Iterations of the functor of naive $\mathbb A^1$-connected components of varieties

Abstract

For any sheaf of sets $\mathcal F$ on $Sm/k$, it is well known that the universal $\mathbb A^1$-invariant quotient of $\mathcal F$ is given as the colimit of sheaves $\mathcal S^n(\mathcal F)$ where $\mathcal S(F)$ is the sheaf of naive $\mathbb A^1$-connected components of $\mathcal F$. We show that these infinite iterations of naive $\mathbb A^1$-connected components in the construction of universal $\mathbb A^1$-invariant quotient for a scheme are certainly required. For every $n$, we construct an $\mathbb A^1$-connected variety $X_n$ such that $\mathcal S^n(X_n)\neq \mathcal S^{n+1}(X_n)$ and $\mathcal S^{n+2}(X_n)=*$.