give a rigorous treatment of these finite path integrals and to generalize them. We give some indications in §\[sec:5\] and §\[cs8\]; we hope to develop these ideas in detail elsewhere.

That structure theorem is the Baez-Dolan cobordism hypothesis, for which we include an exposition in \[subsec:2.2\]. A much more detailed account may be found in [@L], which also contains a detailed sketch of the proof. This theorem asserts that a fully extended TQFT—that is, a TQFT extended down to points—is determined by its value on a point. Furthermore, it characterizes the possible values on a point. This theorem provides one possible construction of many of the theories in this paper.[^7]

The anomaly theory ${\mathscr{A}}$ which appears in Chern-Simons theory has a classical description as a 4-dimensional field theory. In \[subsec:6.1\] we sketch two such classical theories, each based on a 2-groupoid of gerbes (rather than a 1-groupoid of connections, which is what appears in gauge theory). These are free theories, so the path integral on a 4-manifold is Gaussian and can be carried out explicitly. Presumably this is a classical description of the theories put forward by Crane and Yetter [@CY], though we do not attempt to make the connection. 