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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="functorial_quantum_field_theory">Functorial quantum field theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></strong></p> <h2 id="contents">Contents</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bordism+categories+following+Stolz-Teichner">Riemannian bordism category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+tangle+hypothesis">generalized tangle hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">classification of TQFTs</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+functorial+field+theory">extended functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">CFT</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/vertex+operator+algebra">vertex operator algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+model">Reshetikhin-Turaev model</a> / <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>, <a class="existingWikiWord" href="/nlab/show/Gromov-Witten+theory">Gromov-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p>FQFT and <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+models+for+tmf">geometric models for tmf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle of higher category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle">holographic principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AdS%2FCFT+correspondence">AdS/CFT correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantization+via+the+A-model">quantization via the A-model</a></p> </li> </ul> </li> </ul> </div></div> <h4 id="physics">Physics</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#GeneralIdea'>Idea</a></li> <ul> <li><a href='#as_an_axiomatization_of_the_path_integral_and_smatrix'>As an axiomatization of the path integral and S-Matrix</a></li> <li><a href='#as_the_dual_axiomatization_to_aqft'>As the dual axiomatization to AQFT</a></li> <li><a href='#AndAndLocalVersion'>Global (“non-covariant”) and local (“covariant”) version</a></li> </ul> <li><a href='#Introduction'>Exposition and Introduction</a></li> <ul> <li><a href='#QuantumMechanicsInSchroedingerPicture'>Quantum mechanics in Schrödinger picture</a></li> <ul> <li><a href='#TimeEvolutionAndFunctors'>Time evolution and Functors</a></li> <li><a href='#entanglement_and_monoidal_structure'>Entanglement and Monoidal structure</a></li> <li><a href='#brakets_and_dual_objects'>Bra-Kets and Dual objects</a></li> <li><a href='#partition_functions_and_monoidal_string_diagrams'>Partition functions and Monoidal string diagrams</a></li> </ul> <li><a href='#PathIntegralQuantizationof1dGaugeTheory'>Path integral quantization of 1d Gauge theory</a></li> <ul> <li><a href='#GroupoidsAndBasicHomotopy1TypeTheory'>Gauge transformation and Groupoids</a></li> <li><a href='#CorrespondencesOfGroupoids'>Trajectories of fields and Correspondences of groupoids</a></li> <li><a href='#1dDWLocalFieldTheory'>Action functionals and Slice groupoids</a></li> </ul> <li><a href='#QMWithInteractionAndFeynmanDiagrams'>Quantum mechanics with interactions and Feynman diagrams</a></li> <ul> <li><a href='#spectral_triples_and_graph_representations'>Spectral triples and Graph representations</a></li> <li><a href='#feynman_diagram_in_worldline_formalism_and_monoidal_string_diagrams'>Feynman diagram in worldline formalism and Monoidal string diagrams</a></li> </ul> <li><a href='#QuantumTopologicalString'>Quantum topological string</a></li> <ul> <li><a href='#Global2dTQFT'>Global 2d TQFT and Frobenius algebra</a></li> <li><a href='#Local2dTQFT'>Cohomological 2d TQFT and Calabi-Yau <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>-algebras</a></li> <li><a href='#local_2d_tqft_and_2modules'>Local 2d TQFT and 2-Modules</a></li> </ul> <li><a href='#IntroductionGeneralFormulation'>General local TQFT</a></li> <ul> <li><a href='#covariant_quantization_and_directed_homotopy_types'>Covariant quantization and Directed homotopy types</a></li> <li><a href='#spaces_of_states_and_the_cobordism_hypothesis'><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>-Spaces of states and the Cobordism hypothesis</a></li> </ul> <li><a href='#Global3dTQFT'>3d TQFT</a></li> <ul> <li><a href='#chernsimons_theory'>Chern-Simons theory</a></li> <li><a href='#Local3dTQFT'>Modular functor</a></li> </ul> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#terminology'>Terminology</a></li> <li><a href='#formalization_of_sewing_and_locality_in_terms_of_functoriality'>Formalization of sewing and locality in terms of functoriality</a></li> <li><a href='#ReferencesRelationBetweenAQFTAndFQFT'>Relation between algebraic and functorial field theory</a></li> <li><a href='#nontopological_fqfts_especially_conformal'>Non-topological FQFTs (especially conformal)</a></li> <li><a href='#Extended'>Extended (multi-tiered) FQFT</a></li> <li><a href='#extended_fqft_from_background_fields_models'>(extended) FQFT from background fields: <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>-models</a></li> <li><a href='#super_qft'>Super QFT</a></li> <li><a href='#homological_2d_fqft_and_tcft'>homological 2d FQFT (and TCFT)</a></li> <li><a href='#2dCFTAsFunctorialQFTReferences'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">D=2</annotation></semantics></math> CFT as functorial field theory</a></li> <li><a href='#TheoryXAsAnFQFTReferences'>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">D=6</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}=(2,0)</annotation></semantics></math> SCFT as an extended functorial field theory</a></li> </ul> </ul> </div> <h2 id="GeneralIdea">Idea</h2> <p><em>Functorial quantum field theory</em> or FQFT for short, is one of the two approaches of providing a precise <a class="existingWikiWord" href="/nlab/show/mathematics">mathematical</a> formulation of and of <a class="existingWikiWord" href="/nlab/show/axiom">axiomatizing</a> <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. FQFT formalizes the <em><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a></em> of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> (generalized to <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>) where <a class="existingWikiWord" href="/nlab/show/spaces+of+quantum+states">spaces of quantum states</a> are assigned to <a class="existingWikiWord" href="/nlab/show/space">space</a> and where linear maps are assigned to <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a>/<a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a> interpolating between these spaces.</p> <h3 id="as_an_axiomatization_of_the_path_integral_and_smatrix">As an axiomatization of the path integral and S-Matrix</h3> <p>The axioms of FQFT may be understood as formulating the basic properties that the <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> or <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> invoked in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> ought to satisfy, if they had been given a precise definition.</p> <p>Much work in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is based on arguments invoking the concept of the <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a>. While in the physics literature this is usually not a well defined object, it is generally assumed to satisfy a handful of properties, notably the <em>sewing laws</em>. These say, roughly, that the path integral over a domain <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> which decomposes into subdomains <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math> is the same as the path integral over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_1</annotation></semantics></math> composed with that over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Sigma_2</annotation></semantics></math>.</p> <p>Accordingly it is the <em><a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a></em> that is manifestly incarnated in the Atiyah-Segal picture of functorial QFT:</p> <p>Here a <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> is given by a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">monoidal</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a></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 lspace="verythinmathspace">:</mo><msubsup><mi>Bord</mi> <mi>d</mi> <mi>S</mi></msubsup><mo>⟶</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> Z \colon Bord_d^S \longrightarrow Vect </annotation></semantics></math></div> <p>from a suitable <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+of+cobordisms">category of cobordisms</a> to a suitable <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>.</p> <ul> <li> <p>To a codimension-1 slice <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">M_{d-1}</annotation></semantics></math> of space this assigns a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><msub><mi>M</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(M_{d-1})</annotation></semantics></math> – the (Hilbert) <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">M_{d-1}</annotation></semantics></math>;</p> </li> <li> <p>to a <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/boundaries">boundaries</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">\partial M</annotation></semantics></math> one assigns the quantum propagator which is the linear map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Z</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>in</mi></msub><mi>M</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Z</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>out</mi></msub><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(M) : Z(\partial_{in} M) \to Z(\partial_{out} M)</annotation></semantics></math> that takes incoming states to outgoing states via propagation along the spacetime/worldvolume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. This <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(M)</annotation></semantics></math> is alternatively known as the the <em><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></em> or <em>S-matrix</em> for propagation from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>in</mi></msub><mi>M</mi></mrow><annotation encoding="application/x-tex">\partial_{in}M</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>out</mi></msub><mi>M</mi></mrow><annotation encoding="application/x-tex">\partial_{out}M</annotation></semantics></math> along a process of shape <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>.</p> </li> </ul> <p>Now for genuine <a class="existingWikiWord" href="/nlab/show/topological+field+theories">topological field theories</a> all spaces of quantum states are <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> and hence we can equivalently consider the <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a> (using that finite dimensional vector spaces form a <a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a>). Doing so the propagator map</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 stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Z</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>in</mi></msub><mi>M</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Z</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>out</mi></msub><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Z(M) : Z(\partial_{in}M) \to Z(\partial_{out}M) </annotation></semantics></math></div> <p>equivalently becomes a linear map of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo>→</mo><mi>Z</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>out</mi></msub><mi>M</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>Z</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>in</mi></msub><mi>M</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><mi>Z</mi><mo stretchy="false">(</mo><mo>∂</mo><mi>M</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{C} \to Z(\partial_{out}M) \otimes Z(\partial_{in}M)^\ast = Z(\partial M) \,. </annotation></semantics></math></div> <p>Notice that such a linear map from the canonical 1-dimensional complex vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> to some other vector space is equivalently just a choice of element in that vector space. It is in this sense that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(M)</annotation></semantics></math> is equivalently a vector in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>out</mi></msub><mi>M</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>Z</mi><mo stretchy="false">(</mo><msub><mo>∂</mo> <mi>in</mi></msub><mi>M</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mo>=</mo><mi>Z</mi><mo stretchy="false">(</mo><mo>∂</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z(\partial_{out}M) \otimes Z(\partial_{in}M)^\ast = Z(\partial M)</annotation></semantics></math>.</p> <p>In this form in physics the propagator is usually called the <em><a class="existingWikiWord" href="/nlab/show/correlator">correlator</a></em> or <em><a class="existingWikiWord" href="/nlab/show/n-point+function">n-point function</a></em> .</p> <p>The axioms of (<a href="#Segal04">Segal 04</a>) for <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a> (<a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a> in this case) were originally explicitly about the propagators/S-matrices, while (<a href="#Atiyah88">Atiyah 88</a>) formulated it in terms of the correlators this way. Both perspectives go over into each other under duality as above.</p> <p>Notice that this kind of discussion is not restricted to topological field theory. For instance already plain quantum mechanics is usefully formulated this way, that’s the point of <a class="existingWikiWord" href="/nlab/show/finite+quantum+mechanics+in+terms+of+dagger-compact+categories">finite quantum mechanics in terms of dagger-compact categories</a>.</p> <h3 id="as_the_dual_axiomatization_to_aqft">As the dual axiomatization to AQFT</h3> <p>Historically older is the proposal for axiomatizing QFT that is known as <em><a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a></em>, short for <em>algebraic quantum field theory</em>. This formalizes the <em><a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a></em> of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>, as do modern variants such as <a class="existingWikiWord" href="/nlab/show/factorization+algebras">factorization algebras</a>. Here the basic assignment is that of <a class="existingWikiWord" href="/nlab/show/algebras+of+observables">algebras of observables</a> to regions of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>.</p> <p>In principle <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> and FQFT should be two sides of the same medal, and in special cases this has been made precise (see for instance at <em><a class="existingWikiWord" href="/nlab/show/topological+chiral+homology">topological chiral homology</a></em>) but generally, much as the formulation of FQFT and AQFT themselves remains in progress, so does their precise relation.</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/duality+between+algebra+and+geometry">duality between</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></strong></p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/category">category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dual+category">dual category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a></mtext></mover><msubsup><mi>Alg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a></mtext></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup><mo>,</mo><mi>comm</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">comm. C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncomm. topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCTopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">NCTopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>general <a class="existingWikiWord" href="/nlab/show/C-star-algebra">C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>Schemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}Schemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op} </annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A} \phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncomm. algebraic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCSchemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">NCSchemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mrow><mi>fin</mi><mo>,</mo><mi>red</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/finitely+generated+algebra">fin. gen.</a> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothManifolds</mi></mrow><annotation encoding="application/x-tex">SmoothManifolds</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mi>comm</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Pursell's theorem</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>SuperSpaces</mi> <mi>Cart</mi></msub></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mphantom><mtext>Pursell's theorem</mtext></mphantom></mover></mtd> <mtd><msubsup><mi>Alg</mi> <mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mphantom><mi>AAAA</mi></mphantom></mrow> <mi>op</mi></msubsup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>q</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \overset{\phantom{\text{Pursell's theorem}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/formal+moduli+problem">formal</a> <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>(<a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super Lie theory</a>)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mi>Super</mi><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>fin</mi></msub></mtd></mtr> <mtr><mtd><mi>𝔤</mi></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><mtext><a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a></mtext><mphantom><mi>A</mi></mphantom></mrow></mover></mtd> <mtd><msup><mi>sdgcAlg</mi> <mi>op</mi></msup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">FDAs</a>”)</td></tr> </tbody></table> <p><strong>in <a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>:</p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/space+of+states">space of states</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/n-plectic+manifold">n-plectic manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/En-algebras">En-algebras</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Beilinson-Drinfeld+algebra">BD</a>-<a class="existingWikiWord" href="/nlab/show/BV+quantization">BV quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+algebra+of+observables">factorization algebra of observables</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/extended+quantum+field+theory">extended quantum field theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+homology">factorization homology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism representation</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div> <h3 id="AndAndLocalVersion">Global (“non-covariant”) and local (“covariant”) version</h3> <p>Functorial QFT in any dimension <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> was originally formulated (<a href="#Atiyah88">Atiyah 88</a>, <a href="#Segal04">Segal 04</a>) as a 1-<a class="existingWikiWord" href="/nlab/show/functor">functor</a> on a 1-<a class="existingWikiWord" href="/nlab/show/category+of+cobordisms">category of cobordisms</a>. In this formulation there is a <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> assigned to every global spatial slice of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>, which is then propagated in time/along a parameter. In physics jargon this corresponds to “non-covariant” <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a>, in that the slicing of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>/<a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a> into space and time components breaks <a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a> which is the hallmark specifically of the <a class="existingWikiWord" href="/nlab/show/topological+quantum+field+theories">topological quantum field theories</a> to which the methods of FQFT apply most immediately.