nLab tangle hypothesis




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

Tangles with structure

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)

Tangles with singularities

The tangle hypothesis can further be generalized to allow for ‘tangles with singularities’ (also called stratified tangles). The idea is analogous to that of cobordisms with singularities. The tangle hypothesis for tangles with singularities includes the case of GG-structured tangles (which requires choosing generators for MGM G, that then yield a datum of singularity types, see (Lurie 09, Sec. 4.3)).

The generalized tangle hypothesis is closely related to the generalized Thom-Pontryagin construction which relates homotopy classes of maps from manifolds MM into CW complexes XX to cobordism classes of XX-stratifications of MM. (The relation was discussed on the nnCafé here and here.)

Directed variants

Manifold diagrams are stratified tangles ‘without critical points’. The relation of stratified tangles to presented higher categories with duals is analogous to the relation of manifold diagrams to presented higher categories (without the need for duals). This yields a “directed” version of the generalized tangle hypothesis.

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 (namely, for cobordisms with singularities).


For a discussion of the generalized tangle hypothesis see

Last revised on June 1, 2023 at 16:57:00. See the history of this page for a list of all contributions to it.