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More Information: <a href="http://lussumo.com/docs">Documentation</a>, <a href="http://lussumo.com/community">Community Support</a>.</p></div> <div id="Content"><div id="NoticeCollector" class="Notices"><div class="Notice"><strong>Welcome to nForum</strong> <br />If you want to take part in these discussions either <a href="/people.php?ReturnUrl=http%3A%2F%2Fnforum.ncatlab.org%2Fdiscussion%2F13677%2F">sign in now</a> (if you have an account), <a href="https://nforum.ncatlab.org/people/?PostBackAction=ApplyForm">apply for one now</a> (if you don't).</div></div><div class="ContentInfo Top"> <h1><a href="https://nforum.ncatlab.org/5/"><a href="https://nforum.ncatlab.org/18/">nLab</a> > </a> <a href="https://nforum.ncatlab.org/5/">Latest Changes</a>: tangle hypothesis</h1> <a href="#pgbottom">Bottom of Page</a> <div class="PageInfo"> <p>1 to 4 of 4</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div> <div id="ContentBody"> <script type="text/javascript"> //<![CDATA[ function toggle_source(id) { var mysrc = document.getElementById("CommentBody_" + id).firstChild; if (mysrc.className == "source") { if (mysrc.style.display == "none") { mysrc.style.display = "block"; } else { mysrc.style.display = "none"; } } } var commentIds = new Array(0); function hide_sources() { for (i = 0; i < commentIds.length; i++) { var myself = document.getElementById("Source" + commentIds[i]); var mycmt = document.getElementById("CommentBody_" + commentIds[i]); if (mycmt.firstChild.className != "source") { myself.style.display = "none"; } } } window.onload = hide_sources; //]]> </script> <ol id="Comments"><li id="Comment_96573"> <a id="Item_1"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>1.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/20/">David_Corfield</a></li> <li><span>CommentTime</span>Nov 9th 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/13677/tangle-hypothesis/?Focus=96573#Comment_96573">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_96573"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/20/">David_Corfield</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Corrected 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></code></div><div> <p>Corrected the reference to where Lurie shows that the tangle hypothesis may be deduced from the cobordism hypothesis.</p> <p><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></p> </div> </div> </li><li id="Comment_96600" class="Alternate"> <a id="Item_2"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>2.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/356/">Dmitri Pavlov</a></li> <li><span>CommentTime</span>Nov 10th 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/13677/tangle-hypothesis/?Focus=96600#Comment_96600">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_96600"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/356/">Dmitri Pavlov</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>Renamed 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></code></div><div> <p>Renamed to the name actually used in the literature. Added redirects and pointers to other entries. Various corrections.</p> <p><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></p> </div> </div> </li><li id="Comment_96602"> <a id="Item_3"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>3.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/20/">David_Corfield</a></li> <li><span>CommentTime</span>Nov 10th 2021</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/13677/tangle-hypothesis/?Focus=96602#Comment_96602">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_96602"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/20/">David_Corfield</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>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 [[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></code></div><div> <p>Much of the content of the page seemed to derive from very early edits and concerns the <a href="https://ncatlab.org/nlab/show/cobordism hypothesis">cobordism hypothesis</a> and so ought to be there. I’ve edited heavily and removed the following which is covered already at <a href="https://ncatlab.org/nlab/show/cobordism hypothesis">cobordism hypothesis</a>.