Dear Andrew, My statement about the theory of lambda ring was wrong. I would like to correct it. Let me first recall the original statement.
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.
I woke up in the middle of the following night with the strong impression that it could not work. Here is the problem: the theory of natural operations on the functor V=Z[-] is uncountable, whilst the theory of lambda-rings is countable. To see this, let analyse the set of unary operations on the functor Z[-]. For any monoid M, the algebra Z[M] is equipped with a natural augmentation e:Z[M]--->Z from which we obtain a natural transformation e:Z[-]--->Z from the functor Z[-] to the constant functor Z. The natural transformation exibits Z as the colimit of the functor Z[-], since we have Z[1]=Z and since the monoid 1 is terminal in the category of commutative monoids CMon. Let me denote by Z[M]_n the fiber of the map e:Z[M]--->Z at n\in Z. The (set valued) functor Z[-]_n is connected, since Z[1]_n={n.1}. The decomposition Z[-]=disjoint union_n Z[-]_n coincide with the canonical decomposition of the functor Z[-] as a disjoint union of connected components. Notices that the functor Z[-]_n is isomorphic to the functor Z[-]_0 since we have n.1+Z[M]_0=Z[M]_n for any monoid M. Hence the functor Z[-] is isomorphic to the functor Z\times Z[-]_0. It then follows from the connectedness of the functor Z[-]_0 that we have a bijection End(Z[-])=End(Z\times Z[-]_0)=Z^Z\times End(Z[-]_0)^Z Hence the set End(Z[-]) is uncountable, since the set Z^Z is uncountable. It seems that the basic idea can be saved by making a slight modification to the functor V. Let me say that an element z in a monoid M is a ZERO ELEMENT if we have zx=xz=z for every x\in M. A zero element is unique when it exists. I will denote a zero element by 0. Let me denote by CMonz the category of commutative monoids with zero element, where a map M--->N should preserve the zero elements. Then the obvious forgetful functor CRing--->CMonz has a left adjoint which associates to M a commutative ring Z[M']. The additive group of Z[M'] is the free abelian group on M'=M\setminus{0}. If we compose the functot Z['] with the forgetful functor U:CRing --->Set we obtain a functor V':CMonz --->Set. I conjecture that the theory of natural operations on the functor V' is the algebraic theory of lambda-rings. 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 (notices that 1+t0=1). Best, André -------- Message d'origine-------- De: Joyal, André Date: mer. 16/12/2009 08:08 À: Andrew Stacey; categories@mta.ca Objet : RE : categories: Re: A well kept secret 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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]