From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4390 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: Re: Further to my question on adjoints Date: Mon, 12 May 2008 20:42:29 +0200 Message-ID: NNTP-Posting-Host: main.gmane.org Mime-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Trace: ger.gmane.org 1241019915 13082 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 14:15:08 2008 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 13 May 2008 14:15:08 -0300 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1JvxtW-000693-6n for categories-list@mta.ca; Tue, 13 May 2008 14:02:30 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 13 Original-Lines: 64 Xref: news.gmane.org gmane.science.mathematics.categories:4390 Archived-At: Dear Michael, Let C be the category of commutative rings (with 1), let t be the unique positive real number with tttt = 2 (if I knew TeX better, I would probably write t^4 = 2), and E be the smallest subfield in the field of real numbers that contains t. Then: (a) Every power of E has exactly one element x such that xx = 2 and there exists y with x = yy. Let us call this x the positive square root of 2. (b) Every morphism between powers of E preserves the positive square root of 2. (c) Therefore every equalizer of two arrows between powers of E has an element x with xx = 2 (note that I am not saying anything about the existence of y, since y above is not determined uniquely!). (d) Therefore the field Q of rational numbers cannot be presented as an equalizer of two arrows between powers of E. (e) On the other hand Q can be presented as an equalizer of two arrows between two objects in C that are equalizers of two arrows between powers of E. Indeed: the equalizer of the identity morphism of E and the unique non-identity morphism of E is the subfield D in E generated by tt (which is just the square root of 2); and the equalizer of the identity morphism of D and the unique non-identity morphism of D is Q. (f) This also gives negative answer to the question about "internally complete", since no arrow of our subcategory composed with the two morphisms D ---> D above will give the same result. This story is of course based on the fact that there are Galois field extensions L/K and M/L, for which M/K is not a Galois extension. Best regards, George ----- Original Message ----- From: "Michael Barr" To: "Categories list" Sent: Monday, May 12, 2008 2:34 PM Subject: categories: Further to my question on adjoints > 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 > > >