Purity for Algebraic Stacks

In previous works, Khan-Ravi and Chowdhury constructed the "Borel" motivic homotopy category for a broad class of algebraic stacks. In a joint paper with C. Chowdhury, we presented a third approach to construct SH(\X), which takes a more classical perspective. We will examine various Nisnevich-like topologies on stacks and demonstrate that any stack can be covered by a smooth-Nisnevich schematic atlas—a result also independently established by J. Hall. Finally, we will explore ambidexterity and purity results, concluding with a non-representable purity statement for smooth maps of algebraic stacks.