Dear Andrew The conceptual definition of lamba ring given by Andre', (for which a presentation by useful but complicated identities is theorem rather than a definition) solves an important one of a generic class of problems that I proposed in the article that is bundled with my thesis in TAC Reprints. A simple pedagogically convincing example is given by the following dialogue : I have an example of a general category C and a functor U from it to finite 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 specific assignment of an algebraic theory to any morphism of algebraic theories (from Andre's example it should be clear which assignment) and asks 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 coequalizers and that some examples are representable or otherwise computable. Bill On Tue 12/15/09 3:14 PM , Joyal, André joyal.andre@uqam.ca sent:
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-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-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 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.
Best, André
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