The cobordism hypothesis gives a functorial bijection between oriented  n-dimensional fully local topological field theories, valued in some  higher category C, and the fully dualizable object of C equipped with  the structure of SO(n)-fixed point.  In this talk I'll explain recent  works of Haugseng, Johnson-Freyd and Scheimbauer which construct a  Morita 4-category of braided tensor categories, and I'll report on joint  work with Brochier and Snyder which identifies two natural subcategories  therein which are 3-...