On the Motivic Homotopy Type of Algebraic Stacks (with an appendix joint with Jack Hall)

We construct smooth presentations of algebraic stacks that are local epimorphisms in the Morel-Voevodsky ${\mathbb{A}}^1$-homotopy category. As a consequence we show that the motive of a smooth stack (in Voevodsky’s triangulated category of motives) has many of the same properties as the motive of a smooth scheme.