The generalized tangle hypothesis is a refinement of the cobordism hypothesis.
The original tangle hypothesis was formulated in
The -category of framed -tangles in dimensions is -equivalent to the free weak -tuply monoidal -category with duals on one object.
In the limit , this gives:
The -category of cobordisms is the free stable -category with duals on one object (the point).
Putting the extended TQFT hypothesis and the cobordism hypothesis together, we obtain:
An -dimensional unitary extended TQFT is completely described by the -Hilbert space it assigns to a point.
Further discussion can be found here:
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:
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.
The -tuply monoidal -category of -structured -tangles in the -cube is the fundamental -category with duals of .