# nLab generalized tangle hypothesis

The generalized tangle hypothesis is a refinement of the cobordism hypothesis.

# History

The original tangle hypothesis was formulated in

as follows:

###### Tangle Hypothesis

The $n$-category of framed $n$-tangles in $n+k$ dimensions is $(n+k)$-equivalent to the free weak $k$-tuply monoidal $n$-category with duals on one object.

In the limit $k \to \infty$, this gives:

###### Cobordism Hypothesis

The $n$-category $n Cob$ of cobordisms is the free stable $n$-category with duals on one object (the point).

In extended toplogical quantum field theory, which is really the representation theory of these cobordism $n$-categories, we expect:

###### 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:

###### 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)

More recently 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:

While the tangle hypothesis and its generalizations are refinements of the cobordism hypothesis and its generalizations, Lurie shows (Sec 4.4. of TQFT) that the former may be deduced from the latter when expressed in a sufficiently general form.

Lecture notes for Lurie’s talks should eventually appear at the Geometry Research Group website.

# Statement of the generalized tangle hypothesis

The $k$-tuply monoidal $n$-category of $G$-structured $n$-tangles in the $(n + k)$-cube is the fundamental $(n + k)$-category with duals of $(M G,Z)$.

• $M G$ is the Thom space of group $G$.
• $G$ can be any group equipped with a homomorphism to $O(k)$. (comment)

# Further resources

## Discussion

Revised on October 14, 2014 08:44:09 by David Corfield (129.12.18.116)