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href="/schreiber/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> <p><a class="existingWikiWord" href="/schreiber/show/%E2%88%9E-Wess-Zumino-Witten+theory">∞-Wess-Zumino-Witten theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory+introduction">∞-Chern-Weil theory introduction</a></p> </li> </ul> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+an+%28%E2%88%9E%2C1%29-topos">differential cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>, <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a>, <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></li> </ul> </li> </ul> <h2 id="connection">Connection</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid+valued+differential+forms">∞-Lie algebroid valued differential forms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connection+on+a+principal+%E2%88%9E-bundle">∞-connection on a principal ∞-bundle</a></p> </li> </ul> <h2 id="curvature">Curvature</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bianchi+identity">Bianchi identity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+derivative">covariant derivative</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+form">Chern-Simons form</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Gauss-Bonnet+theorem">Chern-Gauss-Bonnet theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/infinity-Chern-Weil+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="differential_cohomology">Differential cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> </ul> <h2 id="connections_on_bundles">Connections on bundles</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/curvature">curvature</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/curvature+characteristic+form">curvature characteristic form</a>, <a class="existingWikiWord" href="/nlab/show/Chern+character">Chern character</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+homomorphism">Chern-Weil homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/secondary+characteristic+class">secondary characteristic class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characteristic+class">differential characteristic class</a></p> </li> </ul> </li> </ul> <h2 id="higher_abelian_differential_cohomology">Higher abelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+function+complex">differential function complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+orientation">differential orientation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+Thom+class">differential Thom class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+characters">differential characters</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe with connection</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+elliptic+cohomology">differential elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> <h2 id="higher_nonabelian_differential_cohomology">Higher nonabelian differential cohomology</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">connection on a 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory+in+Smooth%E2%88%9EGrpd">Chern-Weil theory in Smooth∞Grpd</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+cohomology">∞-Lie algebra cohomology</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Simons+theory">∞-Chern-Simons theory</a></p> </li> </ul> <h2 id="fiber_integration">Fiber integration</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+K-theory">fiber integration in differential K-theory</a></p> </li> </ul> </li> </ul> <h2 id="application_to_gauge_theory">Application to gauge theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>/<a class="existingWikiWord" href="/nlab/show/B-field">B-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/RR-field">RR-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">supergravity</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/differential+cohomology+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#higher_parallel_transport'>Higher parallel transport</a></li> <li><a href='#higher_holonomy'>Higher holonomy</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#for_trivial_circle_bundles__for_forms'>For trivial circle <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>-bundles / for <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>-forms</a></li> <li><a href='#for_circle_bundles_with_connection'>For circle <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>-bundles with connection</a></li> <li><a href='#nonabelian_parallel_transport_in_low_dimension'>Nonabelian parallel transport in low dimension</a></li> <ul> <li><a href='#1transport'>1-Transport</a></li> <li><a href='#2transport'>2-Transport</a></li> <li><a href='#3transport'>3-Transport</a></li> </ul> <li><a href='#FlatInTop'>Flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-parallel transport in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math></a></li> <li><a href='#InLieGrpd'>Flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-parallel transport in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>LieGrpd</mi></mrow><annotation encoding="application/x-tex">\infty LieGrpd</annotation></semantics></math></a></li> <li><a href='#parallel_transport_from_flat_differential_forms_with_values_in_chain_complexes'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Parallel transport from flat differential forms with values in chain complexes</a></li> </ul> <li><a href='#applications'>Applications</a></li> <ul> <li><a href='#in_physics'>In physics</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a> induces a notion of <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a> over <em>paths</em> . A <a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">connection on a 2-bundle</a> induces a generalization of this to a notion of parallel transport over <em>surfaces</em> . Similarly a <a class="existingWikiWord" href="/nlab/show/connection+on+a+3-bundle">connection on a 3-bundle</a> induces a notion of parallel transport over 3-dimensional volumes.</p> <p>Generally, a <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a> induces a notion of parallel transport in arbitrary dimension.</p> <h2 id="definition">Definition</h2> <p>The higher notions of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> and <a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a> make sense in any <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mi>Disc</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>Disc</mi></mover></mover><mover><mo>→</mo><mi>Π</mi></mover></mover><mn>∞</mn><mi>Grpd</mi><mo>≃</mo><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Pi \dashv Disc \dashv \Gamma) : \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{Disc}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \simeq Top \,. </annotation></semantics></math></div> <p>In every such there is a notion of <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a> and of its higher parallel transport.</p> <p>A typical context considered (more or less explicitly) in the literature is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, the cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">smooth ∞-groupoids</a>. But other choices are possible. (See also the <a href="#Examples">Examples</a>.)</p> <h3 id="higher_parallel_transport">Higher parallel transport</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a> such that morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">X \to A</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> classify the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s under consideration. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">A_{conn}</annotation></semantics></math> for the differential refinement described at <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra+valued+form">∞-Lie algebra valued form</a>, such that lifts</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∇</mo></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && A_{conn} \\ & {}^{\mathllap{\nabla}}\nearrow & \downarrow \\ X &\stackrel{g}{\to}& A } </annotation></semantics></math></div> <p>describe <a class="existingWikiWord" href="/nlab/show/connections+on+%E2%88%9E-bundles">connections on ∞-bundles</a>.