</p> <p>A local (“<a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended</a>”, “multi-tiered”) refinement of this is naturally given by passing from 1-functors to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-functors">(∞,n)-functors</a> on <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-categories+of+cobordisms">(∞,n)-categories of cobordisms</a>. This formulation was vaguely suggested in (<a href="#BaezDolan95">Baez-Dolan 95</a>) (“<a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>”) and formalized in (<a href="#Luire09">Lurie 09</a>). It captures what in physics jargon would be called “covariant” quantum field theory, in that the “localization down to the point” means that the formalism knows how to glue/propagate in spatial directions just as in time directions, in fact that no such distinction is retained.</p> <h2 id="Introduction">Exposition and Introduction</h2> <blockquote> <p>under construction</p> </blockquote> <p>We give here motivation for, introduction to and an exposition of the ideas of <a class="existingWikiWord" href="/nlab/show/local+quantum+field+theory">local</a> (<a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended</a>) functorial field theory.</p> <p>We start in</p> <ul> <li><em><a href="#QuantumMechanicsInSchroedingerPicture">Quantum mechanics in Schrödinger picture</a></em></li> </ul> <p>by showing how all the basic category-theoretic ideas are already right beneath the surface of the traditional textbook discussion of quantum mechanics. Following that in</p> <ul> <li><a href="#PathIntegralQuantizationof1dGaugeTheory">Path integral quantization of 1d gauge theory</a></li> </ul> <p>we show for the simple case of 1-dimensional finite gauge theory how also <a class="existingWikiWord" href="/nlab/show/path+integral+quantization">path integral quantization</a> of <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum</a> (<a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical</a>) data is naturally organized by monoidal category theory with first bits of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> showing up (that will play a more paramount role as one goes up in dimension).</p> <p>Then in</p> <ul> <li><em><a href="#QMWithInteractionAndFeynmanDiagrams">Quantum mechanics with interaction and Feynman diagrams</a></em></li> </ul> <p>this becomes all the more pronounced when one considers quantum mechanics with interaction as in the <a class="existingWikiWord" href="/nlab/show/worldline+formalism">worldline formalism</a> and hence when one considers <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> as diagrams of interactions of <a class="existingWikiWord" href="/nlab/show/particles">particles</a>.</p> <p>This 1-dimensional functorial description worldline quantum mechanics has an evident generalization to a <a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a> formulation of <a class="existingWikiWord" href="/nlab/show/2d+topological+field+theory">2d topological field theory</a>. This original 1-functorial TQFT axiomatics due to Atiyah and Segal we review in</p> <ul> <li><em><a href="#Global2dTQFT">Naive generalization: Global 2d TQFT</a></em></li> </ul> <p>However, this “naive” generalization is not quite refined enough. Physically one sees this from the fact that the <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>/<a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>, which is the archetype of a <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> in physics, is not actually an instance of the Atiyah-Segal axiomatics. Mathematically one sees it from the fact that the 1-category theoretic formulation of 2d <a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a> is clearly lacking a “categorical dimension” in order to be satisfactory.</p> <p>The correct refinement of 2d TQFT to a <a class="existingWikiWord" href="/nlab/show/cohomological+field+theory">cohomological field theory</a> or “<a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>” with coefficients not just in vector spaces but in <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> of vector spaces we then consider in</p> <ul> <li><em><a href="#Local2dTQFT">Local2dTQFT – The topological string</a></em></li> </ul> <p>This gives a natural conceptual home to the <a class="existingWikiWord" href="/nlab/show/derived+categories">derived categories</a> of <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> famous from <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a>. But it is still not quite the fully general functorial formalization of quantum field theory.</p> <p>To get a feeling for what is missing, we next consider <a class="existingWikiWord" href="/nlab/show/3d+TQFT">3d TQFT</a></p> <ul> <li> <p><em><a href="#Global3dTQFT">Global 3d TQFT – Chern-Simons theory</a></em></p> </li> <li> <p><em><a href="#Local3dTQFT">Local 3d TQFT – Modular functor and Wess-Zumino-Witten theory</a></em></p> </li> </ul> <p>This finally is enough information to naturally motivate the full formulation of the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a> in <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28infinity%2Cn%29-category">symmetric monoidal (infinity,n)-category</a> theory in</p> <ul> <li><em><a href="#IntroductionGeneralFormulation">General local TQFT</a></em></li> </ul> <h3 id="QuantumMechanicsInSchroedingerPicture">Quantum mechanics in Schrödinger picture</h3> <p>This section introduces the observation that the basic structures in <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> are accurately reflected in <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>, by explaining the following dictionary:</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a></th><th><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <a class="existingWikiWord" href="/nlab/show/category+theory">theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/object">object</a> in a <a class="existingWikiWord" href="/nlab/show/category">category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in a <a class="existingWikiWord" href="/nlab/show/category">category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/local+quantum+field+theory">local</a> <a class="existingWikiWord" href="/nlab/show/dynamics">time evolution</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/functor">functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/compound+system">compound system</a> <a class="existingWikiWord" href="/nlab/show/entanglement">entanglement</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">non-cartesian</a> <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal structure</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bra">bra</a>/<a class="existingWikiWord" href="/nlab/show/ket">ket</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dual+objects">dual objects</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a></td><td style="text-align: left;">monoidal <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a></td></tr> </tbody></table> <h4 id="TimeEvolutionAndFunctors">Time evolution and Functors</h4> <p>The basic idea of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> in the “<a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a>” is to describe a <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a> (such as an <a class="existingWikiWord" href="/nlab/show/electron">electron</a> in the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a> of a <a class="existingWikiWord" href="/nlab/show/proton">proton</a>) by</p> <ol> <li> <p>assigning to each time <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">t \in \mathbb{R}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> (<a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">V_t</annotation></semantics></math> to be thought of as the <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> (of <a class="existingWikiWord" href="/nlab/show/pure+states">pure states</a>, that is) of the system at that time;</p> </li> <li> <p>assigning to each pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[t_1,t_2]</annotation></semantics></math> of times a <a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a> (<a class="existingWikiWord" href="/nlab/show/unitary+operator">unitary operator</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msub><mi>V</mi> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow></msub><mo>→</mo><msub><mi>V</mi> <mrow><msub><mi>t</mi> <mn>2</mn></msub></mrow></msub></mrow><annotation encoding="application/x-tex">U(t_1,t_2) \colon V_{t_1} \to V_{t_2}</annotation></semantics></math> to be thought of as encoding the <a class="existingWikiWord" href="/nlab/show/time">time</a> evolution of quantum states</p> </li> </ol> <p>such that</p> <ul> <li> <p>this assignment is <em>local</em> in <a class="existingWikiWord" href="/nlab/show/time">time</a> in that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mn>1</mn></msub><mo>≤</mo><msub><mi>t</mi> <mn>2</mn></msub><mo>≤</mo><msub><mi>t</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex">t_1 \leq t_2 \leq t_3</annotation></semantics></math> one has</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>∘</mo><mi>U</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U(t_1, t_3) = U(t_2, t_3) \circ U(t_1, t_2) \,. </annotation></semantics></math></div></li> </ul> <p>In basic quantum mechanics one also demands that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>id</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U(t,t) = id \,. </annotation></semantics></math></div> <p>(While this looks like the most innocent condition, this has technical subtleties for genuine <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>, to deal with which however there exist established tools.)</p> <p>The locality condition intuitively says that “all global effects arise by integrating up local effects”. Indeed, when assuming in addition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(-,-)</annotation></semantics></math> depends <a class="existingWikiWord" href="/nlab/show/smooth+function">smoothly</a> on the time arguments, then the locality condition is equivalent to (see at <em><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a></em>) the existence of a <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">H_t</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/self-adjoint+operator">self-adjoint operator</a> depending smoothly on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math>, such that time evolution is given by the <a class="existingWikiWord" href="/nlab/show/Dyson+formula">Dyson formula</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><mi>exp</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><msubsup><mo>∫</mo> <mrow><msub><mi>t</mi> <mn>1</mn></msub></mrow> <mrow><msub><mi>t</mi> <mn>2</mn></msub></mrow></msubsup><msub><mi>H</mi> <mi>t</mi></msub><mspace width="thinmathspace"></mspace><mi>d</mi><mi>t</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U(t_1,t_2) = P \exp\left( \tfrac{i}{\hbar} \int_{t_1}^{t_2} H_t \, d t \right) \,. </annotation></semantics></math></div> <p>(Here the notation on the right denotes the “path ordered exponential”, see at <em><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a></em>.) In the special case that the Hamiltonian is time-independent, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><msub><mi>H</mi> <mi>t</mi></msub></mrow><annotation encoding="application/x-tex">H = H_t</annotation></semantics></math>, then this reduces to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>exp</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>2</mn></msub><mo>−</mo><msub><mi>t</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>H</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U(t_1,t_2) = \exp\left( \tfrac{i}{\hbar} (t_2 - t_1) H \right) \,. </annotation></semantics></math></div> <p>This is the way that quantum mechanical time evolution is traditionally introduced in the textbooks.</p> <p>But the equivalent formulation above in terms of locality of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(-,-)</annotation></semantics></math> is noteworthy. The condition of locality here is precisely what in <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> is called <em><a class="existingWikiWord" href="/nlab/show/functor">functoriality</a></em>: the condition that a system of <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> (here: linear/unitary operator) depends on another system of “directed data” (here: the time intervals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[t_1,t_2]</annotation></semantics></math>) such that <a class="existingWikiWord" href="/nlab/show/composition">composition</a> is respected.</p> <p>More specifically, one says that the collection <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> (or <a class="existingWikiWord" href="/nlab/show/Hilb">Hilb</a>) of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>) forms a <em><a class="existingWikiWord" href="/nlab/show/category">category</a></em> whose <em><a class="existingWikiWord" href="/nlab/show/objects">objects</a></em> are vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> and whose <em><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a></em> are linear maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f \colon V_1 \to V_2</annotation></semantics></math>; where the point is that these morphisms may be <a class="existingWikiWord" href="/nlab/show/associativity">associatively</a> composed whenever their <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a>/<a class="existingWikiWord" href="/nlab/show/domain">domain</a> matches:</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>V</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>V</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>V</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && V_2 \\ & {}^{\mathllap{f_1}}\nearrow && \searrow^{\mathrlap{f_2}} \\ V_1 && \stackrel{f_2 \circ f_1}{\longrightarrow} && V_3 } \,. </annotation></semantics></math></div> <p>Similarly, there is a category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>1</mn> <mi>Riem</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord_1^{Riem}</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are instants of time <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">t \in \mathbb{R}</annotation></semantics></math>, whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are time intervals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[t_1,t_2]</annotation></semantics></math>, and whose <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation is concatenation of time intervals</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>t</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>t</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>t</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && t_2 \\ & \nearrow && \searrow \\ t_1 && \stackrel{}{\longrightarrow} && t_3 } </annotation></semantics></math></div> <p>Given two categories like this, then a function that takes morphisms of one to morphisms of the other such that <a class="existingWikiWord" href="/nlab/show/composition">composition</a> is respected is called a <em><a class="existingWikiWord" href="/nlab/show/functor">functor</a></em>.</p> <p>In this language, the above locality condition of <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> says that quantum time evolution is a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Bord</mi> <mn>1</mn> <mi>Riem</mi></msubsup><mo>⟶</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> U \;\colon\; Bord_1^{Riem} \longrightarrow Vect </annotation></semantics></math></div> <p>that takes</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>t</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>t</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>t</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>V</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>V</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>U</mi><mo stretchy="false">(</mo><msub><mi>t</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>t</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>V</mi> <mn>3</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && t_2 \\ & \nearrow && \searrow \\ t_1 && \stackrel{}{\longrightarrow} && t_3 } \;\;\;\;\;\; \mapsto \;\;\;\;\;\; \array{ && V_2 \\ & {}^{\mathllap{U(t_1,t_2)}}\nearrow && \searrow^{\mathrlap{U(t_2,t_3)}} \\ V_1 && \stackrel{U(t_1,t_3)}{\longrightarrow} && V_3 } </annotation></semantics></math></div> <p>If this looks like a trivial reformulation of textbook material, then this is because it is a trivial reformulation of textbook material. But introducing such category-theoretic language for making the locality principle in quantum mechanics fully manifest turns out to be rather useful for capturing the full locality of <a class="existingWikiWord" href="/nlab/show/local+quantum+field+theory">local quantum field theory</a>, which is not in the traditional textbooks. This we come to below.</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/physics">physics</a></th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/local+quantum+field+theory">locality</a> of time evolution</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/functor">functor</a></td></tr> </tbody></table> <h4 id="entanglement_and_monoidal_structure">Entanglement and Monoidal structure</h4> <p>Besides the time evolution, there is the theory of <a class="existingWikiWord" href="/nlab/show/composite+systems">composite systems</a>.</p> <p>Given two <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+systems">quantum mechanical systems</a> (e.g. of two electrons orbiting the same atomic nucleus), with <a class="existingWikiWord" href="/nlab/show/spaces+of+quantum+states">spaces of quantum states</a> (<a class="existingWikiWord" href="/nlab/show/pure+states">pure states</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>V</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde V</annotation></semantics></math>, respectively, then the space of quantum states of the compound system is given by the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of vector spaces</a> (of <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊗</mo><mover><mi>V</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><mi>Vect</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V \otimes \tilde V \in Vect \,. </annotation></semantics></math></div> <p>This should be compared with the way compound systems are formed in <a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_2</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/configuration+spaces">configuration spaces</a>/<a class="existingWikiWord" href="/nlab/show/phase+spaces">phase spaces</a> of two classical <a class="existingWikiWord" href="/nlab/show/mechanical+systems">mechanical systems</a>, then the configuration space/phase space of their compound is the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \times X_2</annotation></semantics></math>.