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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 9th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexCorrected the reference to where Lurie shows that the tangle hypothesis may be deduced from the cobordism hypothesis. <a href="https://ncatlab.org/nlab/revision/diff/generalized+tangle+hypothesis/22">diff</a>, <a href="https://ncatlab.org/nlab/revision/generalized+tangle+hypothesis/22">v22</a>, <a href="https://ncatlab.org/nlab/show/generalized+tangle+hypothesis">current</a>

   Corrected the reference to where Lurie shows that the tangle hypothesis may be deduced from the cobordism hypothesis.

   diff, v22, current

2. • CommentRowNumber2.
   • CommentAuthorDmitri Pavlov
   • CommentTimeNov 10th 2021
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexRenamed to the name actually used in the literature. Added redirects and pointers to other entries. Various corrections. <a href="https://ncatlab.org/nlab/revision/diff/tangle+hypothesis/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/tangle+hypothesis/23">v23</a>, <a href="https://ncatlab.org/nlab/show/tangle+hypothesis">current</a>

   Renamed to the name actually used in the literature. Added redirects and pointers to other entries. Various corrections.

   diff, v23, current

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 10th 2021
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexMuch of the content of the page seemed to derive from very early edits and concerns the [[cobordism hypothesis]] and so ought to be there. I've edited heavily and removed the following which is covered already at [[cobordism hypothesis]]. In [[FQFT|extended topological quantum field theory]], which is really the [[representation theory]] of these cobordism $n$-categories, we expect: +-- {: .un_prop} ###### Extended TQFT Hypothesis An $n$-dimensional unitary extended TQFT is a weak $n$-[[n-functor|functor]], preserving all levels of duality, from the $n$-category $n Cob$ of cobordisms to $n Hilb$, the $n$-category of $n$-[[n-Hilbert space|Hilbert spaces]]. =-- Putting the extended TQFT hypothesis and the cobordism hypothesis together, we obtain: +-- {: .un_prop} ###### The primacy of the point An $n$-dimensional unitary extended TQFT is completely described by the $n$-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](http://arxiv.org/abs/0901.3975)) Around 2009, [[Mike Hopkins]] and [[Jacob Lurie]] have claimed (see [Hopkins-Lurie on Baez-Dolan](http://golem.ph.utexas.edu/category/2008/05/hopkinslurie_on_baezdolan.html)) to have formalized and proven this hypothesis in the context of [[(n,r)-category|(infinity,n)-categories]] modeled on [[complete Segal space]]s. See: * [[Jacob Lurie]], _On the classification of topological field theories_ ([pdf](http://www.math.harvard.edu/~lurie/papers/cobordism.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: * [[Jacob Lurie]], _TQFT and the cobordism hypothesis_, videos of 4 lectures at the [Geometry Research Group](http://www.ma.utexas.edu/users/plowrey/dev/rtg/perspectives.html), Mathematics Department, University of Texas Austin. Lecture notes for Lurie's talks are available at the [Geometry Research Group website](http://www.ma.utexas.edu/users/plowrey/dev/rtg/perspectives.html). <a href="https://ncatlab.org/nlab/revision/diff/tangle+hypothesis/24">diff</a>, <a href="https://ncatlab.org/nlab/revision/tangle+hypothesis/24">v24</a>, <a href="https://ncatlab.org/nlab/show/tangle+hypothesis">current</a>

   Much of the content of the page seemed to derive from very early edits and concerns the cobordism hypothesis and so ought to be there. I’ve edited heavily and removed the following which is covered already at cobordism hypothesis.

   In extended topological quantum field theory, which is really the representation theory of these cobordism $n$-categories, we expect: +– {: .un_prop}

   Extended TQFT Hypothesis

   An $n$-dimensional unitary extended TQFT is a weak $n$-functor, preserving all levels of duality, from the $n$-category $n Cob$ of cobordisms to $n Hilb$, the $n$-category of $n$-Hilbert spaces. =–

   Putting the extended TQFT hypothesis and the cobordism hypothesis together, we obtain: +– {: .un_prop}

   The primacy of the point

   An $n$-dimensional unitary extended TQFT is completely described by the $n$-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)

   Around 2009, 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:

   • Jacob Lurie, TQFT and the cobordism hypothesis, videos of 4 lectures at the Geometry Research Group, Mathematics Department, University of Texas Austin.

   Lecture notes for Lurie’s talks are available at the Geometry Research Group website.

   diff, v24, current

4. • CommentRowNumber4.
   • CommentAuthorChristoph Dorn
   • CommentTimeJun 1st 2023
   • PermaLink
   Author: Christoph Dorn
   Format: MarkdownItex* added the case of tangles with singularities (aka. stratified tangles) * added "directed variant" <a href="https://ncatlab.org/nlab/revision/diff/tangle+hypothesis/27">diff</a>, <a href="https://ncatlab.org/nlab/revision/tangle+hypothesis/27">v27</a>, <a href="https://ncatlab.org/nlab/show/tangle+hypothesis">current</a>

   • added the case of tangles with singularities (aka. stratified tangles)
   • added “directed variant”

   diff, v27, current