</p> <p>In <a href="https://ncatlab.org/nlab/show/FQFT">extended topological quantum field theory</a>, which is really the <a href="https://ncatlab.org/nlab/show/representation theory">representation theory</a> of these cobordism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-categories, we expect: +– {: .un_prop}</p> <h6>Extended TQFT Hypothesis</h6> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional unitary extended TQFT is a weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a href="https://ncatlab.org/nlab/show/n-functor">functor</a>, preserving all levels of duality, from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>Cob</mi></mrow><annotation encoding="application/x-tex">n Cob</annotation></semantics></math> of cobordisms to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mi>Hilb</mi></mrow><annotation encoding="application/x-tex">n Hilb</annotation></semantics></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a href="https://ncatlab.org/nlab/show/n-Hilbert space">Hilbert spaces</a>. =–</p> <p>Putting the extended TQFT hypothesis and the cobordism hypothesis together, we obtain: +– {: .un_prop}</p> <h6>The primacy of the point</h6> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional unitary extended TQFT is completely described by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-Hilbert space it assigns to a point. =–</p> <p>Further discussion can be found here:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Bruce Bartlett">Bruce Bartlett</a>, <em>On unitary 2-representations of finite groups and topological quantum field theory</em>. PhD thesis, Sheffield (2008) (<a href="http://arxiv.org/abs/0901.3975">arXiv</a>)</li> </ul> <p>Around 2009, <a href="https://ncatlab.org/nlab/show/Mike Hopkins">Mike Hopkins</a> and <a href="https://ncatlab.org/nlab/show/Jacob Lurie">Jacob Lurie</a> have claimed (see <a href="http://golem.ph.utexas.edu/category/2008/05/hopkinslurie_on_baezdolan.html">Hopkins-Lurie on Baez-Dolan</a>) to have formalized and proven this hypothesis in the context of <a href="https://ncatlab.org/nlab/show/(n,r)-category">(infinity,n)-categories</a> modeled on <a href="https://ncatlab.org/nlab/show/complete Segal space">complete Segal space</a>s. See:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Jacob Lurie">Jacob Lurie</a>, <em>On the classification of topological field theories</em> (<a href="http://www.math.harvard.edu/~lurie/papers/cobordism.pdf">pdf</a>)</li> </ul> <p>where an <a href="https://ncatlab.org/nlab/show/(infinity,n)-category of cobordisms">(infinity,n)-category of cobordisms</a> is defined and shown to lead to a formalization and proof of the <em>cobordism hypothesis</em>. Lurie explains his work here:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Jacob Lurie">Jacob Lurie</a>, <em>TQFT and the cobordism hypothesis</em>, videos of 4 lectures at the <a href="http://www.ma.utexas.edu/users/plowrey/dev/rtg/perspectives.html">Geometry Research Group</a>, Mathematics Department, University of Texas Austin.</li> </ul> <p>Lecture notes for Lurie’s talks are available at the <a href="http://www.ma.utexas.edu/users/plowrey/dev/rtg/perspectives.html">Geometry Research Group website</a>.</p> <p><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></p> </div> </div> </li><li id="Comment_110216" class="Alternate"> <a id="Item_4"></a> <div class="CommentHeader"> <ul> <li><span>CommentRowNumber</span>4.</li><li><span>CommentAuthor</span><a href="https://nforum.ncatlab.org/account/1858/">Christoph Dorn</a></li> <li><span>CommentTime</span>Jun 1st 2023</li></ul></div> <div class="CommentActions"> <div class="CommentActionsInner"><ul class="CommentActionsList"><li><a href="https://nforum.ncatlab.org/discussion/13677/tangle-hypothesis/?Focus=110216#Comment_110216">PermaLink</a></li></ul></div></div><div class="CommentBody" id="CommentBody_110216"><div class="source" style="display: none;"><span class="sourceType">Author: <a href="https://nforum.ncatlab.org/account/1858/">Christoph Dorn</a></span><br/><span class="sourceType">Format: MarkdownItex</span><code>* 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></code></div><div> <ul> <li>added the case of tangles with singularities (aka. stratified tangles)</li> <li>added “directed variant”</li> </ul> <p><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></p> </div> </div> </li></ol> </div><div class="ContentInfo Middle"> <div class="PageInfo"> <p>1 to 4 of 4</p> <ol class="PageList PageListEmpty"> <li> </li> </ol> </div> </div></div> <a id="pgbottom" > </a> </div> </div></body> </html>