nLab tangle hypothesis

Contents

Contents

Statement

The tangle hypothesis (Baez and Dolan 95) is 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 k-tuply monoidal n-category with duals on one object.

The tangle hypothesis may be seen as a refinement of the cobordism hypothesis in the sense that the latter arises from the former in the limit kk \to \infty:

Cobordism Hypothesis

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

Generalized tangle hypothesis

The tangle hypothesis has been generalized to allow certain structures on the tangles.

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)

The tangle hypothesis as a consequence of the cobordism hypothesis

While the tangle hypothesis and its generalizations may be seen as refinements of the cobordism hypothesis and its generalizations, Lurie shows (Lurie 09, Sec. 4.4) that the former may be deduced from the latter when expressed in a sufficiently general form.

References

For a discussion of the generalized tangle hypothesis see

Last revised on November 10, 2021 at 15:28:42. See the history of this page for a list of all contributions to it.