From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/4393 Path: news.gmane.org!not-for-mail From: "George Janelidze" Newsgroups: gmane.science.mathematics.categories Subject: re: Further to my question on adjoints Date: Tue, 13 May 2008 01:43:29 +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 1241019917 13092 80.91.229.2 (29 Apr 2009 15:45:17 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Wed, 29 Apr 2009 15:45:17 +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 1Jvxvp-0006NG-5C for categories-list@mta.ca; Tue, 13 May 2008 14:04:53 -0300 Original-Sender: cat-dist@mta.ca Precedence: bulk X-Keywords: X-UID: 16 Original-Lines: 134 Xref: news.gmane.org gmane.science.mathematics.categories:4393 Archived-At: Dear Michael, Sorry to say, I have no time now - so, just briefly: I know that "positive" is just an illusion and I only used it to make things seem more obvious. Moreover, "fields" is also an illusion, since all the rings involved are (quasi-) separable Q-algebras - and in fact one should put things inside the dual category of G-sets, where, say, G is a finite group that has a subgroup whose normalizer is a normal subgroup in G different from G itself. This would imply that the phenomenon you were looking for can even be found in a category dual to a Boolean topos. However I still have to study what you say in the second paragraph of your message... Thank you for an interesting question- George ----- Original Message ----- From: "Michael Barr" To: "George Janelidze" Cc: "Categories list" ; "John F. Kennison" ; "Bob Raphael" Sent: Monday, May 12, 2008 9:27 PM Subject: Re: categories: Further to my question on adjoints > I have checked this carefully and it works. To summarize, let F = > Q[2^{1/2}] and E = Q[2^{1/4}]. Then any power of E contains a square > whose square is a square root of 2 and any ring homomorphism between > powers of E preserves it. (Incidentally, although it may help your > intuition to take the positive fourth of 2, the various fourth roots of 2 > are indistinguishable algebraically.) Thus any ring in EqP(E) contains a > square root of 2 (although not necessarily a fourth root). Now F is the > equalizer of the two distinct maps E to E, while Q is the equalizer of the > two distinct maps F to F. > > This now gives a counter-example for my original question. Let C be the > category of commutative rings, F = Hom(-,E) : C ---> Set\op and U = E^{-}: > Set\op ---> C are adjoint. If T is the resultant triple, then F ---> E > ===> E is an equalizer between two values of U, while not being the > canonical equalizer. TF = E x E and T^2F = E x E x E x E. I haven't done > the computation, but I believe the equalizer of TF ===> T^2 is F x F. > > Thanks George, > > Michael > > > On Mon, 12 May 2008, George Janelidze wrote: > > > 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 > >> > >> > >> > > >