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

In the limit kk \to \infty, this gives:

Cobordism Hypothesis

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

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

Extended TQFT Hypothesis

An nn-dimensional unitary extended TQFT is a weak nn-functor, preserving all levels of duality, from the nn-category nCobn Cob of cobordisms to nHilbn Hilb, the nn-category of nn-Hilbert spaces?.

Putting the extended TQFT hypothesis and the cobordism hypothesis together, we obtain:

The primacy of the point

An nn-dimensional unitary extended TQFT is completely described by the nn-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:

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

Statement of the generalized tangle hypothesis

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

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

Further resources

Discussion

Revised on November 29, 2009 20:56:00 by Toby Bartels (173.60.119.197)