Motivic classifying $\infty$-topoi and spectral stacks

Abstract

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 $\infty$-categories, $\infty$-topoi, and classifying $\infty$-topoi. We then define motivic spectral schemes and motivic spectral Deligne--Mumford stacks in terms of structured motivic $\infty$-topoi. The main result establishes the existence of a motivic stackification functor: a geometric morphism between compatible motivic classifying \(\infty\)-topoi induces a pullback functor on structured motivic topoi, and this functor admits a left adjoint relative to the underlying motivic $\infty$-topos.