</p> <p>These Cartesian products and tensor products extend to <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">f\colon V \to V</annotation></semantics></math> is a linear operator acting on the first system and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo><mover><mi>V</mi><mo stretchy="false">˜</mo></mover><mo>→</mo><mover><mi>V</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f \colon \tilde V\to \tilde V</annotation></semantics></math> is one acting on the second system, then there is a tensor product morphism</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>f</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>⊗</mo><mover><mi>V</mi><mo stretchy="false">˜</mo></mover><mo>⟶</mo><mi>V</mi><mo>⊗</mo><mover><mi>V</mi><mo stretchy="false">˜</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (f_1\otimes f_2) \;\colon\; V \otimes \tilde V \longrightarrow V \otimes \tilde V \,. </annotation></semantics></math></div> <p>Hence (Cartesian or non-cartesian) <a class="existingWikiWord" href="/nlab/show/tensor+products">tensor products</a> are something like product operations on sets, but on whole <a class="existingWikiWord" href="/nlab/show/categories">categories</a>. Since binary associative product operations on sets are sometimes called <em><a class="existingWikiWord" href="/nlab/show/monoids">monoids</a></em>, one says that the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> of vector spaces when equipped with the tensor product of vector spaces is a <em><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></em>.</p> <p>Similarly, the categories <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> or <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> or just bare <a class="existingWikiWord" href="/nlab/show/sets">sets</a>, carry a monoidal structure given simply by the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of sets. This is called a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal structure</a>.</p> <p>The characteristic property of Cartesian products <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \times X_2</annotation></semantics></math> is that elements of these are equivalently pairs of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_2</annotation></semantics></math>, respectively. This reflects in turn the characteristic property of compound classical mechanical systems: a state of these is simply a pair of states of the two subsystems.</p> <p>The tensor product on <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> however is not Cartesian: an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi><mo>∈</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\Psi \in V_1 \otimes V_2</annotation></semantics></math> need not be of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ψ</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>ψ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\psi_1\otimes \psi_2</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ψ</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>V</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\psi_i \in V_i</annotation></semantics></math>. Instead, in general it is a <a class="existingWikiWord" href="/nlab/show/sum">sum</a> of such elements</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>j</mi></munder><mo stretchy="false">(</mo><msub><mi>Ψ</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>j</mi></msub><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>Ψ</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>j</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Psi = \underset{j}{\sum} (\Psi_1)_j \otimes (\Psi_2)_j \,. </annotation></semantics></math></div> <p>In terms of <a class="existingWikiWord" href="/nlab/show/physics">physics</a> such non-cartesian vectors are <a class="existingWikiWord" href="/nlab/show/quantum+states">quantum states</a> that exhibit <a class="existingWikiWord" href="/nlab/show/entanglement">entanglement</a>. This hallmark property of quantum mechanics is hence accurately reflected by the abstract property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Vect</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Vect,\otimes)</annotation></semantics></math> being a non-cartesian <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>.</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/physics">physics</a></th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/entanglement">entanglement</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">non-cartesian</a> <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> of <a class="existingWikiWord" href="/nlab/show/spaces+of+states">spaces of states</a></td></tr> </tbody></table> <p>For more exposition of this point see (<a href="#Baez04">Baez 04</a>).</p> <h4 id="brakets_and_dual_objects">Bra-Kets and Dual objects</h4> <p>Consider now for simplicity of notation an application in <a class="existingWikiWord" href="/nlab/show/quantum+computing">quantum computing</a>/<a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a>, where the <a class="existingWikiWord" href="/nlab/show/spaces+of+states">spaces of states</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> involved are <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a> (spaces of <a class="existingWikiWord" href="/nlab/show/qubits">qubits</a>), such as for instance in the topological sector of the <a class="existingWikiWord" href="/nlab/show/quantum+Hall+effect">quantum Hall system</a>.</p> <p>Then for every space of states <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> there is the <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^\ast</annotation></semantics></math>. In <a class="existingWikiWord" href="/nlab/show/physics">physics</a> notation the states in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/kets">kets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>Ψ</mi><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\vert \Psi \rangle</annotation></semantics></math>, while those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^\ast</annotation></semantics></math> are the “bra”s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">⟨</mo><mi>Ψ</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">\langle \Psi \vert</annotation></semantics></math>.</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/physics">physics</a></th><th><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bra">bra</a>/<a class="existingWikiWord" href="/nlab/show/ket">ket</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dual+objects">dual objects</a></td></tr> </tbody></table> <p>Essentially all of <a class="existingWikiWord" href="/nlab/show/quantum+information+theory">quantum information theory</a> has a slick reformulation in terms of <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> for <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">with dual objects</a>. More on this is at <em><a class="existingWikiWord" href="/nlab/show/finite+quantum+mechanics+in+terms+of+dagger-compact+categories">finite quantum mechanics in terms of dagger-compact categories</a></em>.</p> <p>While the introduction of bra-ket notation by <a class="existingWikiWord" href="/nlab/show/Paul+Dirac">Paul Dirac</a> was (while just notation) already quite useful for thinking about the subject, the language of monoidal categories in fact reflects the actual physical processes involved even better.</p> <p>For instance, in quantum mechanics textbooks one often sees the following manipulation of symbols for expressing a <a class="existingWikiWord" href="/nlab/show/trace">trace</a> in terms of a sum over <a class="existingWikiWord" href="/nlab/show/basis+of+a+vector+space">basis</a> elements</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mo stretchy="false">⟨</mo><mi>Ψ</mi><mo stretchy="false">|</mo><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><mi>H</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>Ψ</mi><mo stretchy="false">⟩</mo></mtd> <mtd><mo>=</mo><mo stretchy="false">⟨</mo><mi>Ψ</mi><mo stretchy="false">|</mo><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><mi>H</mi><mi>t</mi><mo stretchy="false">)</mo><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>j</mi></munder><mo stretchy="false">|</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo stretchy="false">|</mo><mo>)</mo></mrow><mo stretchy="false">|</mo><mi>Ψ</mi><mo stretchy="false">⟩</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>j</mi></munder><mo stretchy="false">⟨</mo><mi>Ψ</mi><mo stretchy="false">|</mo><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><mi>H</mi><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo stretchy="false">|</mo><mi>Ψ</mi><mo stretchy="false">⟩</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>j</mi></munder><mo stretchy="false">⟨</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo stretchy="false">|</mo><mrow><mo>(</mo><mo stretchy="false">|</mo><mi>Ψ</mi><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><mi>Ψ</mi><mo stretchy="false">|</mo><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><mi>H</mi><mi>t</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo stretchy="false">|</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo stretchy="false">⟩</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \langle \Psi \vert \exp(\tfrac{i}{\hbar} H t ) \vert \Psi \rangle & = \langle \Psi \vert \exp(\tfrac{i}{\hbar} H t ) \left( \sum_j \vert \Psi_j \rangle \langle \Psi_j \vert \right) \vert \Psi \rangle \\ & = \sum_j \langle \Psi \vert \exp(\tfrac{i}{\hbar} H t ) \vert \Psi_j \rangle \langle \Psi_j \vert \Psi \rangle \\ & = \sum_j \langle \Psi_j \vert \left( \vert \Psi \rangle \langle \Psi \vert \exp(\tfrac{i}{\hbar} H t ) \right) \vert \Psi_j \rangle \end{aligned} \,. </annotation></semantics></math></div> <p>This “rotation” operation where symbols are cyclically permuted reflects the fact that indeed traces as in <a class="existingWikiWord" href="/nlab/show/partition+functions">partition functions</a> reflect actual physical circular processes.</p> <p>In “<a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>”-notation of <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>-theory this is reflected as follows. The unit map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>↦</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>j</mi></munder><mo stretchy="false">|</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex"> 1 \mapsto \underset{j}{\sum} \vert \Psi_j \rangle \langle \Psi_j \vert </annotation></semantics></math></div> <p>is depicted as</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><mo>⟵</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>V</mi> <mo>*</mo></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \longleftarrow \\ & \swarrow && \nwarrow \\ V && && V^\ast } </annotation></semantics></math></div> <p>and the counit map</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>Ψ</mi> <mi>j</mi></msub><mo stretchy="false">⟩</mo><mo stretchy="false">⟨</mo><msub><mi>Ψ</mi> <mi>k</mi></msub><mo stretchy="false">|</mo><mo>↦</mo><mo stretchy="false">⟨</mo><msub><mi>Ψ</mi> <mi>k</mi></msub><mo stretchy="false">|</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex"> \vert \Psi_j \rangle \langle \Psi_k \vert \mapsto \langle \Psi_k \vert \Psi_j \rangle </annotation></semantics></math></div> <p>as</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>V</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>V</mi> <mo>*</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ V && && V^\ast \\ & \searrow && \nearrow \\ && \longrightarrow } \,. </annotation></semantics></math></div> <h4 id="partition_functions_and_monoidal_string_diagrams">Partition functions and Monoidal string diagrams</h4> <p>Given a <a class="existingWikiWord" href="/nlab/show/Hamiltonian">Hamiltonian</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of the <a class="existingWikiWord" href="/nlab/show/quantum+mechanical+system">quantum mechanical system</a> is the <a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>t</mi></msub><mo>≔</mo><mi>Tr</mi><mrow><mo>(</mo><mi>exp</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><mi>H</mi><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Z_t \coloneqq Tr\left( \exp\left(\tfrac{i}{\hbar} H t \right) \right) \,. </annotation></semantics></math></div> <p>In bra-ket notation this is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mi>t</mi></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>j</mi></munder><mrow><mo>⟨</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo>|</mo></mrow><mi>exp</mi><mrow><mo>(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><mi>H</mi><mi>t</mi><mo>)</mo></mrow><mrow><mo>|</mo><msub><mi>Ψ</mi> <mi>j</mi></msub><mo>⟩</mo></mrow></mrow><annotation encoding="application/x-tex"> Z_t = \underset{j}{\sum} \left\langle \Psi_j \right\vert \exp\left(\tfrac{i}{\hbar} H t \right) \left| \Psi_j \right\rangle </annotation></semantics></math></div> <p>In terms of monoidal category-theoretic notation (<a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a>) this same expression reads as follows</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><mo>⟵</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>V</mi> <mo>*</mo></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mstyle displaystyle="false"><mfrac><mi>i</mi><mi>ℏ</mi></mfrac></mstyle><mi>H</mi><mi>t</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>id</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>V</mi> <mo>*</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \longleftarrow \\ & \swarrow && \nwarrow \\ V && && V^\ast \\ {}^{\mathllap{\exp(\tfrac{i}{\hbar} H t)}}\downarrow && && \uparrow^{\mathrlap{id}} \\ V && && V^\ast \\ & \searrow && \nearrow \\ && \longrightarrow } </annotation></semantics></math></div> <p>This is striking, because this <em>picture</em> is an accurate reflection of the physical process that the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> describes, for the partition function is the <a class="existingWikiWord" href="/nlab/show/correlator">correlator</a> of a particle with a closed circular <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a>.</p> <p>In fact, the monoidal category theoretic <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>-notation is essentially the <a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>-notation. This we turn to <a href="#QMWithInteractionAndFeynmanDiagrams">below</a>.</p> <p>For a 1-dimensional <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a> the Hamiltonian above vanishes. (Or maybe more interestingly: for <a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a> the Hamiltonian may not vanish, but in the <a class="existingWikiWord" href="/nlab/show/super+trace">super trace</a> in the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> all non-zero energy eigenmodes cancel out by supersymmetry, and only the topological part is left after all.)</p> <p>In this case the partition function reduces to</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><mo>⟵</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>V</mi> <mo>*</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \longleftarrow \\ & \swarrow && \nwarrow \\ V && && V^\ast \\ & \searrow && \nearrow \\ && \longrightarrow } </annotation></semantics></math></div> <p>which is just the trace on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>.</p> <p>To reflect this in the functorial notation from <a href="#TimeEvolutionAndFunctors">above</a>, notice that also the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>1</mn> <mi>Riem</mi></msubsup></mrow><annotation encoding="application/x-tex">Bord_{1}^{Riem}</annotation></semantics></math> from above is naturally a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> if we take its objects to consist not just of single points, but of arbitrary collections of points, and to have its morphisms consist of all 1-dimensional <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a>. Then a monoidal structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Bord</mi> <mn>1</mn> <mi>Riem</mi></msubsup><msup><mo stretchy="false">)</mo> <mo lspace="thinmathspace" rspace="thinmathspace">∐</mo></msup></mrow><annotation encoding="application/x-tex">(Bord_1^{Riem})^\coprod</annotation></semantics></math> is given by <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a>.</p> <p>A 1d TQFT with values in vector spaces is then a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</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><msubsup><mi>Bord</mi> <mn>1</mn> <mi>Riem</mi></msubsup><msup><mo stretchy="false">)</mo> <mo lspace="thinmathspace" rspace="thinmathspace">∐</mo></msup><mo>⟶</mo><msup><mi>Vect</mi> <mo>⊗</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Bord_1^{Riem})^\coprod \longrightarrow Vect^{\otimes} \,. </annotation></semantics></math></div> <p>Other processes that such a 1d TQFT encodes include</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>V</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>V</mi> <mo>*</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⟵</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>V</mi> <mo>*</mo></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V && && V^\ast \\ & \searrow && \nearrow \\ && \longrightarrow \\ && \longleftarrow \\ & \swarrow && \nwarrow \\ V && && V^\ast } </annotation></semantics></math></div> <p>In this way a 1d TQFT is entirely encoded in the operation that exhibit a <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional vector space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math> 1d TQFT with coefficients in <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dualizable+objects">dualizable objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≃</mo></mrow><annotation encoding="application/x-tex">\simeq</annotation></semantics></math></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\{</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\}</annotation></semantics></math></p> <p>This is the simplest incarnation of the statement that for higher dimensional <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended TQFT</a> becomes the <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>. We come to this <a href="#IntroductionGeneralFormulation">below</a>.</p> <h3 id="PathIntegralQuantizationof1dGaugeTheory">Path integral quantization of 1d Gauge theory</h3> <p>The <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> tends to be as suggestive in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> as it is invoked ubiquitously, and FQFT may be understood as being precisely the axiomatics that a would-be path integral ought to satisfy, thereby decoupling its construction as a suitably regularized integral from its operational definition as yielding a consistent <a class="existingWikiWord" href="/nlab/show/S-matrix">S-matrix</a> for the QFT.