  We study the question of dualizability in higher Morita categories of locally
presentable tensor categories and braided tensor categories. Our main results
are that the 3-category of rigid tensor categories with enough compact
projectives is 2-dualizable, that the 4-category of rigid braided tensor
categories with enough compact projectives is 3-dualizable, and that (in
characteristic zero) the 4-category of braided fusion categories is
4-dualizable. Via the cobordism hypothesis, this produces respectively 2, 3 and
4-dimensional framed local topological field theories. In particular, we
produce a framed 3-dimensional local TFT attached to the category of
representations of a quantum group at any value of $q$.
