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• CommentRowNumber1.
• CommentAuthorDavid_Corfield
• CommentTimeNov 9th 2021

Corrected the reference to where Lurie shows that the tangle hypothesis may be deduced from the cobordism hypothesis.

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeNov 10th 2021

Renamed to the name actually used in the literature. Added redirects and pointers to other entries. Various corrections.

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeNov 10th 2021

Much of the content of the page seemed to derive from very early edits and concerns the cobordism hypothesis and so ought to be there. I’ve edited heavily and removed the following which is covered already at cobordism hypothesis.

In extended topological quantum field theory, which is really the representation theory of these cobordism $n$-categories, we expect: +– {: .un_prop}

###### Extended TQFT Hypothesis

An $n$-dimensional unitary extended TQFT is a weak $n$-functor, preserving all levels of duality, from the $n$-category $n Cob$ of cobordisms to $n Hilb$, the $n$-category of $n$-Hilbert spaces. =–

Putting the extended TQFT hypothesis and the cobordism hypothesis together, we obtain: +– {: .un_prop}

###### The primacy of the point

An $n$-dimensional unitary extended TQFT is completely described by the $n$-Hilbert space it assigns to a point. =–

Further discussion can be found here:

• Bruce Bartlett, On unitary 2-representations of finite groups and topological quantum field theory. PhD thesis, Sheffield (2008) (arXiv)

Around 2009, Mike Hopkins and Jacob Lurie have claimed (see Hopkins-Lurie on Baez-Dolan) to have formalized and proven this hypothesis in the context of (infinity,n)-categories modeled on complete Segal spaces. See:

• Jacob Lurie, On the classification of topological field theories (pdf)

where an (infinity,n)-category of cobordisms is defined and shown to lead to a formalization and proof of the cobordism hypothesis. Lurie explains his work here:

Lecture notes for Lurie’s talks are available at the Geometry Research Group website.