From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5373 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: =?iso-8859-1?Q?RE=A0=3A_categories=3A_Re=3A_A_well_kept_secret?= Date: Wed, 16 Dec 2009 08:08:00 -0500 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1261025719 9463 80.91.229.12 (17 Dec 2009 04:55:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 17 Dec 2009 04:55:19 +0000 (UTC) To: "Andrew Stacey" , Original-X-From: categories@mta.ca Thu Dec 17 05:55:12 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NL8ON-0004OA-Qe for gsmc-categories@m.gmane.org; Thu, 17 Dec 2009 05:55:12 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NL7rz-0006H1-AF for categories-list@mta.ca; Thu, 17 Dec 2009 00:21:43 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5373 Archived-At: Dear Andrew, Please disregard my suggestion about a new definition of lambda ring! My memory may have failed me! I am not sure the new definition is right! Best, Andr=E9 -------- Message d'origine-------- De: categories@mta.ca de la part de Joyal, Andr=E9 Date: mar. 15/12/2009 15:14 =C0: Andrew Stacey; categories@mta.ca Objet : categories: Re: A well kept secret =20 Dear Andrew, You wrote >Let me make these remarks a little more concrete with a request (or >a challenge if you prefer). In my department, the colloquium is called >"Mathematical Pearls" (gosh, I actually wrote "Perls" first time round; = I've >been writing too many scripts lately!). I'm giving this talk in = January. My >original plan was to say something nice and differential, with lots of = fun >pictures of manifolds deforming or knots unknotting, or something like = that. >However, the discussion here has set me to thinking about saying = something >instead about category theory. It is a pearl of mathematics, it does = have >a certain beauty, there's certainly a lot that can be said, even to a = fairly >applied audience as we tend to have here (it is the Norwegian = university of >Science and Technology, after all), even without talking about = programming >(about which I know nothing). >But for such a talk, I need a story. I don't mean a historical one = (I'm not >much of a mathematical historian anyway), I mean a mathematical one. I = want >some simple problem that category theory solves in an elegant fashion. = It >would be nice if there was one that used category theory in a = surprising way; >beyond the idea that categories are places in which things happen (so = perhaps >about small categories rather than large ones). A colloquium is a good place for expressing wild ideas. But they must be related to something everyone can understand and touch. I suggest you talk about "The field with one element" if you think=20 the subject can fit your audience. http://en.wikipedia.org/wiki/Field_with_one_element Many things in this subject are very speculative but there are also a few concrete developpements.=20 One is the algebraic geometry "under SpecZ" of Toen and Vaqui=E9. Another due to Borger is using lambda-rings. What is a lambda-ring? In their book "Riemann-Roch-Algebra" Fulton and Lang define a = lambda-ring to be a pre-lambda-ring satisfying two complicated identities [(1.4) and = (1.5)] [Beware that F&L are using an old terminology: they call a lambda-ring a = "special lambda-ring" and they call a pre-lambda-ring a "lambda-ring"] The notion of lambda-ring (ie of "special lambda-ring" in the = terminology of F&L) can be defined in a natural way if we use category theory.=20 Let Z[]:CMon ---> CRing be the functor which associates to a commutative = monoid M the=20 ring Z[M] freely generated by M (it is the left adjoint to the forgetful = functor in the opposite direction). If we compose the functot Z[] with the forgetful = functor U:CRing --->Set we obtain a functor V:CMon --->Set. The algebraic theory of lambda-rings = can be defined to be the theory of natural operations on the functor V. The total lambda operation V(M)--->V(M)[[t]] is the group homomorphism = Z[M]--->1+tZ[M][[t]]=20 which takes an element x\in M to the power series 1+tx. Best, Andr=E9 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]