FQFT and cohomology
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:
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 .