</p> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> <strong>admits parallel <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>-transport</strong> if for all <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <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> and all morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \phi : \Sigma \to X </annotation></semantics></math></div> <p>we have that the pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mo>∇</mo><mo>:</mo><mi>Σ</mi><mover><mo>→</mo><mi>ϕ</mi></mover><mi>X</mi><mover><mo>→</mo><mo>∇</mo></mover><msub><mi>A</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \phi^* \nabla : \Sigma \stackrel{\phi}{\to} X \stackrel{\nabla}{\to} A_{conn} </annotation></semantics></math></div> <p><strong>flat</strong> in that it factors through the canonical inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><mi>A</mi><mo>→</mo><msub><mi>A</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{\flat}A \to A_{conn}</annotation></semantics></math>.</p> <p>In other words: if all the lower <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> <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>-forms, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1 \leq k \leq n</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mo>∇</mo></mrow><annotation encoding="application/x-tex">\phi^* \nabla</annotation></semantics></math> vanish (the higher ones vanish automatically for dimensional reasons).</p> </div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mo>♭</mo></mstyle><mi>A</mi><mo>=</mo><mo stretchy="false">[</mo><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{\flat}A = [\mathbf{\Pi}(-),A]</annotation></semantics></math> is the coefficient for <a class="existingWikiWord" href="/schreiber/show/path+%E2%88%9E-groupoid">flat differential A-cohomology</a>.</p> <div class="un_defn"> <h6 id="remark">Remark</h6> <p>This condition is automatically satisfied for ordinary <a class="existingWikiWord" href="/nlab/show/connections+on+bundles">connections on bundles</a>, hence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">A = \mathbf{B}G</annotation></semantics></math> with <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> an ordinary <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>: because in that case there is only a single curvature form, namely the ordinary <a class="existingWikiWord" href="/nlab/show/curvature">curvature</a> 2-form.</p> <p>But for a <a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a> with connection there is in general a 2-form curvature and a 3-form curvature. A 2-connection therefore admits parallel transport only if its 2-form curvature is constrained to vanish.</p> <p>Notice however that if the coefficient object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> happens to be <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><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/connected">connected</a> – for instance if it is an <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+object">Eilenberg-MacLane object</a> in degree <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>, then there is no extra condition and <em>every</em> connection admits parallel transport. This is notably the case for <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a>.</p> </div> <div class="un_defn"> <h6 id="definition_3">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>:</mo><mi>X</mi><mo>→</mo><msub><mi>A</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla : X \to A_{conn}</annotation></semantics></math> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-connection that admits parallel <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>-transport, this is for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi : \Sigma \to X</annotation></semantics></math> the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> \mathbf{\Pi}(\Sigma) \to A </annotation></semantics></math></div> <p>that corresponds to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mo>∇</mo></mrow><annotation encoding="application/x-tex">\phi^* \nabla</annotation></semantics></math> under the equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mstyle mathvariant="bold"><mo>♭</mo></mstyle><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(\Sigma, \mathbf{\flat}A ) \simeq \mathbf{H}(\mathbf{\Pi}(\Sigma), A) \,. </annotation></semantics></math></div></div> <div class="un_defn"> <h6 id="remark_2">Remark</h6> <p>The objects of the <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#Paths">path ∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}(\Sigma)</annotation></semantics></math> are points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, the morphisms are paths in there, the 2-morphisms surfaces between these paths, and so on. Hence a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}(\Sigma) \to A</annotation></semantics></math> assigns fibers in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to points in <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>, and equivalences between these fibers to paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, and so on.</p> </div> <h3 id="higher_holonomy">Higher holonomy</h3> <p>We now define the higher <a class="existingWikiWord" href="/nlab/show/analogs">analogs</a> of <a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a> for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is closed.</p> <div class="un_defn"> <h6 id="definition_4">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>:</mo><mi>X</mi><mo>→</mo><msub><mi>A</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla : X \to A_{conn}</annotation></semantics></math> be a connection with parallel <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>-transport and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi : \Sigma \to X</annotation></semantics></math> a morphism from a <em>closed</em> <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/manifold">manifold</a>.</p> <p>Then the <strong><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>-holonomy</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is the image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mo>∇</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\phi^* \nabla]</annotation></semantics></math> of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mo>∇</mo><mo>:</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi^* \nabla : \Pi(\Sigma) \to \Gamma(A) </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mo>∇</mo><mo stretchy="false">]</mo><mo>∈</mo><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [\phi^* \nabla] \in [\Pi(\Sigma), \Gamma(A)] </annotation></semantics></math></div></div> <h2 id="Examples">Examples</h2> <h3 id="for_trivial_circle_bundles__for_forms">For trivial circle <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>-bundles / for <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>-forms</h3> <p>The simplest example is the parallel transport in a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a> over a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <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> whose underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^{n-1}U(1)</annotation></semantics></math>-bundle is trivial. This is equivalently given by a degree <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/differential+form">differential form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in \Omega^n(X)</annotation></semantics></math>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msub><mi>Σ</mi> <mi>n</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi : \Sigma_n \to X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> from an <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>-dimensional manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, the corresponding parallel transport is simply the <a class="existingWikiWord" href="/nlab/show/integral">integral</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">tra</mo> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><msub><mo>∫</mo> <mi>Σ</mi></msub><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>A</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tra_A(\Sigma) = \exp(i \int_\Sigma \phi^* A) \;\;\; \in \;\; U(1) \,. </annotation></semantics></math></div> <p>One can understand higher parallel transport therefore as a generalization of integration of differential <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>-forms to the case where</p> <ul> <li> <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>-form is not globally defined;</p> </li> <li> <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>-form takes values not in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> but more generally is an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid+valued+differential+form">∞-Lie algebroid valued differential form</a>.