</p> <p>We discuss now the simplest non-trivial example of a <a class="existingWikiWord" href="/nlab/show/path+integral+quantization">path integral quantization</a>, namely for 1-dimensional finite <a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, the 1-dimensional <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+model">Dijkgraaf-Witten model</a>. This has the two-fold purpose of</p> <ol> <li> <p>indicating how not just the final quantum theory but also its <a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum data</a> is naturally organized by monoidal category theory;</p> </li> <li> <p>motivating and introducing elements of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> which become crucial for the understanding of FQFT as one moves up in dimension.</p> </li> </ol> <p>A more detailed version of this section is at <em><a href="prequantum+field+theory#1dDWTheory">Local prequantum field theory – id Dijkgraaf-Witten theory</a></em>.</p> <h4 id="GroupoidsAndBasicHomotopy1TypeTheory">Gauge transformation and Groupoids</h4> <p>The following is a quick review of basics of <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> and their <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+1-type">homotopy 1-type</a>-theory), geared towards the constructions and facts needed for 1-dimensional Dijkgraaf-Witten theory.</p> <div class="num_defn" id="Groupoid"> <h6 id="definition">Definition</h6> <p>A (<a class="existingWikiWord" href="/nlab/show/small+category">small</a>) <em><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_\bullet</annotation></semantics></math> is</p> <ul> <li> <p>a pair of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>0</mn></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}_0 \in Set </annotation></semantics></math> (the set of <a class="existingWikiWord" href="/nlab/show/objects">objects</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\mathcal{G}_1 \in Set</annotation></semantics></math> (the set of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a>)</p> </li> <li> <p>equipped with <a class="existingWikiWord" href="/nlab/show/functions">functions</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><msub><mi>𝒢</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow></msub><msub><mi>𝒢</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mo>∘</mo></mover></mtd> <mtd><msub><mi>𝒢</mi> <mn>1</mn></msub></mtd> <mtd><mover><mover><munder><mo>⟶</mo><mi>s</mi></munder><mover><mo>←</mo><mi>i</mi></mover></mover><mover><mo>⟶</mo><mi>t</mi></mover></mover></mtd> <mtd><msub><mi>𝒢</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{G}_1 \times_{\mathcal{G}_0} \mathcal{G}_1 &\stackrel{\circ}{\longrightarrow}& \mathcal{G}_1 & \stackrel{\overset{t}{\longrightarrow}}{\stackrel{\overset{i}{\leftarrow}}{\underset{s}{\longrightarrow}}}& \mathcal{G}_0 }\,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> on the left is that over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>t</mi></mover><msub><mi>𝒢</mi> <mn>0</mn></msub><mover><mo>←</mo><mi>s</mi></mover><msub><mi>𝒢</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_1 \stackrel{t}{\to} \mathcal{G}_0 \stackrel{s}{\leftarrow} \mathcal{G}_1</annotation></semantics></math>,</p> </li> </ul> <p>such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> takes values in <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a>;</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∘</mo><mi>i</mi><mo>=</mo><mi>s</mi><mo>∘</mo><mi>i</mi><mo>=</mo><msub><mi>id</mi> <mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow></msub><mo>,</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex"> t \circ i = s \circ i = id_{\mathcal{G}_0}, \;\;\; </annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math> defines a partial <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation which is <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unital</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(\mathcal{G}_0)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/identities">identities</a>; in particular</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s (g \circ f) = s(f)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo>∘</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><mi>t</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">t (g \circ f) = t(g)</annotation></semantics></math>;</p> </li> <li> <p>every morphism has an <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> under this composition.</p> </li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This data is visualized as follows. The set of morphisms is</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> <mn>1</mn></msub><mo>=</mo><mrow><mo>{</mo><msub><mi>ϕ</mi> <mn>0</mn></msub><mover><mo>→</mo><mi>k</mi></mover><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathcal{G}_1 = \left\{ \phi_0 \stackrel{k}{\to} \phi_1 \right\} </annotation></semantics></math></div> <p>and the set of pairs of composable morphisms is</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> <mn>2</mn></msub><mo>≔</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><munder><mo>×</mo><mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow></munder><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>ϕ</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>ϕ</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>k</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>k</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd></mtd> <mtd><msub><mi>ϕ</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{G}_2 \coloneqq \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 = \left\{ \array{ && \phi_1 \\ & {}^{\mathllap{k_1}}\nearrow && \searrow^{\mathrlap{k_2}} \\ \phi_0 && \stackrel{k_2 \circ k_1}{\to} && \phi_2 } \right\} \,. </annotation></semantics></math></div> <p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>p</mi> <mn>2</mn></msub><mo>,</mo><mo>∘</mo><mo lspace="verythinmathspace">:</mo><msub><mi>𝒢</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>𝒢</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1, p_2, \circ \colon \mathcal{G}_2 \to \mathcal{G}_1</annotation></semantics></math> are those which send, respectively, these triangular diagrams to the left morphism, or the right morphism, or the bottom morphism.</p> </div> <div class="num_example" id="SetAsGroupoid"> <h6 id="example">Example</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/set">set</a>, it becomes a groupoid by taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to be the set of objects and adding only precisely the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> morphism from each object to itself</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>X</mi><mover><mover><munder><mo>⟶</mo><mi>id</mi></munder><mover><mo>⟵</mo><mi>id</mi></mover></mover><mover><mo>⟶</mo><mi>id</mi></mover></mover><mi>X</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( X \stackrel {\overset{id}{\longrightarrow}} { \stackrel{\overset{id}{\longleftarrow}}{\underset{id}{\longrightarrow}} } X \right) \,. </annotation></semantics></math></div></div> <div class="num_example" id="DeloopingGroupoid"> <h6 id="example_2">Example</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/group">group</a>, its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">(\mathbf{B}G)_\bullet</annotation></semantics></math> has</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">(\mathbf{B}G)_0 = \ast</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">(\mathbf{B}G)_1 = G</annotation></semantics></math>.</p> </li> </ul> <p>For <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> and <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> two groups, group homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">f \colon G \to K</annotation></semantics></math> are in <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> with groupoid homomorphisms</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>f</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>→</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>K</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{B}f)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}K)_\bullet \,. </annotation></semantics></math></div> <p>In particular a <a class="existingWikiWord" href="/nlab/show/group+character">group character</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \colon G \to U(1)</annotation></semantics></math> is equivalently a groupoid homomorphism</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>c</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>→</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{B}c)_\bullet \;\colon\; (\mathbf{B}G)_\bullet \to (\mathbf{B}U(1))_\bullet \,. </annotation></semantics></math></div> <p>Here, for the time being, all groups are <a class="existingWikiWord" href="/nlab/show/discrete+groups">discrete groups</a>. Since the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> <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> also has a standard structure of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, and since later for the discussion of Chern-Simons type theories this will be relevant, we will write from now on</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mi>Grp</mi></mrow><annotation encoding="application/x-tex"> \flat U(1) \in Grp </annotation></semantics></math></div> <p>to mean explicitly the <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> underlying the circle group. (Here “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</annotation></semantics></math>” denotes the “<a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a>”.)</p> </div> <div class="num_example" id="ActionGroupoid"> <h6 id="example_3">Example</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/set">set</a>, <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> a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\rho \colon X \times G \to X</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/action">action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> 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> (a <a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation representation</a>), the <strong><a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a></strong> or <strong><a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> by <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 the groupoid</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 stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ρ</mi></msub><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>X</mi><mo>×</mo><mi>G</mi><mover><munder><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></munder><mover><mo>⟶</mo><mi>ρ</mi></mover></mover><mi>X</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> X//_\rho G = \left( X \times G \stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}} X \right) </annotation></semantics></math></div> <p>with composition induced by the product in <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>. Hence this is the groupoid whose objects are the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and where <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mover><mo>→</mo><mi>g</mi></mover><msub><mi>x</mi> <mn>2</mn></msub><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> x_1 \stackrel{g}{\to} x_2 = \rho(x_1)(g) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x_1, x_2 \in X</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math>.</p> </div> <p>As an important special case we have:</p> <div class="num_example" id="BGGroupoidAsActionGroupoid"> <h6 id="example_4">Example</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/discrete">discrete</a> group and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> the trivial action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math> (the singleton set), the coresponding <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> according to def. <a class="maruku-ref" href="#ActionGroupoid"></a> is the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> according to def. <a class="maruku-ref" href="#DeloopingGroupoid"></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>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>=</mo><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\ast //G)_\bullet = (\mathbf{B}G)_\bullet \,. </annotation></semantics></math></div> <p>Another canonical action is the action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on itself by right multiplication. The corresponding action groupoid we write</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><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><mo>≔</mo><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{E}G)_\bullet \coloneqq G//G \,. </annotation></semantics></math></div> <p>The constant map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">G \to \ast</annotation></semantics></math> induces a canonical morphism</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>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd> <mtd><mo>≃</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ G//G & \simeq & \mathbf{E}G \\ \downarrow && \downarrow \\ \ast //G & \simeq & \mathbf{B}G } \,. </annotation></semantics></math></div> <p>This is known as the <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>-<a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>. See below in <a class="maruku-ref" href="#PullbackOfEGGroupoidAsHomotopyFiberProduct"></a> for more on this.</p> </div> <div class="num_example"> <h6 id="example_5">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/interval">interval</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is the groupoid with</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mn>0</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">I_0 = \{a,b\}</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mn>1</mn></msub><mo>=</mo><mo stretchy="false">{</mo><msub><mi mathvariant="normal">id</mi> <mi>a</mi></msub><mo>,</mo><msub><mi mathvariant="normal">id</mi> <mi>b</mi></msub><mo>,</mo><mi>a</mi><mo>→</mo><mi>b</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">I_1 = \{\mathrm{id}_a, \mathrm{id}_b, a \to b \}</annotation></semantics></math>.</li> </ul> </div> <div class="num_example" id="FundamentalGroupoid"> <h6 id="example_6">Example</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, its <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(\Sigma)</annotation></semantics></math> is</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">\Pi_1(\Sigma)_0 = </annotation></semantics></math> 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>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">\Pi_1(\Sigma)_1 = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a> paths 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> modulo <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> that leaves the endpoints fixed.</li> </ul> </div> <div class="num_example" id="PathSpaceGroupoid"> <h6 id="example_7">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_\bullet</annotation></semantics></math> any groupoid, there is the <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒢</mi> <mo>•</mo> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathcal{G}^I_\bullet</annotation></semantics></math> with</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒢</mi> <mn>0</mn> <mi>I</mi></msubsup><mo>=</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>ϕ</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>k</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>ϕ</mi> <mn>1</mn></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex">\mathcal{G}^I_0 = \mathcal{G}_1 = \left\{ \array{ \phi_0 \\ \downarrow^{\mathrlap{k}} \\ \phi_1 } \right\}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒢</mi> <mn>1</mn> <mi>I</mi></msubsup><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{G}^I_1 = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting squares</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_\bullet</annotation></semantics></math> = <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>ϕ</mi> <mn>0</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>h</mi> <mn>0</mn></msub></mrow></mover></mtd> <mtd><msub><mover><mi>ϕ</mi><mo stretchy="false">˜</mo></mover> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>k</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mover><mi>k</mi><mo stretchy="false">˜</mo></mover></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>ϕ</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>h</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mover><mi>ϕ</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\{ \array{ \phi_0 &\stackrel{h_0}{\to}& \tilde \phi_0 \\ {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{\tilde k}} \\ \phi_1 &\stackrel{h_1}{\to}& \tilde \phi_1 } \right\} \,. </annotation></semantics></math></p> </li> </ul> <p>This comes with two canonical homomorphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝒢</mi> <mo>•</mo> <mi>I</mi></msubsup><mover><munder><mo>⟶</mo><mrow><msub><mi>ev</mi> <mn>0</mn></msub></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>ev</mi> <mn>1</mn></msub></mrow></mover></mover><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex"> \mathcal{G}^I_\bullet \stackrel{\overset{ev_1}{\longrightarrow}}{\underset{ev_0}{\longrightarrow}} \mathcal{G}_\bullet </annotation></semantics></math></div> <p>which are given by endpoint evaluation, hence which send such a commuting square to either its top or its bottom hirizontal component.