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Hazewinkel talks about both the ‘orthodox’ and ‘heterodox’ approaches to -rings. Indeed he starts out with a lot of material on Witt vectors and their relation to -adics. The Lambda-rings only make their debut on page 87, where the operation of ‘taking the Witt vectors’ of a commutative ring is revealed to be the right adjoint to the forgetful functor from -rings to commutative rings.
He then goes ahead and defines -rings on page 88. At first his definition looks a bit frustrating, because Hazewinkel defines ‘-ring’ using the concept of ‘morphism of -rings’! But it’s not actually circular; it’s really just a trick to spare us certain ugly equations that appear in the usual definition.
Just for fun, note the unusual remark in footnote 62 on page 88: warning the reader to ‘steer clear’ of a certain book by two famous authors.
On page 92, Hazewinkel proves the Wilkerson theorem getting -rings from rings equipped with Adams operations . And then, at the bottom of page 94, he goes heterodox and defines ‘-rings’ to be commutative rings equipped with Adams operations — and notes that over a field of characteristic zero, -rings are the same as -rings.
On page 97, he describes ‘taking the Witt vectors’ as a comonad on the category of commutative rings. This is just another way of talking about the right adjointness property he mentioned on page 87; now he’s proving it.
On page 102 starts explaining ‘plethysm’.