Model Topoi and Motivic Homotopy Theory

Abstract

Given a small simplicial category $\mathcal{C}$ whose underlying ordinary category is equipped with a Grothendieck topology $\tau$, we construct a model structure on the category of simplicially enriched presheaves on $\mathcal{C}$ where the weak equivalences are the local weak equivalences of the underlying (non-enriched) simplicial presheaves. We show that this model category is a $t$-complete model topos and describe the Grothendieck topology $[\tau ]$ on the homotopy category of $\mathcal{C}$ that corresponds to this model topos. After we first review a proof showing that the motivic homotopy theory is not a model topos, we specialize this construction to the category of smooth schemes of finite type, which is simplicially enriched using the standard algebraic cosimplicial object, and compare the result with the motivic homotopy theory. We also collect some partial positive results on the exactness properties of the motivic localization functor.