Schematic Functorialities of Birational Motivic Homotopy Categories

We promote the $n$-birational motivic homotopy category $S\mapsto \mathcal{H}^n(S)$ to a $Pr^L$-valued presheaf on $Corr(\mathrm{Sch})_{uglt,sm}$. As a consequence, for any unibranch scheme $X$, its (zeroth) birational motivic homotopy category $\mathcal{H}^{b\mathbb{A}^1}(X)$ decomposes as the product of the birational motivic homotopy categories of its function fields; in particular, for a variety $V$, $\mathcal{H}^{b\mathbb{A}^1}(V) \simeq \mathcal{H}^{b\mathbb{A}^1}(k(V))$. This implies that birational equivalences of unibranch schemes in $Sm_S$ can be detected by the birational contractibility of their generic fibers. Finally, we establish that stably birational morphisms and purely transcendental field extensions induce fully faithful embeddings of birational motivic homotopy categories.