</p> </li> </ul> <h3 id="for_circle_bundles_with_connection">For circle <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>-bundles with connection</h3> <p>We show how 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>-holonomy of <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a> is reproduced from the above.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mo>∇</mo><mo>:</mo><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi^* \nabla : \mathbf{\Pi}(\Sigma) \to \mathbf{B}^n U(1)</annotation></semantics></math> be the parallel transport for a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a> over a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi : \Sigma \to X</annotation></semantics></math>.</p> <p>This is equivalent to a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>ℬ</mi> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>,</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi(\Sigma) \to \mathcal{B}^n U(1) ,. </annotation></semantics></math></div> <p>We may map this further to its <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>dim</mi><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n-dim \Sigma)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/truncated">truncation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>:</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>ℬ</mi> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>τ</mi> <mrow><mi>n</mi><mo>−</mo><mi>dim</mi><mi>Σ</mi></mrow></msub><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>ℬ</mi> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> :\infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) \to \tau_{n-dim \Sigma} \infty Grpd(\Pi(X), \mathcal{B}^n U(1)) \,. </annotation></semantics></math></div> <div class="un_theorem"> <h6 id="theorem">Theorem</h6> <p>We have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mi>n</mi><mo>−</mo><mi>dim</mi><mi>Σ</mi></mrow></msub><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>ℬ</mi> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mi>dim</mi><mi>Σ</mi></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tau_{n-dim\Sigma} \infty Grpd(\Pi(\Sigma), \mathcal{B}^n U(1)) \simeq \mathbf{B}^{n-dim \Sigma} U(1) \,. </annotation></semantics></math></div></div> <p>(This is due to an observation by <a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>.)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>By general abstract reasoning (recalled at <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> and <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a>) we have for the <a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a>s that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>i</mi></msub><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>ℬ</mi> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_i \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1)) \simeq H^{n-i}(\Sigma, U(1)) \,. </annotation></semantics></math></div> <p>Now use the <a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a>, which asserts that we have an <a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>Hom</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 0 \to Ext^1(H_{n-i-1}(\Sigma,\mathbb{Z}),U(1)) \to H^{n-i}(\Sigma,U(1)) \to Hom(H_{n-i}(\Sigma,\mathbb{Z}),U(1)) \to 0 \,. </annotation></semantics></math></div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/injective+object">injective</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ext</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ext^1(-,U(1))=0 \,. </annotation></semantics></math></div> <p>This means that we have an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>Ab</mi></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>,</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> H^{n-i}(\Sigma,U(1)) \simeq Hom_{Ab}(H_{n-i}(\Sigma,\mathbb{Z}),U(1)) </annotation></semantics></math></div> <p>that identifies the <a class="existingWikiWord" href="/nlab/show/cohomology+group">cohomology group</a> in question with the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> from the integral <a class="existingWikiWord" href="/nlab/show/homology">homology</a> group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo><</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>dim</mi><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i\lt (n-dim \Sigma)</annotation></semantics></math>, the right hand is zero, so that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mi>i</mi></msub><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>for</mi><mi>i</mi><mo><</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>dim</mi><mi>Σ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_i \infty Grpd(\Pi(\Sigma),\mathbf{B}^n U(1)) =0 \;\;\;\; for i\lt (n-dim \Sigma) \,. </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>=</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>dim</mi><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i=(n-dim \Sigma)</annotation></semantics></math>, instead, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H_{n-i}(\Sigma,\mathbb{Z})\simeq \mathbb{Z}</annotation></semantics></math>, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is a closed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mi>Σ</mi></mrow><annotation encoding="application/x-tex">dim \Sigma</annotation></semantics></math>-manifold and so</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>dim</mi><mi>Σ</mi><mo stretchy="false">)</mo></mrow></msub><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><msup><mi>ℬ</mi> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_{(n-dim\Sigma)} \infty Grpd(\Pi(\Sigma),\mathcal{B}^n U(1))\simeq U(1) \,. </annotation></semantics></math></div></div> <div class="un_def"> <h6 id="definition_5">Definition</h6> <p>The resulting morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><msub><mi>A</mi> <mi>conn</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mi>dim</mi><mi>Σ</mi></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(\Sigma, A_{conn}) \stackrel{\exp(i S(-))}{\to} \mathbf{B}^{n-dim\Sigma} U(1) </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> we call the <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Chern-Simons action</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>.</p> </div> <p>Here in the language of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/object">object</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><msub><mi>A</mi> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(\Sigma,A_{conn})</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/gauge+field">gauge field</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><msub><mi>A</mi> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(\Sigma, A_{conn})</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>s;.</p> </li> </ul> <h3 id="nonabelian_parallel_transport_in_low_dimension">Nonabelian parallel transport in low dimension</h3> <p>At least in low categorical dimension one has the definition of the <a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{P}_n(X)</annotation></semantics></math> of a smooth manifold, whose <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>-morphisms are <a class="existingWikiWord" href="/nlab/show/thin+homotopy">thin homotopy</a>-classes of smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">[0,1]^n \to X</annotation></semantics></math>. Parallel <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>-transport with only 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><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n+1)</annotation></semantics></math>-curvature form possibly nontrivial and all the lower curvature degree 1- to <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>-forms nontrivial may be expressed in terms of smooth <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>-functors out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{P}_n</annotation></semantics></math> (<a href="#SWI">SWI</a>, <a href="#SWII">SWII</a>, <a href="#MartinsPickenI">MartinsPickenI</a>, <a href="#MartinsPickenII">MartinsPickenII</a>).</p> <h4 id="1transport">1-Transport</h4> <p>See <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>.