  We consider $p$-weak differentiable structures that were recently introduced
by the first and last named authors, and prove that the product of $p$-weak
charts is a $p$-weak chart. This implies that the product of two spaces with a
$p$-weak differentiable structure also admits a $p$-weak differentiable
structure. We make partial progress on the tensorization problem of Sobolev
spaces by showing an isometric embedding result. Further, we establish
tensorization when one of the factors is PI.
