Re: A well kept secret

["Joyal, André" <joyal.andre@uqam.ca>]

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



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