Let $\mathscr{C}$ be a symmetric monoidal (weak) $n$-category. A framed extended TQFT of dimension $n$ with values in $\mathscr{C}$ is a symmetric monoidal functor from the framed bordism $n$-category $\textsf{Cob}^{\textsf{fr}}_n(n)$ into $\mathscr{C}$ (cf. Lurie 2008).
Let $Z : \textsf{Cob}^{\textsf{fr}}_n(n) \rightarrow \mathscr{C}$ be a framed extended $\mathscr{C}$-valued TQFT. A statement of the cobordism hypothesis (Baez and Dolan 1995; Lurie 2008) is that the evaluation functor
$$Z \mapsto Z(*)$$
determines a bijection between (isomorphism classes of) framed extended $\mathscr{C}$-valued topological field theories and (isomorphism classes of) fully dualizable objects of $\mathscr{C}$, so that for every object $C$ of $\mathscr{C}$ there is an essentially unique symmetric monoidal functor $Z_C : \textsf{Cob}^{\textsf{fr}}_n(n) \rightarrow \mathscr{C}$ such that $Z_C(*) \equiv C$.
Lurie has outlined a proof of the cobordism hypothesis:
- Jacob Lurie, On the Classification of Topological Field Theories. Current developments in mathematics 2008.1 (2008): 129-280. doi:10.4310/CDM.2008.v2008.n1.a3
Grady and Pavlov have claimed a complete proof of a 'geometric' generalization of it:
- Daniel Grady and Dmitri Pavlov, The geometric cobordism hypothesis. arXiv preprint arXiv:2111.01095. 2021.
The first statement of the cobordism hypothesis by Baez and Dolan is in:
- John C. Baez and James Dolan. Higher‐dimensional algebra and topological quantum field theory. Journal of mathematical physics 36.11 (1995): 6073-6105. doi:10.1063/1.531236
On the physical implications of the cobordism hypothesis, see this MO question.
My question is: What is the status of the cobordism hypothesis, and in particular of Grady and Pavlov's claimed proof?
