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tangle hypothesis in nLab

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content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="functorial_quantum_field_theory">Functorial quantum field theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></strong></p> <h2 id="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a></p> </li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#statement'>Statement</a></li> <ul> <li><a href='#tangle_hypothesis'>Tangle Hypothesis</a></li> <li><a href='#'><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">Cobordism Hypothesis</a></a></li> </ul> <li><a href='#generalized_tangle_hypothesis'>Generalized tangle hypothesis</a></li> <ul> <li><a href='#tangles_with_structure'>Tangles with structure</a></li> <li><a href='#tangles_with_singularities'>Tangles with singularities</a></li> <li><a href='#directed_variants'>Directed variants</a></li> </ul> <li><a href='#the_tangle_hypothesis_as_a_consequence_of_the_cobordism_hypothesis'>The tangle hypothesis as a consequence of the cobordism hypothesis</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="statement">Statement</h2> <p>The <em>tangle hypothesis</em> (<a href="#BaezDolan95">Baez and Dolan 95</a>) is as follows:</p> <div class="un_prop"> <h6 id="tangle_hypothesis">Tangle Hypothesis</h6> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-category of framed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/tangles">tangles</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n+k</annotation></semantics></math> dimensions is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+k)</annotation></semantics></math>-equivalent to the free <a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a> with duals on one object.</p> </div> <p>The tangle hypothesis may be seen as a refinement of the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> in the sense that the latter arises from the former in the limit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>→</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">k \to \infty</annotation></semantics></math>:</p> <div class="un_prop"> <h6 id=""><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">Cobordism Hypothesis</a></h6> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><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" class="maruku-mathml"><semantics><mrow><mi>n</mi><mi>Cob</mi></mrow><annotation encoding="application/x-tex">n Cob</annotation></semantics></math> of cobordisms is the free stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-category with duals on one object (the point).</p> </div> <h2 id="generalized_tangle_hypothesis">Generalized tangle hypothesis</h2> <h3 id="tangles_with_structure">Tangles with structure</h3> <p>The tangle hypothesis has been generalized to allow certain structures on the tangles.</p> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-tuply monoidal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><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" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structured <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-tangles in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n + k)</annotation></semantics></math>-cube is the <a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>+</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n + k)</annotation></semantics></math>-category with duals of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mi>G</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(M G,Z)</annotation></semantics></math>.</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">M G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Thom+space">Thom space</a> of group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> can be any group equipped with a homomorphism to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">O(k)</annotation></semantics></math>. (<a href="http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c006471">comment</a>)</li> </ul> <h3 id="tangles_with_singularities">Tangles with singularities</h3> <p>The tangle hypothesis can further be generalized to allow for ‘tangles with singularities’ (also called <a class="existingWikiWord" href="/nlab/show/stratified+space">stratified</a> tangles). The idea is analogous to that of <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis#ForCobordismsWithSingularities">cobordisms with singularities</a>. The tangle hypothesis for tangles with singularities includes the case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-structured tangles (which requires choosing generators for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mi>G</mi></mrow><annotation encoding="application/x-tex">M G</annotation></semantics></math>, that then yield a datum of singularity types, see (<a href="#LurieCTFT">Lurie 09, Sec. 4.3</a>)).</p> <p>The generalized tangle hypothesis is closely related to the generalized <a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+collapse+map#ReferencesPontrjaginThomConstruction">Thom-Pontryagin construction</a> which relates homotopy classes of maps from manifolds <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> into CW complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to cobordism classes of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>-stratifications of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. (The relation was discussed on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>Café <a href="https://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c006196">here</a> and <a href="https://golem.ph.utexas.edu/category/2023/03/an_invitation_to_geometric_hig.html">here</a>.)</p> <h3 id="directed_variants">Directed variants</h3> <p><a class="existingWikiWord" href="/nlab/show/manifold+diagram">Manifold diagrams</a> 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.</p> <h2 id="the_tangle_hypothesis_as_a_consequence_of_the_cobordism_hypothesis">The tangle hypothesis as a consequence of the cobordism hypothesis</h2> <p>While the tangle hypothesis and its generalizations may be seen as refinements of the cobordism hypothesis and its generalizations, Lurie shows (<a href="#LurieCTFT">Lurie 09, Sec. 4.4</a>) that the former may be deduced from the latter when expressed in a sufficiently general form (namely, for cobordisms with singularities).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></li> </ul> <h2 id="references">References</h2> <ul> <li id="BaezDolan95"> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a> and <a class="existingWikiWord" href="/nlab/show/James+Dolan">James Dolan</a>, <em>Higher-dimensional Algebra and Topological Quantum Field Theory</em> 1995 (<a href="http://arxiv.org/abs/q-alg/9503002">arXiv</a>)</p> </li> <li id="LurieCTFT"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">On the Classification of Topological Field Theories</a> (<a href="http://arxiv.org/abs/0905.0465">arXiv:0905.0465</a>)</p> </li> <li id="LurieTQFT"> <p><a class="existingWikiWord" href="/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. 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> </li> </ul> <p>For a discussion of the generalized tangle hypothesis see</p> <ul> <li><a href="http://golem.ph.utexas.edu/category/2006/11/this_weeks_finds_in_mathematic_2.html#c006381">n-Category Café</a>, This Week’s Finds in Mathematical Physics (Week 241)</li> <li><a href="https://golem.ph.utexas.edu/category/2023/03/an_invitation_to_geometric_hig.html">n-Category Café</a>, Invitation to geometric higher categories</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 1, 2023 at 16:57:00. 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