$\mathbb{P}^1$ Homotopy Theory of Schemes

We study the $\mathbb{P}^1$-local motivic homotopy theory under the Nisnevich topology on $Sm_S$. We show that although the homotopy category $\mathcal{H}^{\mathbb{P}^1}(-)$ admits standard schematic functorialities, it fails six-functor gluing, thereby obstructing a full six-functor formalism. On t