Definition 4.1.1.
We started our description of the \(p\)-typical Witt vector functor with the fact that the underlying functor to sets is represented by the ring \(A = \ZZ\{y\}\text{,}\) but crucially we already had produced a functor valued in rings (and even in \(\delta\)-rings). If we had needed to construct from scratch a functor valued in rings, we would have needed structures on \(A\) giving rise to the addition and multiplication maps. These structures are:
- a coaddition morphism \(\Delta^+\colon A \to A \otimes_{\ZZ} A\text{;}\) and
- a comultiplication morphism \(\Delta^\times\colon A \to A \otimes_{\ZZ} A\text{.}\)
A ring \(A\) equipped with these structures represents a functor from rings to sets equipped with two binary operations \(+, \times\text{.}\) A biring is a ring equipped with coaddition and comultiplication operators which are further subject to the axioms that correspond to the ring axioms on \(+, \times\text{.}\) Namely, the coaddition map is cocommutative, coassociative, and admits a counit and an antipode (giving rise to additive inverses); the comultiplication map is cocommutative, coassociative, codistributive over coaddition, and admits a counit.
A shorter way to say this is that a biring is a commutative ring object in the category of affine schemes. (Remember that the functor \(\Spec\colon \Ring \to \Sch\) is contravariant!)