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.

14 pages. Major revision of arXiv:1703.02849v3. The title, abstract, and main results have been substantially revised. The previous material on Grassmannians, algebraic $K$-theory, Thom spaces, and algebraic cobordism has been removed; the paper now focuses on motivic classifying infinity-topoi, spectral stacks, and motivic stackification