  In this paper, we study the Nisnevich sheafification
$\mathcal{H}^1_{\acute{e}t}(G)$ of the presheaf associating to a smooth scheme
the set of isomorphism classes of $G$-torsors, for a reductive group $G$. We
show that if $G$-torsors on affine lines are extended, then
$\mathcal{H}^1_{\acute{e}t}(G)$ is homotopy invariant and show that the sheaf
is unramified if and only if Nisnevich-local purity holds for $G$-torsors. We
also identify the sheaf $\mathcal{H}^1_{\acute{e}t}(G)$ with the sheaf of
$\mathbb{A}^1$-connected components of the classifying space ${\rm
B}_{\acute{e}t}G$. This establishes the homotopy invariance of the sheaves of
components as conjectured by Morel. It moreover provides a computation of the
sheaf of $\mathbb{A}^1$-connected components in terms of unramified $G$-torsors
over function fields whenever Nisnevich-local purity holds for $G$-torsors.
