On the internal homotopy theory of motivic categories

Abstract

We develop a general $\infty$-categorical framework for internal Eilenberg-MacLane objects, internal homotopy groups, and an internal notion of covering spaces, and study their behavior under suitable localizations. Applying this to the ordinary motivic localization, we identify internal $n$-Eilenberg-MacLane objects with strongly $\mathbb{A}^1$-invariant sheaves of (abelian when $n\geq 2$) groups, yielding a formal obstruction to the motivic homotopy category being an $\infty$-topos. We prove that taking $\mathbb{A}^1$-localizations induces an equivalence between classical $\mathbb{A}^1$-coverings of a Nisnevich local space and the internal motivic coverings of its motivic localization, providing a streamlined proof of a generalized motivic Van Kampen theorem. Along the way, we establish a Nisnevich-local-to-global $\mathbb{A}^1$-connectivity criterion: a $k$-scheme is $\mathbb{A}^1$-connected if and only if it admits a Nisnevich cover by $\mathbb{A}^1$-connected schemes such that each pairwise intersection has a ($k$-)point. Finally, we show that over a general Qcqs base, passing to the birational motivic homotopy category recovers certain essential topos-theoretic properties absent in the ordinary $\mathbb{A}^1$-setting. In fact, for a scheme with finitely many generic points, we show that the birational motivic homotopy category is a Postnikov-complete $\infty$-topos of cohomological dimension $0$.