</p> </div> <div class="num_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><msub><mi>f</mi> <mo>•</mo></msub><mo>,</mo><msub><mi>g</mi> <mo>•</mo></msub><mo>:</mo><msub><mi>𝒢</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>𝒦</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">f_\bullet, g_\bullet : \mathcal{G}_\bullet \to \mathcal{K}_\bullet</annotation></semantics></math> two morphisms between groupoids, a <em><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⇒</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \Rightarrow g</annotation></semantics></math> (a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>) is a homomorphism of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mo>•</mo></msub><mo>:</mo><msub><mi>𝒢</mi> <mo>•</mo></msub><mo>→</mo><msubsup><mi>𝒦</mi> <mo>•</mo> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">\eta_\bullet : \mathcal{G}_\bullet \to \mathcal{K}^I_\bullet</annotation></semantics></math> (with <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> the <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒦</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{K}_\bullet</annotation></semantics></math> as in example <a class="maruku-ref" href="#PathSpaceGroupoid"></a>) such that it fits into the diagram as depicted here on the right:</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><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>𝒢</mi></mtd> <mtd><msup><mo>⇓</mo> <mpadded width="0"><mi>η</mi></mpadded></msup></mtd> <mtd><mi>𝒦</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mpadded width="0"><mrow><msub><mi>g</mi> <mo>•</mo></msub></mrow></mpadded></msub></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>𝒦</mi> <mo>•</mo></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mo>•</mo></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>𝒢</mi> <mo>•</mo></msub></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>η</mi> <mo>•</mo></msub></mrow></mover></mtd> <mtd><msubsup><mi>𝒦</mi> <mo>•</mo> <mi>I</mi></msubsup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>g</mi> <mo>•</mo></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>𝒦</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow^{\mathrlap{f_\bullet}} \\ \mathcal{G} &\Downarrow^{\mathrlap{\eta}}& \mathcal{K} \\ & \searrow \nearrow_{\mathrlap{g_\bullet}} } \;\;\;\; \coloneqq \;\;\;\; \array{ && \mathcal{K}_\bullet \\ & {}^{\mathllap{f_\bullet}}\nearrow & \uparrow^{\mathrlap{(ev_1)_\bullet}} \\ \mathcal{G}_\bullet &\stackrel{\eta_\bullet}{\to}& \mathcal{K}^I_\bullet \\ & {}_{\mathllap{g_\bullet}}\searrow & \downarrow^{\mathrlap{(ev_0)_\bullet}} \\ && \mathcal{K} } \,. </annotation></semantics></math></div></div> <div class="num_defn" id="GroupoidsAsHomotopy1Types"> <h6 id="definition_notation">Definition (Notation)</h6> <p>Here and in the following, the convention is that we write</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_\bullet</annotation></semantics></math> (with the subscript decoration) when we regard groupoids with just homomorphisms (<a class="existingWikiWord" href="/nlab/show/functors">functors</a>) between them,</p> </li> <li> <p><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">\mathcal{G}</annotation></semantics></math> (without the subscript decoration) when we regard groupoids with homomorphisms (<a class="existingWikiWord" href="/nlab/show/functors">functors</a>) between them and <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> (<a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a>) between these</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><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mi>g</mi></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow^{\mathrlap{f}} \\ X &\Downarrow& Y \\ & \searrow \nearrow_{g} } \,. </annotation></semantics></math></div></li> </ul> <p>The unbulleted version of groupoids are also called <em><a class="existingWikiWord" href="/nlab/show/homotopy+1-types">homotopy 1-types</a></em> (or often just their <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>-<a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> are called this way.) Below we generalize this to arbitrary homotopy types (def. <a class="maruku-ref" href="#KanComplexesAsHomotopyTypes"></a>).</p> </div> <div class="num_example" id="MappingGroupoid"> <h6 id="example_8">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math> two groupoids, the <a class="existingWikiWord" href="/nlab/show/internal+hom">mapping groupoid</a> <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,Y]</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Y^X</annotation></semantics></math> is</p> <ul> <li><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><msub><mo stretchy="false">]</mo> <mn>0</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">[X,Y]_0 = </annotation></semantics></math> homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math>;</li> <li><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><msub><mo stretchy="false">]</mo> <mn>1</mn></msub><mo>=</mo></mrow><annotation encoding="application/x-tex">[X,Y]_1 = </annotation></semantics></math> homotopies between such.</li> </ul> </div> <div class="num_defn" id="HomotopyEquivalenceOfGroupoids"> <h6 id="definition_3">Definition</h6> <p>A (<a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy-</a>) <em><a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a></em> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi><mo>→</mo><mi>𝒦</mi></mrow><annotation encoding="application/x-tex">\mathcal{G} \to \mathcal{K}</annotation></semantics></math> which has a left and right <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> up to <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>.</p> </div> <div class="num_example" id="BZIsPiSOne"> <h6 id="example_9">Example</h6> <p>The map</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>B</mi></mstyle><mi>ℤ</mi><mover><mo>→</mo><mrow></mrow></mover><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}\mathbb{Z} \stackrel{}{\to} \Pi(S^1) </annotation></semantics></math></div> <p>which picks any point and sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{Z}</annotation></semantics></math> to the loop based at that point which winds around <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> times, is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+groupoids">equivalence of groupoids</a>.</p> </div> <div class="num_prop" id="DiscreteGroupoidIsDijointUnioonOfDeloopings"> <h6 id="proposition">Proposition</h6> <p>Assuming the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> in the ambient <a class="existingWikiWord" href="/nlab/show/set+theory">set theory</a>, every groupoid is equivalent to a disjoint union of <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoids, example <a class="maruku-ref" href="#DeloopingGroupoid"></a> – a <em><a class="existingWikiWord" href="/nlab/show/skeleton">skeleton</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>The statement of prop. <a class="maruku-ref" href="#DiscreteGroupoidIsDijointUnioonOfDeloopings"></a> becomes false as when we pass to groupoids that are equipped with <a class="existingWikiWord" href="/nlab/show/geometry">geometric</a> structure. This is the reason why for discrete geometry all <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons</a>-type field theories (namely <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a>-type theories) fundamentally involve just groups (and higher groups), while for nontrivial geometry there are genuine groupoid theories, for instance the <a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-models">AKSZ sigma-models</a>. But even so, <a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a> is usefully discussed in terms of groupoid technology, in particular since the choice of equivalence in prop. <a class="maruku-ref" href="#DiscreteGroupoidIsDijointUnioonOfDeloopings"></a> is not canonical.</p> </div> <div class="num_defn" id="HomotopyFiberProductOfGroupoids"> <h6 id="definition_4">Definition</h6> <p>Given two morphisms of groupoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>B</mi><mover><mo>←</mo><mi>g</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\to} B \stackrel{g}{\leftarrow} Y</annotation></semantics></math> their <em><a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a></em></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><munder><mo>×</mo><mi>B</mi></munder><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><munder><mo>→</mo><mi>g</mi></munder></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X \underset{B}{\times} Y &\stackrel{}{\to}& X \\ \downarrow &\swArrow& \downarrow^{\mathrlap{f}} \\ Y &\underset{g}{\to}& B } </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/limit">limit</a> <a class="existingWikiWord" href="/nlab/show/cone">cone</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><msub><mi>X</mi> <mo>•</mo></msub><munder><mo>×</mo><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></munder><msubsup><mi>B</mi> <mo>•</mo> <mi>I</mi></msubsup><munder><mo>×</mo><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></munder><msub><mi>Y</mi> <mo>•</mo></msub></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>X</mi> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msubsup><mi>B</mi> <mo>•</mo> <mi>I</mi></msubsup></mtd> <mtd><munder><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></munder></mtd> <mtd><msub><mi>B</mi> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>Y</mi> <mo>•</mo></msub></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>g</mi> <mo>•</mo></msub></mrow></munder></mtd> <mtd><msub><mi>B</mi> <mo>•</mo></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X_\bullet \underset{B_\bullet}{\times} B^I_\bullet \underset{B_\bullet}{\times} Y_\bullet &\to& &\to& X_\bullet \\ \downarrow && && \downarrow^{\mathrlap{f_\bullet}} \\ && B^I_\bullet &\underset{(ev_0)_\bullet}{\to}& B_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ Y_\bullet &\underset{g_\bullet}{\to}& B_\bullet } \,, </annotation></semantics></math></div> <p>hence the ordinary iterated <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> over the <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> groupoid, as indicated.</p> </div> <div class="num_remark" id="FiberProductsOfGroupoidsComponentwise"> <h6 id="remark_3">Remark</h6> <p>An ordinary fiber product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><munder><mo>×</mo><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></munder><msub><mi>Y</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet \underset{B_\bullet}{\times}Y_\bullet</annotation></semantics></math> of groupoids is given simply by the fiber product of the underlying sets of objects and morphisms:</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>X</mi> <mo>•</mo></msub><munder><mo>×</mo><mrow><msub><mi>B</mi> <mo>•</mo></msub></mrow></munder><msub><mi>Y</mi> <mo>•</mo></msub><msub><mo stretchy="false">)</mo> <mi>i</mi></msub><mo>=</mo><msub><mi>X</mi> <mi>i</mi></msub><munder><mo>×</mo><mrow><msub><mi>B</mi> <mi>i</mi></msub></mrow></munder><msub><mi>Y</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X_\bullet \underset{B_\bullet}{\times}Y_\bullet)_i = X_i \underset{B_i}{\times} Y_i \,. </annotation></semantics></math></div></div> <div class="num_example" id="PullbackOfEGGroupoidAsHomotopyFiberProduct"> <h6 id="example_10">Example</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, <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> a <a class="existingWikiWord" href="/nlab/show/group">group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \to \mathbf{B}G</annotation></semantics></math> a map into its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>, the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <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> of the <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>-<a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a> of example <a class="maruku-ref" href="#BGGroupoidAsActionGroupoid"></a> is equivalently the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with the point over <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></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>≃</mo><mi>X</mi><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> P \simeq X \underset{\mathbf{B}G}{\times} \ast \,. </annotation></semantics></math></div> <p>Namely both squares in the following diagram are pullback squares</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><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mo>*</mo> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msubsup><mo stretchy="false">)</mo> <mo>•</mo> <mi>I</mi></msubsup></mtd> <mtd><munder><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></munder></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mo>•</mo></msub></mtd> <mtd><munder><mo>→</mo><mrow></mrow></munder></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ P &\to& \mathbf{E}G &\to& \ast_\bullet \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& (\mathbf{B}G)_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ X_\bullet &\underset{}{\to}& (\mathbf{B}G)_\bullet } \,. </annotation></semantics></math></div> <p>(This is the first example of the more general phenomenon of <a class="existingWikiWord" href="/nlab/show/universal+principal+infinity-bundles">universal principal infinity-bundles</a>.)</p> </div> <div class="num_example" id="LoopSpaceGroupoid"> <h6 id="example_11">Example</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\ast \to X</annotation></semantics></math> a point in it, we call</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi><mo>≔</mo><mo>*</mo><munder><mo>×</mo><mi>X</mi></munder><mo>*</mo></mrow><annotation encoding="application/x-tex"> \Omega X \coloneqq \ast \underset{X}{\times} \ast </annotation></semantics></math></div> <p>the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space groupoid</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>For <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> a group 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></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid from example <a class="maruku-ref" href="#DeloopingGroupoid"></a>, we have</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>Ω</mi><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>=</mo><mo>*</mo><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G \simeq \Omega \mathbf{B}G = \ast \underset{\mathbf{B}G}{\times} \ast \,. </annotation></semantics></math></div> <p>Hence <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 the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a> of its own <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>, as it should be.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We are to compute the ordinary limiting cone <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub></mrow></munder><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msup><mi>G</mi> <mi>I</mi></msup><msub><mo stretchy="false">)</mo> <mo>•</mo></msub><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub></mrow></munder><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast \underset{\mathbf{B}G_\bullet}{\times} (\mathbf{B}G^I)_\bullet \underset{\mathbf{B}G_\bullet}{\times} \ast</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></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msubsup><mo stretchy="false">)</mo> <mo>•</mo> <mi>I</mi></msubsup></mtd> <mtd><munder><mo>→</mo><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></munder></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>ev</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><munder><mo>→</mo><mrow></mrow></munder></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ &\to& &\to& \ast \\ \downarrow && && \downarrow^{\mathrlap{}} \\ && (\mathbf{B}G)^I_\bullet &\underset{(ev_0)_\bullet}{\to}& \mathbf{B}G_\bullet \\ \downarrow && \downarrow^{\mathrlap{(ev_1)_\bullet}} \\ \ast &\underset{}{\to}& \mathbf{B}G_\bullet } \,, </annotation></semantics></math></div> <p>In the middle we have the groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msubsup><mo stretchy="false">)</mo> <mo>•</mo> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">(\mathbf{B}G)^I_\bullet</annotation></semantics></math> whose objects are elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and whose morphisms starting at some element are labeled by pairs of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>h</mi> <mn>2</mn></msub><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">h_1, h_2 \in G</annotation></semantics></math> and end at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mn>1</mn></msub><mo>⋅</mo><mi>g</mi><mo>⋅</mo><msub><mi>h</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">h_1 \cdot g \cdot h_2</annotation></semantics></math>. Using remark <a class="maruku-ref" href="#FiberProductsOfGroupoidsComponentwise"></a> the limiting cone is seen to precisely pick those morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mo>•</mo></msub><msubsup><mo stretchy="false">)</mo> <mo>•</mo> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">(\mathbf{B}G_\bullet)^I_\bullet</annotation></semantics></math> such that these two elements are constant on the neutral element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>h</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>h</mi> <mn>2</mn></msub><mo>=</mo><mi>e</mi><mo>=</mo><msub><mi>id</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">h_1 = h_2 = e = id_{\ast}</annotation></semantics></math>, hence it produces just the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> regarded as a groupoid with only identity morphisms, as in example <a class="maruku-ref" href="#SetAsGroupoid"></a>.</p> </div> <div class="num_prop" id="FreeLoopSpaceOfGroupoid"> <h6 id="proposition_2">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a> is</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 stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>X</mi><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></munder><mi>X</mi></mrow><annotation encoding="application/x-tex"> [\Pi(S^1), X] \simeq X \underset{[\Pi(S^0), X]}{\times}X </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>≃</mo><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\Pi_1(S^0) \simeq \ast \coprod \ast</annotation></semantics></math>. Therefore the <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><msubsup><mo stretchy="false">]</mo> <mo>•</mo> <mi>I</mi></msubsup></mrow><annotation encoding="application/x-tex">[\Pi(S^0), X_\bullet]^I_\bullet</annotation></semantics></math> has</p> <ul> <li> <p>objects are pairs of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math>;</p> </li> <li> <p>morphisms are commuting squares of such.</p> </li> </ul> <p>Now the fiber product in def. <a class="maruku-ref" href="#HomotopyFiberProductOfGroupoids"></a> picks in there those pairs of morphisms for which both start at the same object, and both end at the same object. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><msub><mo stretchy="false">]</mo> <mo>•</mo></msub></mrow></munder><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><msubsup><mo stretchy="false">]</mo> <mo>•</mo> <mi>I</mi></msubsup><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mi>X</mi> <mo>•</mo></msub><msub><mo stretchy="false">]</mo> <mo>•</mo></msub></mrow></munder><mi>X</mi></mrow><annotation encoding="application/x-tex">X_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} [\Pi(S^0), X_\bullet]^I_\bullet \underset{[\Pi(S^0), X_\bullet]_\bullet}{\times} X</annotation></semantics></math> is the groupoid whose</p> <ul> <li> <p>objects are diagrams in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> of the form</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><mo>↗</mo><mo>↘</mo></mtd></mtr> <mtr><mtd><msub><mi>x</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd><msub><mi>x</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><mo>↗</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ & \nearrow \searrow \\ x_0 && x_1 \\ & \searrow \nearrow } </annotation></semantics></math></div></li> <li> <p>morphism are cylinder-diagrams over these.</p> </li> </ul> <p>One finds along the lines of example <a class="maruku-ref" href="#BZIsPiSOne"></a> that this is equivalent to maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(S^1)</annotation></semantics></math> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> and homotopies between these.</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>Even though all these models of the circle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi_1(S^1)</annotation></semantics></math> are equivalent, below the special appearance of the circle in the proof of prop. <a class="maruku-ref" href="#FreeLoopSpaceOfGroupoid"></a> as the combination of two semi-circles will be important for the following proofs. As we see in a moment, this is the natural way in which the circle appears as the composition of an <a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a> with a <a class="existingWikiWord" href="/nlab/show/coevaluation+map">coevaluation map</a>.