Remarks on iterations of the $\mathbb A^1$-chain connected components construction

Abstract

We show that the sheaf of $\mathbb A^1$-connected components of a Nisnevich sheaf of sets and its universal $\mathbb A^1$-invariant quotient (obtained by iterating the $\mathbb A^1$-chain connected components construction and taking the direct limit) agree on field-valued points. This establishes an explicit formula for the field-valued points of the sheaf of $\mathbb A^1$-connected components of any space. Given any natural number $n$, we construct an $\mathbb A^1$-connected space on which the iterations of the naive $\mathbb A^1$-connected components construction do not stabilize before the $n$th stage.