I am studying this paper about the unstable motivic homotopy theory. The authors work in the category $Sm_{S}$ of smooth schemes over an arbitrary scheme $S$ and then they consider the "affine line" $\mathbb{A}^{1}$. Is it really meant the $\mathbb{A}^{1}_{S}:=\mathrm{Spec}(\mathbb{Z}[t])\times_{\mathbb{Z}}S$? Because in chapter 4.3 they consider the cosimplicial scheme
$$\mathbb{\Delta}^{n}=\mathrm{Spec}\ k[x_{0},\dots,x_{n}]/(x_{0}+\dots +x_{n}=1)$$
and then they define $$\mathrm{Sing}^{\mathbb{A}^{1}}X:=|X(-\times \mathbb{\Delta}^{\bullet})|$$
But I suppose the scheme $\mathbb{\Delta}^{n}$ should be over $S$, right?