From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5372 Path: news.gmane.org!not-for-mail From: F William Lawvere Newsgroups: gmane.science.mathematics.categories Subject: Re: A well kept secret Date: Wed, 16 Dec 2009 12:17:39 -0500 Message-ID: Reply-To: F William Lawvere NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1261025682 9378 80.91.229.12 (17 Dec 2009 04:54:42 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Thu, 17 Dec 2009 04:54:42 +0000 (UTC) To: "Andrew Stacey" , Original-X-From: categories@mta.ca Thu Dec 17 05:54:35 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 1NL8Nk-0004Dg-Na for gsmc-categories@m.gmane.org; Thu, 17 Dec 2009 05:54:32 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NL7t1-0006JD-5x for categories-list@mta.ca; Thu, 17 Dec 2009 00:22:47 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5372 Archived-At: Dear Andrew The conceptual definition of lamba ring given by Andre', (for which a presentation by useful but complicated identities=20 is theorem rather than a definition) solves an important one of a generic class of problems that I proposed in the article that is=20 bundled with my thesis in TAC Reprints. A simple pedagogically convincing example is given by the=20 following dialogue : I have an example of a general category C and a functor U from it to finit= e sets; moreover I have a particular object X in C : what information about X can I find using U ? Well, you could count the points of U(X).Yes but that is by no means all. The functor U has a group G of all natural automorphisms, and so U can be lifted across the category of G-sets, thus that number is actually a sum of more refined invariants indexed by the subgroups of G. The generic problem (for the doctrine of algebraic theories rather than for the subdoctrine of permutation representations) considers a=20 specific assignment of an algebraic theory to any morphism of algebraic=20 theories (from Andre's example it should be clear which assignment) and ask= s=20 for specific calculation (eg a presentation in terms of given presentations= , or indeed any information). The construction involves the natural structure of a functor that is not representable in general but hope comes from the fact that this functor preserves filtered colimits and reflexive coequ= alizers and that some examples are representable or otherwise computable. Bill On Tue 12/15/09 3:14 PM , Joyal, Andr=C3=A9 joyal.andre@uqam.ca sent:=20 > Dear Andrew, >=20 > You wrote >=20 > >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 p= earl 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 (i= t is > the Norwegian university of>Science and Technology, after all), even with= out > talking about programming>(about which I know nothing). >=20 > >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 solv= es > in an elegant fashion. It>would be nice if there was one that used categ= ory > 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 the subject can fit your audience. >=20 > 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=C3=83=C2=A9.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-ringto 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-r= ing > 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 the= ory.=20 > Let Z[]:CMon ---> CRing be the functor which associates to a commutative > monoid M the ring Z[M] freely generated by M (it is the left adjoint to t= he forgetful > functor in theopposite direction). If we compose the functot Z[] with the= forgetful > functor U:CRing --->Setwe 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]] which takes an element x\in M to the power series 1+= tx. >=20 >=20 > Best, > Andr=C3=83=C2=A9 >=20 >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]