A surgery presentation for the n-dimensional bordism 2-category

In previous work, a classification of 3-dimensional once-extended topological quantum field theories has been obtained. However, this work relied on a presentation for the n-dimensional bordism 2-category, which was claimed but not proved, and filling in the details has proved difficult. The basic idea is to formulate parameterized Morse theory at codimension 2 in terms of surgery data, then arrange this data into a 2-category, and then to simplify this data in dimension 3. I will explain these steps and some recent progress I have made in each of them. Joint work with Chris Douglas, Chris Schommer-Pries and Jamie Vicary.