</p> </div> <div class="num_example" id="AdjointActionGroupoidFromFreeLoopSpaceObject"> <h6 id="example_12">Example</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, the <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a> of its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <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></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ad</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">G//_{ad} G</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a>, def. <a class="maruku-ref" href="#ActionGroupoid"></a>, of the <a class="existingWikiWord" href="/nlab/show/adjoint+action">adjoint action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on itself:</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 stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ad</mi></msub><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Pi(S^1), \mathbf{B}G] \simeq G//_{ad} G \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_13">Example</h6> <p>For an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\flat U(1)</annotation></semantics></math> we have</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 stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>≃</mo><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ad</mi></msub><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><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><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><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(S^1), \mathbf{B}\flat U(1)] \simeq \flat U(1)//_{ad} \flat U(1) \simeq (\flat U(1)) \times (\mathbf{B}\flat U(1)) \,. </annotation></semantics></math></div></div> <div class="num_example" id="GroupCharacterAsClassFunctionByFreeLoopSpace"> <h6 id="example_14">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \colon G \to \flat U(1)</annotation></semantics></math> be a group homomorphism, hence a <a class="existingWikiWord" href="/nlab/show/group+character">group character</a>. By example <a class="maruku-ref" href="#DeloopingGroupoid"></a> this has a <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> to a groupoid homomorphism</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>B</mi></mstyle><mi>c</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><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"> \mathbf{B}c \;\colon\; \mathbf{B}G \to \mathbf{B}\flat U(1) \,. </annotation></semantics></math></div> <p>Under the <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a> construction this becomes</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 stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>c</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><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"> [\Pi(S^1), \mathbf{B}c] \;\colon\; [\Pi(S^1), \mathbf{B}G] \to [\Pi(S^1), \mathbf{B}\flat U(1)] </annotation></semantics></math></div> <p>hence</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 stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>c</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ad</mi></msub><mi>G</mi><mo>→</mo><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><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(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to \flat U(1) \times \mathbf{B}U(1) \,. </annotation></semantics></math></div> <p>So by postcomposing with the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> on the first factor we recover from the general <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of groupoids the statement that a group character is a <a class="existingWikiWord" href="/nlab/show/class+function">class function</a> on <a class="existingWikiWord" href="/nlab/show/conjugacy+classes">conjugacy classes</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 stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>c</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo stretchy="false">/</mo><msub><mo stretchy="false">/</mo> <mi>ad</mi></msub><mi>G</mi><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(S^1), \mathbf{B}c] \;\colon\; G//_{ad}G \to U(1) \,. </annotation></semantics></math></div></div> <h4 id="CorrespondencesOfGroupoids">Trajectories of fields and Correspondences of groupoids</h4> <p>With some basic <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> in hand, we can now talk about <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a> in finite gauge theories, namely about <a class="existingWikiWord" href="/nlab/show/spans">spans</a>/<a class="existingWikiWord" href="/nlab/show/correspondences">correspondences</a> of groupoids and their composition. These correspondences of groupoids encode <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a>/histories of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field configurations</a>.</p> <p>Namely consider a groupoid to be called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>∈</mo></mrow><annotation encoding="application/x-tex">\mathbf{Fields} \in</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a>, to be thought of as the <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> in some field theory, or equivalently and specifically as the <a class="existingWikiWord" href="/nlab/show/target+space">target space</a> of a <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> field theory. This just means that for <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> any <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> thought of as <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> or <a class="existingWikiWord" href="/nlab/show/worldvolume">worldvolume</a>, the space of fields <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Fields}(\Sigma)</annotation></semantics></math> of the field theory 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> is the <a class="existingWikiWord" href="/nlab/show/mapping+stack">mapping stack</a> (<a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a>) from <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> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields}</annotation></semantics></math>, which means here for DW theory that it is the mapping groupoid, def. <a class="maruku-ref" href="#MappingGroupoid"></a>, out of the fundamental groupoid, def. <a class="maruku-ref" href="#FundamentalGroupoid"></a>, 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>:</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>Fields</mi></mstyle><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields}(\Sigma) = [\Pi_1(\Sigma), \mathbf{Fields}] \,. </annotation></semantics></math></div> <p>We think of the <a class="existingWikiWord" href="/nlab/show/objects">objects</a> of the groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Pi_1(\Sigma), \mathbf{Fields}]</annotation></semantics></math> as being the fields themselves, and of the <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> as being the <a class="existingWikiWord" href="/nlab/show/gauge+transformations">gauge transformations</a> between them.</p> <p>The example to be of interest in a moment is that where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{Fields} = \mathbf{B}G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid as in def. <a class="maruku-ref" href="#DeloopingGroupoid"></a>, in which case the fields are equivalently <a class="existingWikiWord" href="/nlab/show/flat+connection">flat</a> <a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a>. In fact in the discrete and 1-dimensional case currently considered this is essentially the only example, due to prop. <a class="maruku-ref" href="#DiscreteGroupoidIsDijointUnioonOfDeloopings"></a>, but for the general idea and for the more general cases considered further below, it is useful to have the notation allude to more general moduli spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Fields}</annotation></semantics></math>.</p> <p>The simple but crucial observation that shows why <a class="existingWikiWord" href="/nlab/show/spans">spans</a>/<a class="existingWikiWord" href="/nlab/show/correspondences">correspondences</a> of groupoids show up in prequantum field theory is the following.</p> <div class="num_example"> <h6 id="example_15">Example</h6> <p>If <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 <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>, hence a <a class="existingWikiWord" href="/nlab/show/manifold+with+boundary">manifold with boundary</a> with incoming <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> component <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>in</mi></msub><mo>↪</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma_{in} \hookrightarrow \Sigma</annotation></semantics></math> and outgoing boundary components <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>out</mi></msub><mo>↪</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma_{out} \hookrightarrow \Sigma</annotation></semantics></math>, then the resulting <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a> of manifolds</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><mi>Σ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><msub><mi>Σ</mi> <mi>in</mi></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>Σ</mi> <mi>out</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \Sigma \\ & \nearrow && \nwarrow \\ \Sigma_{in} && && \Sigma_{out} } </annotation></semantics></math></div> <p>is sent under the operation of mapping into the moduli space of fields</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> <mn>1</mn></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Mfds</mi> <mi>op</mi></msup><mo>→</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> [\Pi_1(-), \mathbf{Fields}] \;\colon\; Mfds^{op} \to Grpd </annotation></semantics></math></div> <p>to a <a class="existingWikiWord" href="/nlab/show/span">span</a> of groupoids</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><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mi>in</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mi>out</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && [\Pi_1(\Sigma), \mathbf{Fields}] \\ & \swarrow && \searrow \\ [\Pi_1(\Sigma_{in}), \mathbf{Fields}] && && [\Pi_1(\Sigma_{out}), \mathbf{Fields}] } \,. </annotation></semantics></math></div> <p>Here the left and right homomorphisms are those which take a field configuration 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> and restrict it to the incoming and to the outgoing field configuration, respectively. (And this being a homomorphism of groupoids means that everything respects the <a class="existingWikiWord" href="/nlab/show/gauge+symmetry">gauge symmetry</a> on the fields.) Hence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>Σ</mi> <mrow><mi>in</mi><mo>,</mo><mi>out</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Pi_1(\Sigma_{in,out}),\mathbf{Fields}]</annotation></semantics></math> is thought of as the spaces of incoming and outgoing field configurations, respectively, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Pi_1(\Sigma), \mathbf{Fields}]</annotation></semantics></math> is to be interpreted as the space of <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a> (sometimes: <em>histories</em>) of field cofigurations over <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a>/<a class="existingWikiWord" href="/nlab/show/worldvolumes">worldvolumes</a> of shape <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>This should make it plausible that specifying the field content of a 1-dimensional discrete gauge field theory is a functorial assignment</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>Fields</mi></mstyle><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Bord</mi> <mn>1</mn></msub><mo>→</mo><mi>Span</mi><mo stretchy="false">(</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \;\colon\; Bord_1 \to Span(Grpd) </annotation></semantics></math></div> <p>from a <a class="existingWikiWord" href="/nlab/show/category+of+cobordisms">category of cobordisms</a> of dimension one into a category of such <a class="existingWikiWord" href="/nlab/show/spans">spans</a> of groupoids. It sends points to spaces of field configurations on the point and 1-dimensional manifolds such as the circle to spaces of trajectories of field configurations on them.</p> <p>Moreover, for a <em>local</em> field theory it should be true that the field configurations on the circle, say, are determined from gluing the field configurations on any decomposition of the circle, notably a decomposition into two semi-circles. But since we are dealing with a <a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological field theory</a>, its field configurations on a contractible interval such as the semicircle will be equivalent to the field configurations on the point itself.</p> <p>The way that the fields on higher spheres in a topological field theory are induced from the fields on the point is by an analog of <em><a class="existingWikiWord" href="/nlab/show/traces">traces</a></em> for spaces of fields, and higher traces of such correspondences (the “<a class="existingWikiWord" href="/nlab/show/span+trace">span trace</a>”). This is because by the <a class="existingWikiWord" href="/nlab/show/cobordism+theorem">cobordism theorem</a>, the field configurations on, notably, the <a class="existingWikiWord" href="/nlab/show/n-sphere">n-sphere</a> are given by 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>-fold <a class="existingWikiWord" href="/nlab/show/span+trace">span trace</a> of the field configurations on the point, the trace of the traces of the … of the 1-trace. This is because for instance the 1-sphere, hence the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> is, regarded as a 1-dimensional <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> itself pretty much manifestly a <a class="existingWikiWord" href="/nlab/show/trace">trace</a> on the point in the <a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a> formulation of traces.</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><msup><mo>*</mo> <mo>−</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo>*</mo> <mo>+</mo></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \ast^- \\ & \swarrow & & \nwarrow \\ \downarrow && && \uparrow \\ & \searrow && \nearrow \\ && \ast^+ } \,. </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>*</mo> <mo>+</mo></msup></mrow><annotation encoding="application/x-tex">\ast^+</annotation></semantics></math> is the point with its positive orientation, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>*</mo> <mo>−</mo></msup></mrow><annotation encoding="application/x-tex">\ast^-</annotation></semantics></math> is its <a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a> in the <a class="existingWikiWord" href="/nlab/show/category+of+cobordisms">category of cobordisms</a>, the point with the reverse orientation. Since, by this picture, the construction that produces the circle from the point is one that involves only the <a class="existingWikiWord" href="/nlab/show/coevaluation+map">coevaluation map</a> and <a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a> map on the point regarded as a <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a>, a <a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological field theory</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mo lspace="verythinmathspace">:</mo><msub><mi>Bord</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>Span</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z \colon Bord_n \to Span_n(\mathbf{H})</annotation></semantics></math>, since it respects all this structure, takes the circle to precisely the same kind of diagram, but now in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Span</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><msup><mo stretchy="false">)</mo> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex">Span_n(\mathbf{H})^\otimes</annotation></semantics></math>, where it becomes instead the <a class="existingWikiWord" href="/nlab/show/span+trace">span trace</a> on the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Fields}(\ast)</annotation></semantics></math> over the point. This we discuss now.</p> <p>Before talking about correspondences of groupoids, we need to organize the groupoids themselves a bit more.</p> <div class="num_defn" id="TwoOneCategory"> <h6 id="definition_5">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is</p> <ol> <li> <p>a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{C}_0</annotation></semantics></math> – the “collection of <a class="existingWikiWord" href="/nlab/show/objects">objects</a>”;</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/tuple">tuple</a> <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><mo>∈</mo><msub><mi>𝒞</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>𝒞</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">(X,Y) \in \mathcal{C}_0 \times \mathcal{C}_0</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(X,Y)</annotation></semantics></math> – the <em><a class="existingWikiWord" href="/nlab/show/hom-groupoid">hom-groupoid</a></em> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>;</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/triple">triple</a> <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>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>𝒞</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>𝒞</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>𝒞</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">(X,Y,Z) \in \mathcal{C}_0 \times \mathcal{C}_0 \times \mathcal{C}_0</annotation></semantics></math> a groupoid homomorphism (<a class="existingWikiWord" href="/nlab/show/functor">functor</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∘</mo> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>×</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \circ_{X,Y,Z} \colon \mathcal{C}(X,Y) \times \mathcal{C}(Y,Z) \to \mathcal{C}(X,Z) </annotation></semantics></math></div> <p>called <em><a class="existingWikiWord" href="/nlab/show/composition">composition</a></em> or <em><a class="existingWikiWord" href="/nlab/show/horizontal+composition">horizontal composition</a></em> for emphasis;</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/quadruple">quadruple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo>,</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(W,X,Y,Z,)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> – the <em><a class="existingWikiWord" href="/nlab/show/associator">associator</a></em> –</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>𝒞</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>×</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>𝒞</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>×</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>⇙</mo> <mrow><msub><mi>α</mi> <mrow><mi>W</mi><mo>,</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi></mrow></msub></mrow></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>𝒞</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>𝒞</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathcal{C}(W,X) \times \mathcal{C}(X,Y) \times \mathcal{C}(Y,Z) &\stackrel{}{\to}& \mathcal{C}(W,Y) \times \mathcal{C}(Y,Z) \\ \downarrow &\swArrow_{\alpha_{W,X,Y,Z}}& \downarrow \\ \mathcal{C}(W,X) \times \mathcal{C}(X,Z) &\stackrel{}{\to}& \mathcal{C}(W,Z) } </annotation></semantics></math></div> <p>(…) and similarly a <a class="existingWikiWord" href="/nlab/show/unitality">unitality</a> homotopy (…)</p> </li> </ol> <p>such that for each <a class="existingWikiWord" href="/nlab/show/quintuple">quintuple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo>,</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V,W,X,Y,Z)</annotation></semantics></math> the associators satisfy the <a class="existingWikiWord" href="/nlab/show/pentagon+identity">pentagon identity</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/objects">objects</a> of the <a class="existingWikiWord" href="/nlab/show/hom-groupoid">hom-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(X,Y)</annotation></semantics></math> we call the <em><a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a></em> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, indicated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\to} Y</annotation></semantics></math>, and the <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}(X,Y)</annotation></semantics></math> we call the <a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, indicated by</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></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo><msup><mo>↘</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>⇓</mo></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo><msub><mo>↗</mo> <mpadded width="0"><mi>g</mi></mpadded></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \\ & \nearrow \searrow^{\mathrlap{f}} \\ X &\Downarrow& Y \\ & \searrow \nearrow_{\mathrlap{g}} } \,. </annotation></semantics></math></div> <p>If all associators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> can and are chosen to be the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> then this is called a <em><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict (2,1)-category</a></em>.