On the notion of $\mathbb{A}^1$-coverings

Source abstract

We introduce the notion of $\mathbb{A}^1$-$n$-coverings and begin the development of their theory over an arbitrary base scheme $S$, in analogy to that of higher coverings in an $\infty$-topos. We show that for $n=0$, our notion recovers Morel's $\mathbb{A}^1$-coverings between simplicial sheaves. Under suitable assumptions, we then provide a Galois-theoretic classification of $\mathbb{A}^1$-$n$-coverings. Finally, we answer a question of Morel concerning a geometric criterion for $\mathbb{A}^1$-coverings in the negative.