Affine representability results in A1-homotopy theory, I: Vector bundles

Abstract

We establish a general “affine representability” result in ${\mathbb{A}}^1$-homotopy theory over a general base. We apply this result to obtain representability results for vector bundles in ${\mathbb{A}}^1$-homotopy theory. Our results simplify and significantly generalize Morel’s ${\mathbb{A}}^1$-representability theorem for vector bundles.