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 multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$.

我们研究了较高的森田类别中局部可表示张量类别和编织张量类别中的对偶化问题。我们的主要结果是，具有足够紧致射影的刚性张量类别的3类是2对偶的，具有足够紧致射影的刚性编织张量类别的4类是3对偶的，并且（特征为零）4编织多融合类别的-类别是4对偶的。通过cobordism假设，这分别产生了二维，三维和四维框架的局部拓扑场论。特别是，我们产生了一个框架化的三维局部拓扑场论，该理论附加到任意值qq $的量子组的表示类别上。