Remarks on iterations of the A1-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.