Motivic classifying $\mathrm{\infty }$-topoi and spectral stacks

In this paper, we develop motivic derived algebraic geometry, an enhancement of derived algebraic geometry adapted to the ${\mathbb{A}}^1$-homotopy theory of Morel and Voevodsky. We construct motivic model categories by imposing descent for a Grothendieck topology and invariance with respect to an interval object, and use them to formulate motivic versions of $\mathrm{\infty }$-categories, $\mathrm{\infty }$-topoi, and classifying $\mathrm{\infty }$-topoi. We then define motivic spectral schemes and motivic spectral Deligne–Mumford stacks in terms of structured motivic $\mathrm{\infty }$-topoi. The main result establishes the existence of a motivic stackification functor: a geometric morphism between compatible motivic classifying $\mathrm{\infty }$-topoi induces a pullback functor on structured motivic topoi, and this functor admits a left adjoint relative to the underlying motivic $\mathrm{\infty }$-topos.