</p> <h4 id="2transport">2-Transport</h4> <p>We work now concretely in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>DiffeoGrpd</mi></mrow><annotation encoding="application/x-tex">2DiffeoGrpd</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>s <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> the category of <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>s.</p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mn>2</mn><mi>DiffeoGrpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{P}_2(X) \in 2DiffeoGrpd</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/path+2-groupoid">path 2-groupoid</a>. Let <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> be a <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>∈</mo><mn>2</mn><mi>DiffeoGrpd</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \in 2DiffeoGrpd</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> 1-object 2-groupoid. Write <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">\mathfrak{g}</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a>.</p> <p>Assume now first that <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> is a <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a> given by a <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>G</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>G</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(G_1 \to G_0)</annotation></semantics></math>. Corresponding to this is a <a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>𝔤</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝔤</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{g}_1 \to \mathfrak{g}_0)</annotation></semantics></math>.</p> <p>We describe now how smooth 2-functors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>tra</mi><mo>:</mo><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> tra : \mathbf{P}_2(X) \to \mathbf{B}G </annotation></semantics></math></div> <p>i.e. morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>DiffeoGrpd</mi></mrow><annotation encoding="application/x-tex">2DiffeoGrpd</annotation></semantics></math> are characterized by <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra+valued+differential+forms">Lie 2-algebra valued differential forms</a> on <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>.</p> <div class="un_defn"> <h6 id="definition_6">Definition</h6> <p>Given a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">F : \mathbf{P}_2(X) \to \mathbf{B}G</annotation></semantics></math> we construct a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔤</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathfrak{g}_1</annotation></semantics></math>-valued 2-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>F</mi></msub><mo>∈</mo><msup><mi>Ω</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝔤</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_F \in \Omega^2(X, \mathfrak{g}_1)</annotation></semantics></math> as follows.</p> <p>To find the value of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>F</mi></msub></mrow><annotation encoding="application/x-tex">B_F</annotation></semantics></math> on two vectors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>T</mi> <mi>p</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">v_1, v_2 \in T_p X</annotation></semantics></math> at some point, <em>choose</em> any smooth function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>:</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \Gamma : \mathbb{R}^2 \to X </annotation></semantics></math></div> <p>with</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi></mrow><annotation encoding="application/x-tex">\Gamma(0,0) = p</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>s</mi></mrow></mfrac><msub><mo stretchy="false">|</mo> <mrow><mi>s</mi><mo>=</mo><mn>0</mn></mrow></msub><mi>Γ</mi><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>v</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\frac{d}{d s}|_{s = 0} \Gamma(s,0) = v_1</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><msub><mo stretchy="false">|</mo> <mrow><mi>t</mi><mo>=</mo><mn>0</mn></mrow></msub><mi>Γ</mi><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>v</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\frac{d}{d t}|_{t = 0} \Gamma(0,t) = v_2</annotation></semantics></math>.</p> </li> </ul> <p>Notice that there is a canonical 2-parameter family</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>ℝ</mi></msub><mo>:</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>→</mo><mn>2</mn><mi>Mor</mi><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Sigma_{\mathbb{R}} : \mathbb{R}^2 \to 2Mor \mathbf{P}_2(\mathbb{R}^2) </annotation></semantics></math></div> <p>of classes of bigons on the plane, given by sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">(s,t) \in \mathbb{R}^2</annotation></semantics></math> to the class represented by any bigon (with sitting instants) with straight edges filling the square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>ℝ</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Sigma_{\mathbb{R}}(s,t) = \left( \array{ (0,0) &\to& (0,t) \\ \downarrow && \downarrow \\ (s,0) &\to& (s,t) } \right) \,. </annotation></semantics></math></div> <p>Using this we obtain a smooth function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>Γ</mi></msub><mo>:</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mover><mo>→</mo><mrow><msub><mi>Σ</mi> <mi>ℝ</mi></msub></mrow></mover><mn>2</mn><mi>Mor</mi><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>Γ</mi> <mo>*</mo></msub></mrow></mover><mn>2</mn><mi>Mor</mi><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>F</mi></mover><msub><mi>G</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover><msub><mi>G</mi> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F_\Gamma : \mathbb{R}^2 \stackrel{\Sigma_{\mathbb{R}}}{\to} 2Mor \mathbf{P}_2(\mathbb{R}^2) \stackrel{\Gamma_*}{\to} 2Mor \mathbf{P}_2(X) \stackrel{F}{\to} G_0 \times G_1 \stackrel{p_2}{\to} G_1 \,. </annotation></semantics></math></div> <p>Then set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>w</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>F</mi> <mi>Γ</mi></msub></mrow><mrow><mo>∂</mo><mi>x</mi><mo>∂</mo><mi>y</mi></mrow></mfrac><msub><mo stretchy="false">|</mo> <mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> B_F(v_1, w_1) := \frac{\partial^2 F_\Gamma}{\partial x \partial y}|_{(0,0)} \,. </annotation></semantics></math></div></div> <div class="un_prop"> <h6 id="proposition">Proposition</h6> <p>This is well defined, in that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B_F(v_1,v_2)</annotation></semantics></math> does not depend on the choices made. Moreover, the 2-form defines this way is smooth.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>To see that the definition does not depend on the choice of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>, proceed as follows.</p> <p>For given vectors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>∈</mo><msub><mo lspace="0em" rspace="thinmathspace">T</mo> <mi>X</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">v_1,v_2 \in \T_X X</annotation></semantics></math> let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>,</mo><mi>Γ</mi><mo>′</mo><mo>:</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Gamma, \Gamma' : \mathbb{R}^2 \to X</annotation></semantics></math> be two choices of smooth maps as in the defnition. By restricting, if necessary, to a neighbourhood of the origin of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math>, we may assume without restriction that these maps land in a single coordinate patch in <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>. Using the vector space structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> defined by such a patch, define a smooth homotopy</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>τ</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>3</mn></msup><mo>→</mo><mi>X</mi><mo>:</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>z</mi><mo stretchy="false">)</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mi>z</mi><mi>Γ</mi><mo>′</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tau : [0,1]^3 \to X : (x,y,z) \mapsto (1-z)\Gamma(x,y) + z \Gamma'(x,y) </annotation></semantics></math></div> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>3</mn></msup><mo stretchy="false">|</mo><mn>0</mn><mo>≤</mo><mi>w</mi><mo>≤</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> Z = \{(x,y,w) \in [0,1]^3 | 0 \leq w \leq \frac{1}{2}(x^2 + y^2) \} </annotation></semantics></math></div> <p>and consider the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>3</mn></msup><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">f : [0,1]^3 \to Z</annotation></semantics></math> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f : (x,y,z) \mapsto (x,y, \frac{1}{2}(x^2 + y^2) z) </annotation></semantics></math></div> <p>and the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">g : Z \to X</annotation></semantics></math> given away from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">(x^2 + y^2) = 0</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>w</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mn>2</mn><mfrac><mi>w</mi><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup></mrow></mfrac><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g : (x,y,w) \mapsto \tau(x,y, 2 \frac{w}{x^2 + y^2}) \,. </annotation></semantics></math></div> <p>Using <a class="existingWikiWord" href="/nlab/show/Hadamard%27s+lemma">Hadamard's lemma</a> and the fact that by constructon <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> has vanishing 0th and 1st order differentials at the origin it follows that this is indeed a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>.