</p> </div> <div class="num_defn"> <h6 id="definitionexample">Definition/Example</h6> <p>Write <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> for the <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict</a> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a>, def. <a class="maruku-ref" href="#TwoOneCategory"></a>, whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> <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">\mathcal{G}</annotation></semantics></math>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/functors">functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>𝒢</mi><mo>→</mo><mi>𝒦</mi></mrow><annotation encoding="application/x-tex">f \colon \mathcal{G} \to \mathcal{K}</annotation></semantics></math>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> between these.</p> </li> </ul> </div> <div class="num_defn" id="OneSpansInOneGroupoids"> <h6 id="definitionexample_2">Definition/Example</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Span_1(Grpd)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/spans">spans</a>/<a class="existingWikiWord" href="/nlab/show/correspondences">correspondences</a> of <a class="existingWikiWord" href="/nlab/show/functors">functors</a>, hence</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>A</mi></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>;</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\leftarrow& X &\rightarrow& B } \,; </annotation></semantics></math></div></li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> in <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> of the form</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>X</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>⇘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↖</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && X_1 \\ & \swarrow &\downarrow& \searrow \\ A &\seArrow& \downarrow^{\mathrlap{\simeq}} &\swArrow& B \\ & \nwarrow &\downarrow& \nearrow \\ && X_2 } </annotation></semantics></math></div></li> <li> <p><a class="existingWikiWord" href="/nlab/show/composition">composition</a> is given by forming the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a>, def. <a class="maruku-ref" href="#HomotopyFiberProductOfGroupoids"></a>, of the two adjacent homomorphisms of two spans, hence for two spans</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mrow></mrow></mover><mi>K</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> X \stackrel{}{\leftarrow} K \rightarrow Y </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mover><mo>←</mo><mrow></mrow></mover><mi>L</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex"> Y \stackrel{}{\leftarrow} L \rightarrow Z </annotation></semantics></math></div> <p>their composite is the span which is the outer part of the diagram</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></mtd> <mtd></mtd> <mtd><mi>K</mi><munder><mo>×</mo><mi>Y</mi></munder><mi>L</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>K</mi></mtd> <mtd></mtd> <mtd><mo>⇙</mo></mtd> <mtd></mtd> <mtd><mi>L</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && && K \underset{Y}{\times}L \\ && & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}} \\ && K && \swArrow && L \\ & \swarrow && \searrow && \swarrow && \searrow \\ X && && Y && && Z } \,. </annotation></semantics></math></div></li> </ul> </div> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p>There is the structure of a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%282%2C1%29-category">symmetric monoidal (2,1)-category</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Span_1(Grpd)</annotation></semantics></math> by degreewise <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> in <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</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>X</mi><mo>←</mo><mi>K</mi><mo>→</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo>←</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo>→</mo><mover><mi>Y</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>×</mo><mover><mi>X</mi><mo stretchy="false">˜</mo></mover><mo>←</mo><mi>K</mi><mo>×</mo><mover><mi>K</mi><mo stretchy="false">˜</mo></mover><mo>→</mo><mi>Y</mi><mo>×</mo><mover><mi>Y</mi><mo stretchy="false">˜</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X \leftarrow K \rightarrow Y) \otimes (\tilde X \leftarrow \tilde K \rightarrow \tilde Y) \;\coloneqq\; X \times \tilde X \leftarrow K \times \tilde K \rightarrow Y \times \tilde Y \,. </annotation></semantics></math></div></div> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>An <a class="existingWikiWord" href="/nlab/show/object">object</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> of a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%282%2C1%29-category">symmetric monoidal (2,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^\otimes</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/fully+dualizable+object">fully dualizable</a> if there exists</p> <ol> <li> <p>another object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">X^\ast</annotation></semantics></math>, to be called the <em><a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a></em>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/1-morphism">1-morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>X</mi> <mo>*</mo></msup><mo>⊗</mo><mi>X</mi><mo>→</mo><mi>𝕀</mi></mrow><annotation encoding="application/x-tex">ev_X \colon X^\ast \otimes X \to \mathbb{I}</annotation></semantics></math>, to be called the <em><a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a></em>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/1-morphism">1-morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>coev</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo><mi>𝕀</mi><mo>→</mo><mi>X</mi><mo>⊗</mo><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">coev_X \colon \mathbb{I} \to X \otimes X^\ast</annotation></semantics></math>, to be called the <a class="existingWikiWord" href="/nlab/show/coevaluation+map">coevaluation map</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</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></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mrow><msub><mi>coev</mi> <mrow><mi>tr</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msup></mtd> <mtd><msup><mo>↘</mo> <mrow><msub><mpadded width="0"><mi>id</mi></mpadded> <mi>𝕀</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd><mi>𝕀</mi></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>coev</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><msup><mi>X</mi> <mo>*</mo></msup><mo>⊗</mo><mi>X</mi></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>ev</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><mi>𝕀</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \rightarrow \\ & \nearrow &\Downarrow^{coev_{tr(X)}}& \searrow^{\mathrlap{id}_{\mathbb{I}}} \\ \mathbb{I} &\underset{coev_X}{\to}& X^\ast \otimes X &\underset{ev_X}{\to}& \mathbb{I} } </annotation></semantics></math></div> <p>and</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><mo>→</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo>⇑</mo> <mrow><msub><mi>ev</mi> <mrow><mi>tr</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msup></mtd> <mtd><msup><mo>↘</mo> <mrow><msub><mpadded width="0"><mi>id</mi></mpadded> <mi>𝕀</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd><mi>𝕀</mi></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>coev</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><msup><mi>X</mi> <mo>*</mo></msup><mo>⊗</mo><mi>X</mi></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>ev</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><mi>𝕀</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \rightarrow \\ & \nearrow &\Uparrow^{ev_{tr(X)}}& \searrow^{\mathrlap{id}_{\mathbb{I}}} \\ \mathbb{I} &\underset{coev_X}{\to}& X^\ast \otimes X &\underset{ev_X}{\to}& \mathbb{I} } </annotation></semantics></math></div> <p>and</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><mo>→</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo>⇓</mo> <mrow><mi>sa</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><msup><mo>↘</mo> <mrow><msub><mpadded width="0"><mi>id</mi></mpadded> <mi>𝕀</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd><msup><mi>X</mi> <mo>*</mo></msup><mo>⊗</mo><mi>X</mi></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>ev</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><mi>𝕀</mi></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>coev</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><msup><mi>X</mi> <mo>*</mo></msup><mo>⊗</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \rightarrow \\ & \nearrow &\Downarrow^{sa(X)}& \searrow^{\mathrlap{id}_{\mathbb{I}}} \\ X^\ast \otimes X &\underset{ev_X}{\to}& \mathbb{I} &\underset{coev_X}{\to}& X^\ast \otimes X } </annotation></semantics></math></div> <p>(the <span class="newWikiWord">saddle<a href="/nlab/new/saddle">?</a></span>)</p> <p>and</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><mo>→</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd><msup><mo>⇑</mo> <mrow><mi>cosa</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><msup><mo>↘</mo> <mrow><msub><mpadded width="0"><mi>id</mi></mpadded> <mi>𝕀</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd><msup><mi>X</mi> <mo>*</mo></msup><mo>⊗</mo><mi>X</mi></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>ev</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><mi>𝕀</mi></mtd> <mtd><munder><mo>→</mo><mrow><msub><mi>coev</mi> <mi>X</mi></msub></mrow></munder></mtd> <mtd><msup><mi>X</mi> <mo>*</mo></msup><mo>⊗</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \rightarrow \\ & \nearrow &\Uparrow^{cosa(X)}& \searrow^{\mathrlap{id}_{\mathbb{I}}} \\ X^\ast \otimes X &\underset{ev_X}{\to}& \mathbb{I} &\underset{coev_X}{\to}& X^\ast \otimes X } </annotation></semantics></math></div> <p>(the co-saddle)</p> </li> </ol> <p>such that these exhibit an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> and are themselves adjoint (…).</p> </div> <div class="num_defn"> <h6 id="definition_8">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%282%2C1%29-category">symmetric monoidal (2,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, and a <a class="existingWikiWord" href="/nlab/show/fully+dualizable+object">fully dualizable object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/1-morphism">1-morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon X \to X</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/trace">trace</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> is the <a class="existingWikiWord" href="/nlab/show/composition">composition</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>tr</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝕀</mi><mover><mo>→</mo><mrow><msub><mi>coev</mi> <mi>X</mi></msub></mrow></mover><mi>X</mi><mo>⊗</mo><msup><mi>X</mi> <mo>*</mo></msup><mover><mo>→</mo><mrow><mi>f</mi><mo>⊗</mo><msub><mi>id</mi> <mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow></msub></mrow></mover><mi>X</mi><mo>⊗</mo><msup><mi>X</mi> <mo>*</mo></msup><mover><mo>→</mo><mrow><msub><mi>ev</mi> <mi>x</mi></msub></mrow></mover><mi>𝕀</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> tr(f) \;\colon\; \mathbb{I} \stackrel{coev_X}{\to} X \otimes X^\ast \stackrel{f \otimes id_{X^\ast}}{\to} X \otimes X^\ast \stackrel{ev_x}{\to} \mathbb{I} \,. </annotation></semantics></math></div></div> <div class="num_prop" id="GroupoidInSpanOneIsSelfDual"> <h6 id="proposition_3">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Grpd</mi><mo>↪</mo><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in Grpd \hookrightarrow Span_1(Grpd)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Span_1(Grpd)</annotation></semantics></math>, and in fact is self-dual.</p> <p>The <a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">ev_X</annotation></semantics></math>, hence the possible image of a symmetric monoidal functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bord_1 \to Span_1(Grpd)</annotation></semantics></math> of a cobordism of the form</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><mo>←</mo></mtd> <mtd><msup><mo>*</mo> <mo>−</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mo>*</mo> <mo>+</mo></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \leftarrow & \ast^- \\ & \swarrow \\ \downarrow \\ & \searrow \\ && \rightarrow & \ast^+ } </annotation></semantics></math></div> <p>is given by the <a class="existingWikiWord" href="/nlab/show/span">span</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></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && X \\ & \swarrow && \searrow \\ \ast && && [\Pi_1(S^0),X] &\simeq& X \times X } </annotation></semantics></math></div> <p>and the <a class="existingWikiWord" href="/nlab/show/coevaluation+map">coevaluation map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>coev</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">coev_X</annotation></semantics></math> by the reverse span.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Grpd</mi><mo>↪</mo><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in Grpd \hookrightarrow Span_1(Grpd)</annotation></semantics></math> any object, the <a class="existingWikiWord" href="/nlab/show/trace">trace</a> (“<a class="existingWikiWord" href="/nlab/show/span+trace">span trace</a>”) of the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> on it, hence the image of</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><msup><mo>*</mo> <mo>−</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo>*</mo> <mo>+</mo></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \ast^- \\ & \swarrow & & \nwarrow \\ \downarrow && && \uparrow \\ & \searrow && \nearrow \\ && \ast^+ } </annotation></semantics></math></div> <p>is its <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a>, prop. <a class="maruku-ref" href="#FreeLoopSpaceOfGroupoid"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>tr</mi><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> tr(id_X) \simeq \left( \array{ && [\Pi_1(S^1), X] \\ & \swarrow && \searrow \\ \ast && && \ast } \right) \,. </annotation></semantics></math></div> <p>The second order covaluation map on the <a class="existingWikiWord" href="/nlab/show/span+trace">span trace</a> of the identity is</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></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>←</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && && \ast \\ && && \uparrow \\ && && X \\ && && \downarrow \\ && && [\Pi(S^1), X] \\ && & \swarrow & & \searrow \\ \ast &\leftarrow& X &\rightarrow& [\Pi(S^0), X] &\leftarrow& X &\rightarrow& & \ast } \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By prop. <a class="maruku-ref" href="#GroupoidInSpanOneIsSelfDual"></a> the <a class="existingWikiWord" href="/nlab/show/trace">trace</a> of the identity is given by the composite span</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></mtd> <mtd></mtd> <mtd><mi>X</mi><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></munder><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mo>⇙</mo></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && && X \underset{[\Pi_1(S^1), X]}{\times} X \\ && & \swarrow && \searrow \\ && X && \swArrow && X \\ & \swarrow && \searrow && \swarrow && \searrow \\ \ast && && [\Pi_1(S^0),X] && && \ast } \,. </annotation></semantics></math></div> <p>By prop. <a class="maruku-ref" href="#FreeLoopSpaceOfGroupoid"></a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></munder><mi>X</mi><mo>≃</mo><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \underset{[\Pi_1(S^1), X]}{\times} X \simeq [\Pi_1(S^1), X] \,. </annotation></semantics></math></div> <p>Along these lines one checks the required <a class="existingWikiWord" href="/nlab/show/zig-zag+identities">zig-zag identities</a>.</p> </div> <h4 id="1dDWLocalFieldTheory">Action functionals and Slice groupoids</h4> <p>We have now assembled all the ingredients need in order to formally regard a <a class="existingWikiWord" href="/nlab/show/group+character">group character</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \colon G \to U(1)</annotation></semantics></math> on a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> as a local action functional of a prequantum field theory, hence as a <a class="existingWikiWord" href="/nlab/show/fully+dualizable+object">fully dualizable object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>c</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msub><mi mathvariant="normal">Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><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"> S \;\coloneqq\; \left[ \array{ \mathbf{B}G \\ \downarrow^{\mathrlap{c}} \\ \mathbf{B}\flat U(1) } \right] \;\in \; \mathrm{Span}_1(Grpd, \mathbf{B}\flat U(1)) </annotation></semantics></math></div> <p>in a <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> of correspondences of groupoids as in def. <a class="maruku-ref" href="#OneSpansInOneGroupoids"></a>, but equipped with maps and homotopies between maps to the <a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}\flat U(1)</annotation></semantics></math>. This is described in def. <a class="maruku-ref" href="#OneSpansInOneGroupoidsOverBU"></a> below. Before stating this, we recall for the 1-dimensional case the general story of def. <a class="maruku-ref" href="#LocalPrequantumFieldWithAction"></a>.