Affine representability results in ${\mathbb{A}}^1$-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 F. Morel's ${\mathbb{A}}^1$-representability theorem for vector bundles.

23 pages; Further minor changes; Accepted for publication in Duke Math. J

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