</p> <p>We want to similarly factor the smooth family of bigons <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>3</mn></msup><mo>→</mo><mn>2</mn><mi>Mor</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[0,1]^3 \to 2Mor(\mathbf{P}_2(X))</annotation></semantics></math> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>3</mn></msup><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>2</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> [0,1]^3 \times [0,1]^2 \to X </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↦</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>s</mi><mi>x</mi><mo>,</mo><mi>t</mi><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ((x,y,z),(s,t)) \mapsto \tau(s x, t y, z) </annotation></semantics></math></div> <p>as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>3</mn></msup><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>2</mn></msup><mo>→</mo><mi>Z</mi><mo>×</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>2</mn></msup><mo>→</mo><mi>Z</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">[0,1]^3 \times [0,1]^2 \to Z \times [0,1]^2 \to Z \to X</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>y</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mi>s</mi><mi>x</mi><mo>,</mo><mi>t</mi><mi>y</mi><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>s</mi><mi>x</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>+</mo><mo stretchy="false">(</mo><mi>t</mi><mi>y</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo stretchy="false">)</mo><mi>z</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>τ</mi><mo stretchy="false">(</mo><mi>s</mi><mi>x</mi><mo>,</mo><mi>s</mi><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ((x,y,z),(s,t)) \mapsto ((x, y, \frac{1}{2}(x^2 + y^2)), (s,t)) \mapsto (s x , t y, \frac{1}{2}((s x)^2 + (t y)^2)z) \mapsto \tau(s x, s y, z) \,. </annotation></semantics></math></div> <p>As before using Hadamard’s lemma this is a sequence of smooth functions. To make this qualify as a family of bigons (which are maps from the square that are constant in a <a class="existingWikiWord" href="/nlab/show/neighbourhood">neighbourhood</a> of the left and right boundary of the square) furthermore precompose this with a suitable smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>2</mn></msup><mo>→</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">[0,1]^2 \to [0,1]^2</annotation></semantics></math> that realizes a square-shaped bigon.</p> <p>Under the hom-adjunction it corresponds to a factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>Γ</mi></msub><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>3</mn></msup><mo>→</mo><mn>2</mn><mi>Mor</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G_\Gamma : [0,1]^3 \to 2 Mor(\mathbf{P}_2(X))</annotation></semantics></math> into</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>Γ</mi></msub><mo>:</mo><mo stretchy="false">[</mo><mn>0</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">]</mo> <mn>3</mn></msup><mover><mo>→</mo><mi>f</mi></mover><mi>Z</mi><mo>→</mo><mn>2</mn><mi>Mor</mi><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G_\Gamma : [0,1]^3 \stackrel{f}{\to} Z \to 2 Mor(\mathbf{P}_2(X)) \,. </annotation></semantics></math></div> <p>By the above construction we have the the push-forwards</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mo>:</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>↦</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>w</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f_* : \frac{\partial}{\partial x}(x=0,y=0,z) \mapsto \frac{\partial}{\partial x}(x= 0, y = 0, w = 0) </annotation></semantics></math></div> <p>and similarly for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial}{\partial y}</annotation></semantics></math> are indendent of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math>. It follows by the <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a> that also</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>G</mi> <mi>Γ</mi></msub></mrow><mrow><mo>∂</mo><mi>x</mi><mo>∂</mo><mi>y</mi></mrow></mfrac><msub><mo stretchy="false">|</mo> <mrow><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> \frac{\partial^2 G_\Gamma}{\partial x \partial y}|_{(x=0,y=0)} </annotation></semantics></math></div> <p>is independent of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math>. But at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">z = 0</annotation></semantics></math> this equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>F</mi> <mi>Γ</mi></msub></mrow><mrow><mo>∂</mo><mi>x</mi><mo>∂</mo><mi>y</mi></mrow></mfrac><msub><mo stretchy="false">|</mo> <mrow><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\frac{\partial^2 F_\Gamma}{\partial x \partial y}|_{(x=0,y=0)}</annotation></semantics></math>, while at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">z = 1</annotation></semantics></math> it equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><msub><mi>F</mi> <mrow><mi>Γ</mi><mo>′</mo></mrow></msub></mrow><mrow><mo>∂</mo><mi>x</mi><mo>∂</mo><mi>y</mi></mrow></mfrac><msub><mo stretchy="false">|</mo> <mrow><mo stretchy="false">(</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\frac{\partial^2 F_{\Gamma'}}{\partial x \partial y}|_{(x=0,y=0)}</annotation></semantics></math>. Therefore these two are equal.</p> </div> <h4 id="3transport">3-Transport</h4> <p>see <a class="existingWikiWord" href="/nlab/show/3-groupoid+of+Lie+3-algebra+valued+forms">3-groupoid of Lie 3-algebra valued forms</a></p> <h3 id="FlatInTop">Flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-parallel transport in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math></h3> <p>Even though it is a degenerate case, it can be useful to regard the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> explicitly a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a>. For a discussion of this see <a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Top">Top</a> lots of structure of cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos theory degenerates, since by the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem here the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Π</mi><mo>⊣</mo><mi>Δ</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mover><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>Δ</mi></mover></mover><mover><mo>←</mo><mi>Π</mi></mover></mover><mo>∈</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> (\Pi \dashv \Delta \dashv \Gamma) : Top \stackrel{\overset{\Pi}{\leftarrow}}{\stackrel{\overset{\Delta}{\leftarrow}}{\underset{\Gamma}{\to}}} \in \infty Grpd </annotation></semantics></math></div> <p>an <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equivalence</a>. The abstract <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> is here the ordinary <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo>:</mo><mi>Top</mi><mover><mo>→</mo><mo>≃</mo></mover><mn>∞</mn><mi>Grpd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi : Top \stackrel{\simeq}{\to} \infty Grpd \,. </annotation></semantics></math></div> <p>If both <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es here are <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by their standard <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> models, the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a> and the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">model structure on topological spaces</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> is presented by the <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> functor in a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mo>−</mo><mo stretchy="false">|</mo><mo>⊣</mo><mi>Sing</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Top</mi><mover><mover><mo>→</mo><mrow><msub><mo>≃</mo> <mi>Quillen</mi></msub></mrow></mover><mo>←</mo></mover><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (|-| \dashv Sing) : Top \stackrel{\leftarrow}{\overset{\simeq_{Quillen}}{\to}} Top \,. </annotation></semantics></math></div> <p>This means that in this case many constructions in <a class="existingWikiWord" href="/nlab/show/topology">topology</a> and classical <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> have equivalent reformulations in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-parallel transport.