Do the Witt vectors satisfy a universal property?

18$\begingroup$ [Pedant hat on: any object has a universal property, it's unversal for maps to the object. Pedant hat off.] $\endgroup$– Kevin BuzzardNov 19, 2009 at 21:33

1$\begingroup$ The witt ring are actually the points of an affine scheme whose algebra of functions has the structure of a hopf algebra and as a hopf algebra it is the free commutative hopf algebra on the conilpotent cocommutative cofree coalgebra on one primitive generator. $\endgroup$– Saal HardaliMar 31, 2019 at 13:51
4 Answers
As Morten Brun said, the big Witt vector functor is the right adjoint of the forgetful functor from the category of lambdarings to the category of rings (commutative). But this answer is not completely satisfying in that Witt vectors usually come up in number theory, in contexts that have little direct connection to Ktheory. It's also not clear what the analogue of that statement for the $p$typical Witt vectors is. ($p$ is a prime here. The $p$typical Witt vectors are the usual "nonbig" Witt vectors as defined by Witt and which come up in the theory of local fields, for instance.)
To me, the most satisfying answer to this question is that Witt $W(A)$ gives the universal way of equipping your ring $A$ with lifts of Frobenius maps. For simplicity, let's look at the $p$typical Witt vectors. Then $W(A)$ has a ring endomorphism $F$ which is congruent to the $p$th power map modulo the ideal $pW(A)$. In other words, $W(A)$ has a lift of the Frobenius endomorphism of $W(A)/pW(A)$. There is also a ring map $W(A) \rightarrow A$ given by projection on the the first component.
Now, in what sense is $W(A)$ universal? Suppose $B$ is another ring equipped with a ring map $B \rightarrow A$ and an endomorphism $F:B \rightarrow B$ lifting the Frobenius endomorphism of $B/pB$. Then, assuming $B$ is $p$torsionfree, there exists a unique ring map $B \rightarrow W(A)$ commuting with the two maps $F$ and the two maps to $A$. (This is a theorem called "Cartier's DieudonneDwork lemma" in classical exposition of the Witt vectors, but is essentially true by definition in some more recent ones.) Thus, ignoring the issue of $p$torsion, $W(A)$ is the universal ring mapping to $A$ with a Frobenius lift.
How do we deal with torsion? First, if $A$ itself is $p$torsion free, then so is $W(A)$  it is actually a subring of an infinite product of copies of $A$. So then $W$ is the right adjoint of the forgetful functor from the category of $p$torsionfree rings equipped with a Frobenius lift to the category of $p$torsionfree rings. Now it will one day be clear that the most important uses of $W(A)$ are when $A$ is torsion free, but certainly the most important existing applications are when $A$ is an $\mathbb{F}_p$algebra, where everything is $p$torsion. So it would be nice to have a universal property that works whether there is $p$torsion or not.
Probably the most straightforward way to do this is to use a better definition of "Frobenius lift". If $F$ is a Frobenius lift on $B$ as above and $B$ is $p$torsion free, then $d(x)=(F(x)x^p)/p$ is a welldefined operator on $B$. The condition that $F$ be a ring endomorphism can be of course be expressed in terms of slightly complicated identities on $d$. The key point, then, is that by the magical properties of binomial coefficients modulo $p$, these identities have integral coefficients  there are no $p$'s in the denominators! Then we can define a $d$ring structure on any ring to be an operator $d$ satisfying these conditions. Then you can show by reduction to the $p$torsionfree case that $W$ is the right adjoint of the forgetful functor from the category of $d$rings to the category of rings. The point of all this is to eliminate the existential quantifier hidden in the word "lift" by specifying a $y$ such that $F(x)x^p$ is $p$ times $y$, rather than just saying some such element $y$ exists.
Pretty much everything is the same when dealing with more than one prime, except that the Frobenius lifts are required to commute. The big Witt vectors are what you get when you have commuting Frobenius lifts at all primes.
I think this point of view was first discovered by Joyal. You can also see the first section of my paper "Basic geometry of Witt vectors", which is on the archive. Unlike mine, Joyal's papers on this are wonderfully short. I don't have their precise details, but you can see the references in my paper.

$\begingroup$ Dear Prof, thanks for your answer. It seems clearer now that the Frobenius lift should be replaced by the derived Frobenius lift, which means a ring morphism $\phi\colon R\to R$ with a (homotopy) commutative diagram $$\require{AMScd}\begin{CD}R@>\phi>>R\\ @VVV@VVV\\ \bar R@>\bar\phi>>\bar R\end{CD}$$ where $\bar R:=R\otimes_{\mathbb Z}^{\mathbb L}\mathbb F_p$ considered as a simplicial commutative ring over $\mathbb F_p$ and $\bar\phi$ is the Frobenius on $\bar R$. Derived Frobenius lifts are 11 corresponding to $\delta$ring structures. $\endgroup$– user20948Nov 14, 2018 at 19:22
For ptypical Witt vectors it is left adjoint to the functor 'reduction mod p' from the category of strict prings to the category of perfect F_p algebras.
See 1.1 and 6 of Joe Rabinov's notes
Witt vectors have several universal properties. For example the big Witt vectors are right adjoint to the forgetful functor from lambda rings to commutative rings. You find information like this in the book "Formal Groups and Applications" by M. Hazewinkel.
Another place to look is the paper Plethystic Algebra by James Borger and Ben Wieland. They have an interpretation of Witt vectors in terms of plethories (thatwhichactsonalgebras). That article is on the arxiv as 0407227, and is published in Adv. Math. (MR2139914). There's also some basic information on plethories at the nlab under the name TallWraith monoid (the name is because there is a paper from the 1970s by David Tall and Gavin Wraith which has similar themes).