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, André -------- Message d'origine-------- De: categories@mta.ca de la part de Joyal, André Date: mar. 15/12/2009 15:14 À: Andrew Stacey; categories@mta.ca Objet : categories: Re: A well kept secret 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 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. One is the algebraic geometry "under SpecZ" of Toen and Vaquié. 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. 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 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]] which takes an element x\in M to the power series 1+tx. Best, André [For admin and other information see: http://www.mta.ca/~cat-dist/ ]