</p> <p>For instance: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">F \in Top</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">Aut(F) \in Top</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-fibrations over a base space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">X \in Top</annotation></semantics></math> are classfied by morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>B</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> g : X \to B Aut(F) </annotation></semantics></math></div> <p>into the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(F)</annotation></semantics></math>. The corresponding fibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> itself is the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber">homotopy fiber</a> of this cocycles, given by the <a class="existingWikiWord" href="/nlab/show/homotopy+pullback">homotopy pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &\to& * \\ \downarrow && \downarrow \\ X &\stackrel{g}{\to}& B Aut(F) } </annotation></semantics></math></div> <p>in <a class="existingWikiWord" href="/nlab/show/Top">Top</a>, as described at <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>.</p> <p>Using the <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a> functor we may send this equivalently to a <a class="existingWikiWord" href="/nlab/show/fiber+sequence">fiber sequence</a> in <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi(P) \to \Pi(X) \to B Aut(\Pi(F)) \,. </annotation></semantics></math></div> <p>One may think of the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>B</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(X) \to B Aut(\Pi(F))</annotation></semantics></math> now as the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-parallel transport coresponding to the original fibration:</p> <ul> <li> <p>to each point in <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> it assigns the unique object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B Aut(\Pi(F))</annotation></semantics></math>, which is the fiber <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> itself;</p> </li> <li> <p>to each path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>→</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x \to y)</annotation></semantics></math> in <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> it assigns an equivalence between the fibers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>x</mi></msub><mi>to</mi><msub><mi>F</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">F_x to F_y</annotation></semantics></math> etc.</p> </li> </ul> <p>If one presents <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo>:</mo><mi>Top</mi><mo>→</mo><msub><mi>sSet</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex">Sing : Top \to sSet_{Quillen}</annotation></semantics></math> as above, then one may look for explicit simplicial formulas that express these morphisms. Such are discussed in <a href="#Stasheff">Stasheff</a>.</p> <p>We may embed this example into the smooth context by regarding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(F)</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/discrete+space">discrete</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a> as discussed in the section <a href="#InLieGrpd">Flat ∞-Parallel transport in Smooth∞Grpd</a>.</p> <p>For that purpose let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>Π</mi> <mi>smooth</mi></msub><mo>⊣</mo><msub><mi>Disc</mi> <mi>smooth</mi></msub><mo>⊣</mo><msub><mi>Γ</mi> <mi>smooth</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mn>∞</mn><mi>LieGrpd</mi><mover><mover><munder><mo>→</mo><mrow><msub><mi>Γ</mi> <mi>smooth</mi></msub></mrow></munder><mover><mo>←</mo><mrow><msub><mi>Disc</mi> <mi>smooth</mi></msub></mrow></mover></mover><mover><mo>→</mo><mrow><msub><mi>Π</mi> <mi>smooth</mi></msub></mrow></mover></mover><mn>∞</mn><mi>Grpd</mi><mo>≃</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> (\Pi_{smooth} \dashv Disc_{smooth} \dashv \Gamma_{smooth}) : \infty LieGrpd \stackrel{\overset{\Pi_{smooth}}{\to}}{\stackrel{\overset{Disc_{smooth}}{\leftarrow}}{\underset{\Gamma_{smooth}}{\to}}} \infty Grpd \simeq Top </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-geometric+morphism">(∞,1)-geometric morphism</a> of the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>.</p> <p>We may reflect the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(F)</annotation></semantics></math> into this using the <a class="existingWikiWord" href="/nlab/show/constant+%E2%88%9E-stack">constant ∞-stack</a>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi></mrow><annotation encoding="application/x-tex">Disc</annotation></semantics></math> to get the discrete <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Disc Aut(F)</annotation></semantics></math>. Let then <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> be a <a class="existingWikiWord" href="/nlab/show/paracompact+topological+space">paracompact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, regarded naturally as an object of <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>. Then we can consider <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a>s/classifying morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Disc</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \to \mathbf{B} Disc Aut(F) \,, </annotation></semantics></math></div> <p>now in the smooth context of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>LieGrpd</mi></mrow><annotation encoding="application/x-tex">\infty LieGrpd</annotation></semantics></math>.</p> <div class="un_prop"> <h6 id="proposition_2">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-fibrations in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> is equivalent to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Disc Aut(F)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s in <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>LieGrpd</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Disc</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Top</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>B</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \infty LieGrpd(X, \mathbf{B} Disc Aut(F)) \simeq Top(X, B Aut(F)) \,. </annotation></semantics></math></div> <p>Moreover, all the <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>s classified by the morphisms on the left have canonical extensions to <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity%2C1)-topos#FlatDifferentialCohomology">Flat differential cohomology</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>LieGrpd</mi></mrow><annotation encoding="application/x-tex">\infty LieGrpd</annotation></semantics></math>, in that the flat parallel <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-transport <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mi>flat</mi></msub></mrow><annotation encoding="application/x-tex">\nabla_{flat}</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Disc</mi><mi>Aut</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mrow><msub><mo>∇</mo> <mi>flat</mi></msub></mrow></msub></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{g}{\to}& \mathbf{B} Disc Aut(F) \\ \downarrow & \nearrow_{\nabla_{flat}} \\ \mathbf{\Pi}(X) } </annotation></semantics></math></div> <p>always exists.