Witt sheaves and the $\eta$-inverted sphere spectrum

Ananyevsky has recently computed the stable operations and cooperations of rational Witt theory. These computations enable us to show a motivic analog of Serre’s finiteness result: Theorem: Let $k$ be a field. Then ${\pi }_n^{{\mathbb{A}}^1}({\mathbb{S}}_k^{-}{)}_{\ast }$ is torsion for $n>0$. As an application we define a category of Witt motives over $k$ and show that rationally this category is equivalent to the minus part of $SH(k{)}_{\mathbb{Q}}$.