WhereFrom: Internal X-PMX-Version: 5.4.2.338381, Antispam-Engine: 2.6.0.325393, Antispam-Data: 2008.5.12.83751 X-PerlMx-Spam: Gauge=IIIIIII, Probability=7%, Report='BODY_SIZE_2000_2999 0, BODY_SIZE_5000_LESS 0, __BOUNCE_CHALLENGE_SUBJ 0, __CT 0, __CT_TEXT_PLAIN 0, __HAS_MSGID 0, __MIME_TEXT_ONLY 0, __MIME_VERSION 0, __SANE_MSGID 0' Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 12 Original-Lines: 50 Xref: news.gmane.org gmane.science.mathematics.categories:4389 Archived-At: No. For example, in the category of topological abelian groups, Z is far from injective. Nonetheless, if you say that a group is Z-compact when it is an equalizer of two maps between powers of Z, then an equalizer of two maps between Z-compact abelian groups is again Z-compact. The proof is not direct. As it happens, I am talking on this in our seminar tomorrow. Even though the reals are not injective in hausdorff spaces, a space is realcompact iff it is a closed subspace of a power of R, which turns out to be equivalent to being an equalizer of two maps between powers of R (that is a cokernel pair of such a closed inclusion has enough real-valued functions to separate points) and it is clear that a closed subspace of a realcompact space is again realcompact. Same thing for N-compact. In fact, for every example I have looked at sufficiently closely. Michael On Mon, 12 May 2008, Eduardo Dubuc wrote: > Consider the dual finitary question: In universal algebra in order to show > that finitely presented objects are closed under coequalizers it is > essential that a amorphism of finitely presented objects lift to a > morphism between the free. Is this the only way to prove it ? : > > " but when I look at examples, it has turned out to be true > for other reasons." > > greetings e.d. > > >> >> In March I asked a question on adjoints, to which I have received no >> correct response. Rather than ask it again, I will pose what seems to be >> a simpler and maybe more manageable question. Suppose C is a complete >> category and E is an object. Form the full subcategory of C whose objects >> are equalizers of two arrows between powers of E. Is that category closed >> in C under equalizers? (Not, to be clear, the somewhat different question >> whether it is internally complete.) >> >> In that form, it seems almost impossible to believe that it is, but it is >> surprisingly hard to find an example. When E is injective, the result is >> relatively easy, but when I look at examples, it has turned out to be true >> for other reasons. Probably there is someone out there who already knows >> an example. >> >> Michael >> >> >