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>The first statement is a special case of that spelled out at <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> and <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a>. The second follows using that in a <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">connected</a> <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a> the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Disc</mi></mrow><annotation encoding="application/x-tex">Disc</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+%28%E2%88%9E%2C1%29-functor">full and faithful (∞,1)-functor</a>.</p> </div> <h3 id="InLieGrpd">Flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-parallel transport in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>LieGrpd</mi></mrow><annotation encoding="application/x-tex">\infty LieGrpd</annotation></semantics></math></h3> <p>(…)</p> <h3 id="parallel_transport_from_flat_differential_forms_with_values_in_chain_complexes"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Parallel transport from flat differential forms with values in chain complexes</h3> <p>A typical choice for an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-vector spaces” is that <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by the a <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a> of modules. In a geometric context this may be replaced by some stack of complexes of vector bundles over some site.</p> <p>If we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi></mrow><annotation encoding="application/x-tex">Mod</annotation></semantics></math> for this stack, then the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-parallel transport for a flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-vector bundle on some <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> is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Pi}(X) \to Mod \,. </annotation></semantics></math></div> <p>This is typically given by differential form data with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi></mrow><annotation encoding="application/x-tex">Mod</annotation></semantics></math>.</p> <p>A discussion of how to integrate flat differential forms with values in chain complexes – a representation of the <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a> as discussed at <a class="existingWikiWord" href="/nlab/show/representations+of+%E2%88%9E-Lie+algebroids">representations of ∞-Lie algebroids</a> – to flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-parallel transport <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}(X) \to Mod</annotation></semantics></math> is in (<a href="#abadSchaetz">AbadSchaetz</a>), building on a construciton in (<a href="#Igusa">Igusa</a>).</p> <h2 id="applications">Applications</h2> <h3 id="in_physics">In physics</h3> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a> various <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>s for <a class="existingWikiWord" href="/nlab/show/quantum+field+theories">quantum field theories</a> are nothing but higher parallel transport.</p> <ul> <li> <p>The gauge interaction part of the action functional for the particle charged under a background <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, which is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle bundle with connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>, is the parallel 1-transport of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>.</p> </li> <li> <p>The gauge interaction part of the action functional for the string charged under a background <a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a>, which is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 2-bundle with connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>, is the parallel 2-transport of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>.</p> </li> <li> <p>The gauge interaction part of the action functional for the membrane charged under a background <a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a>, which is a <a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle 3-bundle with connection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>, is the parallel 3-transport of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>.</p> </li> <li> <p>The action functional of <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> is the parallel 3-transport of a <a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle">Chern-Simons circle 3-bundle</a>.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+a+2-bundle">connection on a 2-bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+infinity-bundle">connection on an infinity-bundle</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <strong>higher parallel transport</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy+group">holonomy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a></p> </li> </ul> <h2 id="References">References</h2> <p>For references on ordinary 1-dimensional parallel transport see <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>.</p> <p>For references on parallel 2-transport in <a class="existingWikiWord" href="/nlab/show/bundle+gerbes">bundle gerbes</a> see <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle+gerbe">connection on a bundle gerbe</a>.</p> <p>The description of parallel <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>-transport in terms of <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>-functors on the <a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a> is discussed</p> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n=2</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/connections+on+2-bundles">connections on 2-bundles</a>) in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em>Higher gauge theory</em>, in A. Davydov et al. (eds.), <em>Categories in Algebra, Geometry and Mathematical Physics</em>, Contemp Math 431, AMS, Providence, Rhode Island (2007) pp 7-30 (<a href="http://arxiv.org/abs/math/0511710">arXiv:0511710</a>, <a href="http://arxiv.org/abs/hep-th/0412325">arXiv:hep-th/0412325</a>)</p> </li> <li id="MartinsPickenI"> <p><a class="existingWikiWord" href="/nlab/show/Jo%C3%A3o+Faria+Martins">João Faria Martins</a>, <a class="existingWikiWord" href="/nlab/show/Roger+Picken">Roger Picken</a>: <em>On 2-dimensional holonomy</em>, Trans. Amer. Math. Soc. <strong>362</strong> (2010), 5657-5695 [<a href="http://arxiv.org/abs/0710.4310">arXiv:0710.4310</a>, <a href="https://doi.org/10.1090/S0002-9947-2010-04857-3">doi:10.1090/S0002-9947-2010-04857-3</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jo%C3%A3o+Faria+Martins">João Faria Martins</a>, <a class="existingWikiWord" href="/nlab/show/Roger+Picken">Roger Picken</a>: <em>Surface holonomy for non-abelian 2-bundles via double groupoids</em>, Advances in Mathematics <strong>226</strong> 4 (2011) 3309-3366 [<a href="https://doi.org/10.1016/j.aim.2010.10.017">doi:10.1016/j.aim.2010.10.017</a>]</p> </li> <li id="SchreiberWaldorf11"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>Smooth Functors and Differential Forms</em>, Homology, Homotopy Appl., <strong>13</strong> 1 (2011) 143-203 [<a href="http://arxiv.org/abs/0802.0663">arXiv:0802.0663</a>, <a href="https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-13/issue-1/Smooth-functors-vs-differential-forms/hha/1311953350.full">hha:1311953350</a>]</p> </li> <li id="SWIII"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>Connections on nonabelian gerbes and their holonomy</em>, Theory Appl. Categ. <strong>28</strong> 17 (2013) 476-540 [<a href="http://arxiv.org/abs/0808.1923">arXiv:0808.1923</a>, <a href="http://www.tac.mta.ca/tac/volumes/28/17/28-17abs.html">tac:28-17</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Amnon+Yekutieli">Amnon Yekutieli</a>: <em>Nonabelian Multiplicative Integration on Surfaces</em>, World Scientific (2015) [<a href="https://arxiv.org/abs/1007.1250">arXiv:1007.1250</a>, <a href="https://doi.org/10.1142/9537">doi:10.1142/9537</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>Parallel transport in principal 2-bundles</em>, Higher Structures <strong>2</strong> 1 (2018) 57-115 [<a href="https://arxiv.org/abs/1704.08542">arXiv:1704.08542</a>, <a href="https://higher-structures.math.cas.cz/api/files/issues/Vol2Iss1/Waldorf">pdf</a>]</p> </li> <li id="KimSaemann20"> <p><a class="existingWikiWord" href="/nlab/show/Hyungrok+Kim">Hyungrok Kim</a>, <a class="existingWikiWord" href="/nlab/show/Christian+Saemann">Christian Saemann</a>, <em>Adjusted Parallel Transport for Higher Gauge Theories</em>, J. Phys. A <strong>52</strong> (2020) 445206 [<a href="https://arxiv.org/abs/1911.06390">arXiv:1911.06390</a>, <a href="https://doi.org/10.1088/1751-8121/ab8ef2">doi:10.1088/1751-8121/ab8ef2</a>]</p> <blockquote> <p>(via <a class="existingWikiWord" href="/nlab/show/adjusted+Weil+algebras">adjusted Weil algebras</a>)</p> </blockquote> </li> <li id="LeeOberhauser23"> <p>Darrick Lee, Harald Oberhauser, <em>Random Surfaces and Higher Algebra</em> [<a href="https://arxiv.org/abs/2311.08366">arXiv:2311.08366</a>]</p> </li> </ul> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n = 3</annotation></semantics></math> in</p> <ul> <li id="MartinsPickenII"><a class="existingWikiWord" href="/nlab/show/Jo%C3%A3o+Faria+Martins">João Faria Martins</a>, <a class="existingWikiWord" href="/nlab/show/Roger+Picken">Roger Picken</a>, <em>The fundamental Gray 3-groupoid of a smooth manifold and local 3-dimensional holonomy based on a 2-crossed module</em>, Differential Geometry and its Applications <strong>29</strong> 2 (2011) 179-206 [<a href="http://arxiv.org/abs/0907.2566">arXiv:0907.2566</a>, <a href="https://doi.org/10.1016/j.difgeo.2010.10.002">doi:10.1016/j.difgeo.2010.10.002</a>]</li> </ul> <p>Further discussion and illustration and relation to <a class="existingWikiWord" href="/nlab/show/tensor+networks">tensor networks</a>:</p> <ul> <li id="Parzygnat18"><a class="existingWikiWord" href="/nlab/show/Arthur+Parzygnat">Arthur Parzygnat</a>, <em>Two-dimensional algebra in lattice gauge theory</em>, Journal of Mathematical Physics 60, 043506 (2019) (<a href="https://arxiv.org/abs/1802.01139">arXiv:1802.01139</a>, <a href="https://doi.org/10.1063/1.5078532">doi:10.1063/1.5078532</a>)</li> </ul> <p>Applications</p> <p>to <a class="existingWikiWord" href="/nlab/show/monopoles">monopoles</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Arthur+Parzygnat">Arthur Parzygnat</a>, <em>Gauge invariant surface holonomy and monopoles</em>, Theory and Applications of Categories <strong>30</strong> 42 (2015) 1319-1428 [<a href="http://www.tac.mta.ca/tac/volumes/30/42/30-42abs.html">tac:30-42</a>, <a href="https://arxiv.org/abs/1410.6938">arXiv:1410.6938</a>]</li> </ul> <p>to <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a> of the <a class="existingWikiWord" href="/nlab/show/Knizhnik-Zamolodchikov+connection">Knizhnik-Zamolodchikov connection</a>:</p> <ul> <li>Lucio Simone Cirio, <a class="existingWikiWord" href="/nlab/show/Jo%C3%A3o+Faria+Martins">João Faria Martins</a>: <em>Categorifying the Knizhnik–Zamolodchikov connection</em>, Differential Geometry and its Applications <strong>30</strong> 3 (2012) 238-261 [<a href="https://doi.org/10.1016/j.difgeo.2012.03.004">doi:10.1016/j.difgeo.2012.03.004</a>, <a href="https://arxiv.org/abs/1106.0042">arXiv:1106.0042</a>]</li> </ul> <p>to <a class="existingWikiWord" href="/nlab/show/BFCG+theory">BFCG theory</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jo%C3%A3o+Faria+Martins">João Faria Martins</a>, <a class="existingWikiWord" href="/nlab/show/Aleksandar+Mikovi%C4%87">Aleksandar Miković</a>: <em>Lie crossed modules and gauge-invariant actions for 2-BF theories</em>, Adv. Theor. Math. Phys. <strong>15</strong> 4 (2011) 913-1199 [<a href="https://arxiv.org/abs/1006.0903">arXiv:1006.0903</a>, <a href="http://projecteuclid.org/euclid.atmp/1339438351">euclid:atmp/1339438351</a>]</p> </li> <li> <p>A.D. López-Hernández, Graciela Reyes-Ahumada, Javier Chagoya: <em>Categorical generalization of BF theory coupled to gravity</em> [<a href="https://arxiv.org/abs/2408.02889">arXiv:2408.02889</a>]</p> </li> </ul> <p>to <a class="existingWikiWord" href="/nlab/show/data+science">data science</a>:</p> <ul> <li>Ilya Chevyrev, <a class="existingWikiWord" href="/nlab/show/Joscha+Diehl">Joscha Diehl</a>, <a class="existingWikiWord" href="/nlab/show/Kurusch+Ebrahimi-Fard">Kurusch Ebrahimi-Fard</a>, Nikolas Tapia: <em>A multiplicative surface signature through its Magnus expansion</em> [<a href="https://arxiv.org/abs/2406.16856">arXiv:2406.16856</a>]</li> </ul> <p>In the abelian case, parallel transport for <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a> is discussed generally in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kiyonori+Gomi">Kiyonori Gomi</a>, <a class="existingWikiWord" href="/nlab/show/Yuji+Terashima">Yuji Terashima</a>, <em>Higher dimensional parallel transport</em> Mathematical Research Letters 8, 25–33 (2001) (<a href="http://mrlonline.org/mrl/2001-008-001/2001-008-001-004.pdf">pdf</a>)</p> </li> <li> <p>David Lipsky, <em>Cocycle constructions for topological field theories</em> (2010) (<a class="existingWikiWord" href="/nlab/files/LipskyThesis.pdf" title="pdf">pdf</a>)</p> </li> <li> <p>Nino Scalbi: <em>Differential Cohomology as Diffeological Homotopy Theory</em> [<a href="https://arxiv.org/abs/2408.02593">arXiv:2408.02593</a>]</p> </li> </ul> <p>see also the discussion at <em><a class="existingWikiWord" href="/nlab/show/fiber+integration+in+ordinary+differential+cohomology">fiber integration in ordinary differential cohomology</a></em>.</p> <p>Realization of this as an <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended TQFT</a> is discussed in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ners%C3%A9s+Aramyan">Nersés Aramyan</a>, Research statement (<a href="http://math.illinois.edu/~aramyan2/research.pdf">pdf</a>)Parallel transport with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <a class="existingWikiWord" href="/nlab/show/crossed+complexes">crossed complexes</a>/<a class="existingWikiWord" href="/nlab/show/strict+infinity-groupoids">strict infinity-groupoids</a> is discussed in</p> </li> <li id="Kapranov15"> <p><a class="existingWikiWord" href="/nlab/show/Mikhail+Kapranov">Mikhail Kapranov</a>, <em>Membranes and higher groupoids</em> (<a href="http://arxiv.org/abs/1502.06166">arXiv:1502.06166</a>)</p> </li> </ul> <p>The integration of flat differential forms with values in chain complexes to flat <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-parallel transport on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-vector bundles is in</p> <ul> <li id="AbadSchaetz"><a class="existingWikiWord" href="/nlab/show/Camilo+Arias+Abad">Camilo Arias Abad</a>, <a class="existingWikiWord" href="/nlab/show/Florian+Schaetz">Florian Schaetz</a>, <em>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math> de Rham theorem and integration of representations up to homotopy</em> (<a href="http://arxiv.org/abs/1011.4693">arXiv</a>)</li> </ul> <p>based on</p> <ul> <li id="BlockSmith"><a class="existingWikiWord" href="/nlab/show/Jonathan+Block">Jonathan Block</a>, <a class="existingWikiWord" href="/nlab/show/Aaron+Smith">Aaron Smith</a>, <em>A Riemann Hilbert correspondence for infinity local systems</em> (<a href="http://arxiv.org/abs/0908.2843">arXiv</a>)</li> </ul> <p>in turn based on constructions in</p> <ul id="Igusa"> <li><a class="existingWikiWord" href="/nlab/show/Kiyoshi+Igusa">Kiyoshi Igusa</a>, <em>Iterated integrals of superconnections</em> (<a href="http://arxiv.org/abs/0912.0249">arXiv</a>)</li> </ul> <p>Remarks on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-parallel transport in <a class="existingWikiWord" href="/nlab/show/Top">Top</a> are in</p> <ul> <li id="Stasheff"><a class="existingWikiWord" href="/nlab/show/Jim+Stasheff">Jim Stasheff</a>, <em><a class="existingWikiWord" href="/nlab/files/StasheffParallelTransportv02.pdf" title="Parallel transport, holonomy and all that -- a homotopy point of view">Parallel transport, holonomy and all that – a homotopy point of view</a></em></li> </ul> <p>Understanding <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a> of <a class="existingWikiWord" href="/nlab/show/circle+n-bundles+with+connection">circle n-bundles with connection</a> as an <a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended</a> <a class="existingWikiWord" href="/nlab/show/homotopy+field+theory">homotopy field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lukas+M%C3%BCller">Lukas Müller</a>, <a class="existingWikiWord" href="/nlab/show/Lukas+Woike">Lukas Woike</a>, <em>Parallel Transport of Higher Flat Gerbes as an Extended Homotopy Quantum Field Theory</em>, J. Homotopy Relat. Struct. <strong>15</strong> (2020) 113–142 [<a href="https://arxiv.org/abs/1802.10455">arXiv:1802.10455</a>, <a href="https://doi.org/10.1007/s40062-019-00242-3">doi:10.1007/s40062-019-00242-3</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 19, 2024 at 08:11:17. 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