Definition 3.1.1.
As indicated in Definition 2.4.5, the forgetful functor \(\Ring_\delta \to \Ring\) admits a right adjoint \(W\text{.}\) To identify the image of a ring \(A\) under this functor, we use the set-theoretic identifications
\begin{align*}
W(A) &= \Hom_{\Ring}(\ZZ[y], W(A))\\
&= \Hom_{\Ring_\delta}(\ZZ\{y\}, W(A))\\
&= \Hom_{\Ring}(\ZZ[y_0, y_1, \dots], A)\\
&= A \times A \times \cdots.
\end{align*}
This means that each element of \(W(A)\) has a unique expansion \((y_0, y_1, \dots)\) with each \(y_n \in A\text{;}\) we call the \(y_n\) the \(y\)-coordinates (or Joyal coordinates) of this element of \(W(A)\text{.}\) (This presentation does not directly describe the ring structure on \(W(A)\text{;}\) see Remark 4.2.6.)
In Lemma 3.1.3 below. we will give a second set of generators \(x_0, x_1, \dots\) of the polynomial ring \(\ZZ[y_0, y_1, \dots]\text{.}\) This means that each element of \(W(A)\) has a unique expansion \((x_0, x_1, \dots)\) with each \(x_n \in A\text{;}\) we call the \(x_n\) the \(x\)-coordinates (or Witt coordinates) of this element of \(W(A)\text{.}\) In these coordinates, \(W(A)\) will become none other than the ring of \(p\)-typical Witt vectors over \(A\) via the translation described in Definition 3.2.1.