</p> <div class="num_example" id="HomotopyAsActionFunctional"> <h6 id="example_16">Example</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/discrete+groupoid">discrete groupoid</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>, <a class="existingWikiWord" href="/nlab/show/functions">functions</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \exp(i S) \colon X \to \flat U(1) </annotation></semantics></math></div> <p>are in <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> with <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> of the form</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><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mi>ϕ</mi></msub></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && X \\ & \swarrow && \searrow \\ \ast && \swArrow_{\phi} && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}\flat U(1) } \,, </annotation></semantics></math></div> <p>where the function corresponding to this homotopy is that given by the unique factorization through the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mo>*</mo><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></munder><mo>*</mo></mrow><annotation encoding="application/x-tex">\flat U(1) \simeq \ast \underset{\mathbf{B}\flat U(1)}{\times} \ast</annotation></semantics></math> (example <a class="maruku-ref" href="#LoopSpaceGroupoid"></a>) as shown on the right of</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><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mi>ϕ</mi></msub></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mo>⇙</mo></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && X \\ & \swarrow && \searrow \\ \ast && \swArrow_{\phi} && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}\flat U(1) } \;\;\;\; \simeq \;\;\;\; \array{ && X \\ &\swarrow& \downarrow^{\mathrlap{\exp(i S)}} & \searrow \\ && \flat U(1) \\ \downarrow & \swarrow && \searrow & \downarrow \\ \ast && \swArrow && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}\flat U(1) } \,, </annotation></semantics></math></div></div> <p>This means that if we have an <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> on a space of <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a>, and if these trajectories are given by <a class="existingWikiWord" href="/nlab/show/spans">spans</a>/<a class="existingWikiWord" href="/nlab/show/correspondences">correspondences</a> of groupoids as discussed above, then the action functional is naturally expressed as the homotopy filling a completion of the span to a square diagram over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}\flat U(1)</annotation></semantics></math>. Therefore we cosider the following.</p> <div class="num_defn" id="OneSpansInOneGroupoidsOverBU"> <h6 id="definition_9">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo>,</mo><mo>♭</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Span_1(Grpd, \flat\mathbf{B}U(1))</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</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> equipped with a morpism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex"> \left[ \array{ X \\ \downarrow^{\mathrlap{f}} \\ \mathbf{B}\flat U(1) } \right] </annotation></semantics></math></div></li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are <a class="existingWikiWord" href="/nlab/show/spans">spans</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>←</mo><mi>Y</mi><mo>→</mo><msub><mi>X</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \leftarrow Y \rightarrow X_2</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</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></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mi>ϕ</mi></msub></mtd> <mtd></mtd> <mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && Y \\ & {}^{\mathllap{f_1}}\swarrow && \searrow^{\mathrlap{f_2}} \\ X_1 && \swArrow_{\phi} && X_2 \\ & \searrow && \swarrow \\ && \mathbf{B}\flat U(1) } </annotation></semantics></math></div></li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-morphisms">2-morphisms</a> are morphism of spans compatible with the maps to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}\flat U(1)</annotation></semantics></math> in the evident way.</p> </li> </ul> <p>The operation of <a class="existingWikiWord" href="/nlab/show/composition">composition</a> is as in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Span_1(Grpd)</annotation></semantics></math>, def. <a class="maruku-ref" href="#OneSpansInOneGroupoids"></a> on the upper part of these diagrams, naturally extended to the whole diagrams by composition of the <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> filling the squares that appear.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><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">Span_1(Grpd, \mathbf{B}\flat U(1))</annotation></semantics></math> carries the structure of a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28infinity%2Cn%29-category">symmetric monoidal (2,1)-category</a> where the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mo>⊗</mo><mrow><mo>[</mo><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mrow><mtable><mtr><mtd><msub><mi>X</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left[ \array{ X_1 \\ \downarrow^{\mathrlap{f_1}} \\ \mathbf{B}\flat U(1) } \right] \otimes \left[ \array{ X_2 \\ \downarrow^{\mathrlap{f_2}} \\ \mathbf{B}\flat U(1) } \right] \;\; \coloneqq \;\; \left[ \array{ X_1 \times X_2 \\ \downarrow^{\mathrlap{f_1 \circ p_1 + f_2 \circ p_2}} \\ \mathbf{B}\flat U(1) } \right] \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>There is an evident <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful</a> <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-functor">(2,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Span_1(Grpd, \mathbf{B}\flat U(1)) \to Span_1(Grpd) </annotation></semantics></math></div> <p>which forgets the maps to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}\flat U(1)</annotation></semantics></math> and the homotopies between them. This is a <a class="existingWikiWord" href="/nlab/show/monoidal+%28infinity%2C1%29-functor">monoidal (2,1)-functor</a>.</p> </div> <p>As generalization of prop. <a class="maruku-ref" href="#GroupoidInSpanOneIsSelfDual"></a> we now have the following:</p> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>Every object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mo>∈</mo><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><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"> \left[ \array{ X \\ \downarrow^{\mathrlap{f}} \\ \mathbf{B}\flat U(1) } \right] \in Span_1(Grpd,\mathbf{B}\flat U(1)) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a>, with dual object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mpadded width="0"><mi>f</mi></mpadded></mrow></msup></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>]</mo></mrow></mrow><annotation encoding="application/x-tex"> \left[ \array{ X \\ \downarrow^{-\mathrlap{f}} \\ \mathbf{B}\flat U(1) } \right] </annotation></semantics></math></div> <p>and with <a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a> given by</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><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><msub><mo>⇙</mo> <mpadded width="0"><mo>≃</mo></mpadded></msub></mtd> <mtd></mtd> <mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mn>0</mn></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><mi>f</mi><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>−</mo><mi>f</mi><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && X \\ & \swarrow && \searrow \\ \ast && \swArrow_{\mathrlap{\simeq}} && X \times X \\ & {}_{\mathllap{0}}\searrow && \swarrow_{\mathrlap{f \circ p_1 - f \circ p_2}} \\ && \mathbf{B}\flat U(1) } \,. </annotation></semantics></math></div></div> <p>In conclusion we may now compute what the 1-dimensional prequantum field theory defined by a group character <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \colon G \to U(1)</annotation></semantics></math> regarded as a local action functional assigns to the <a class="existingWikiWord" href="/nlab/show/circle">circle</a>.</p> <div class="num_theorem" id="GroupCharacterPrequantumTheoryOnCircle"> <h6 id="theorem">Theorem</h6> <p>The prequantum field theory defined by a <a class="existingWikiWord" href="/nlab/show/group+character">group character</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>Field</mi></mstyle></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>♭</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>[</mo><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mpadded width="0"><mi>c</mi></mpadded></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>♭</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>]</mo></mrow><mo>∈</mo><msub><mi>Span</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Grpd</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><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"> \left[ \array{ \mathbf{Field} \\ \downarrow^{\mathrlap{\exp(i S)}} \\ \flat \mathbf{B}U(1) } \right] \;\; \coloneqq \;\; \left[ \array{ \mathbf{B}G \\ \downarrow^{\mathrlap{\mathbf{B}\mathrlap{c}}} \\ \flat \mathbf{B}U(1) } \right] \in Span_1(Grpd,\mathbf{B}\flat U(1)) </annotation></semantics></math></div> <p>assigns to the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> the <a class="existingWikiWord" href="/nlab/show/trace">trace</a> of the identity on this object, which under the identifications of example <a class="maruku-ref" href="#LoopSpaceGroupoid"></a>, example <a class="maruku-ref" href="#GroupCharacterAsClassFunctionByFreeLoopSpace"></a>, and example <a class="maruku-ref" href="#HomotopyAsActionFunctional"></a> is the group character itself:</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></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd></mtd> <mtd><mo>⇙</mo></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>×</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mn>0</mn></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>exp</mi><mo stretchy="false">(</mo><mi>i</mi><mi>S</mi><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub><mo>−</mo><mi>i</mi><mi>S</mi><mo>∘</mo><msub><mi>p</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mn>0</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>≃</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>c</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mo>⇙</mo></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>♭</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex"> \array{ && && [\Pi_1(S^1), \mathbf{B}G] \\ && & \swarrow && \searrow \\ && \mathbf{B}G && \swArrow && \mathbf{B}G \\ & \swarrow && \searrow && \swarrow && \searrow \\ \ast && && \mathbf{B}G \times \mathbf{B}G && && \ast \\ &{}_0\searrow & && \downarrow^{\mathrlap{\exp(i S \circ p_1 - i S \circ p_2)}} &&& \swarrow_{0} \\ &&&& \mathbf{B}\flat U(1) } \;\;\; \simeq \;\;\; \array{ && G//G \\ && \simeq \\ && [\Pi(S^1), \mathbf{B}G] \\ && \downarrow^{\mathrlap{c}} \\ && \flat U(1) \\ & \swarrow && \searrow \\ \ast && \swArrow && \ast \\ & \searrow && \swarrow \\ && \mathbf{B}\flat U(1) } \;\;\; </annotation></semantics></math></div> <p>Here the <a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a> on the right sends a <a class="existingWikiWord" href="/nlab/show/field+configuration">field configuration</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi><mo>=</mo><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><msub><mo stretchy="false">]</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">g \in G = [\Pi(S^1), \mathbf{B}G]_0</annotation></semantics></math> to its value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mo>♭</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">c(g) \in U(1) = (\flat \mathbf{B}U(1))_1</annotation></semantics></math> under the <a class="existingWikiWord" href="/nlab/show/group+character">group character</a>.</p> </div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>It follows that in a discussion of <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> the <a class="existingWikiWord" href="/nlab/show/path+integral">path integral</a> for the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of 1d DW theory is given by the <a class="existingWikiWord" href="/nlab/show/Schur+inner+product">Schur integral</a> over the <a class="existingWikiWord" href="/nlab/show/group+character">group character</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mrow><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow></mfrac><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><mi>c</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">⟨</mo><mi>c</mi><mo>,</mo><mn>1</mn><mo stretchy="false">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{1}{\vert G \vert} \underset{g\in G}{\sum} c(g) = \langle c,1\rangle \,. </annotation></semantics></math></div></div> <p>In conclusion, <a class="existingWikiWord" href="/nlab/show/1-dimensional+Dijkgraaf-Witten+theory">1-dimensional Dijkgraaf-Witten theory</a> as a prequantum field theory comes down to be essentially a geometric interpretation of what <a class="existingWikiWord" href="/nlab/show/group+characters">group characters</a> are and do. One may regard this as a simple example of <a class="existingWikiWord" href="/nlab/show/geometric+representation+theory">geometric representation theory</a>. Simple as this example is, it contains in it the seeds of many of the interesting aspects of richer prequantum field theories.</p> <h3 id="QMWithInteractionAndFeynmanDiagrams">Quantum mechanics with interactions and Feynman diagrams</h3> <p>We saw above that <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> theory naturally captures all the key aspects of basic <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a>.</p> <p>This becomes all the more pronounced when one considers quantum mechanics with interaction as in the <a class="existingWikiWord" href="/nlab/show/worldline+formalism">worldline formalism</a> and hence when one considers <a class="existingWikiWord" href="/nlab/show/Feynman+diagrams">Feynman diagrams</a> as diagrams of interactions of <a class="existingWikiWord" href="/nlab/show/particles">particles</a>.</p> <h4 id="spectral_triples_and_graph_representations">Spectral triples and Graph representations</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><msubsup><mi>Cob</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>1</mn></mrow> <mi>Feyn</mi></msubsup></mrow><annotation encoding="application/x-tex">R Cob_{1|1}^{Feyn}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a> of <a class="existingWikiWord" href="/nlab/show/Feynman+graphs">Feynman graphs</a> for the <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a> with a single type of interaction along the lines of <a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a>. So its morphisms are generated from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">|</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(1|1)</annotation></semantics></math>-dimensional super-Riemannian manifolds (i.e. super-intervals) and from a single interaction vertex</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><mo>•</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd> <mtd><mo>→</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><mo>•</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \bullet \\ & \searrow \\ && \bullet & \to \\ & \nearrow \\ \bullet } </annotation></semantics></math></div> <p>subject to the obvious associativity condition.</p> <p>Then a <a class="existingWikiWord" href="/nlab/show/spectral+triple">spectral triple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,H,D)</annotation></semantics></math> is the data encoding a sufficiently nice smooth <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></msub><mo>:</mo><mi>R</mi><msubsup><mi>Cob</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>1</mn></mrow> <mi>Feyn</mi></msubsup><mo>→</mo><mi>sVect</mi></mrow><annotation encoding="application/x-tex"> Z_{(A,H,D)} : R Cob_{1|1}^{Feyn} \to sVect </annotation></semantics></math></div> <p>to the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>s.</p> <p>Here</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><msub><mi>Z</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo>•</mo><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">A = Z_{(A,H,D)}(\bullet)_0</annotation></semantics></math> is the even part of the <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a> assigned by the functor to the point, equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> whose product is given by the image of the interaction vertex</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></msub><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mo>•</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>•</mo></mtd> <mtd><mo>→</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd><mo>•</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> Z_{(A,H,D)} \left( \array{ \bullet \\ & \searrow \\ && \bullet & \to \\ & \nearrow \\ \bullet } \right) </annotation></semantics></math></div></li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is some completion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Z</mi> <mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>D</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo>•</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Z_{(A,H,D)}(\bullet)</annotation></semantics></math> to a <a class="existingWikiWord" href="/nlab/show/super+Hilbert+space">super Hilbert space</a></p> </li> <li> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>∈</mo><mi>End</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D \in End(H)</annotation></semantics></math> is an odd self-adjoint operator on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, which gives the value of the functor on the super-interval <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(t,\theta)</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><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>exp</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>t</mi><msup><mi>D</mi> <mn>2</mn></msup><mo>+</mo><mi>θ</mi><mi>D</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (t,\theta) \mapsto \exp( - t D^2 + \theta D ) \,. </annotation></semantics></math></div></li> </ul> <p>So this is the <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> of a <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a>. In the simplest case this comes from a spinor particle propagating on a <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a> <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian 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>in which case</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>=</mo><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H = L^2(S)</annotation></semantics></math> is the space of square integrable spinor sections;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = C^\infty(X)</annotation></semantics></math> is the space of smooth functions 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> </li> </ul> <p>One point of a <a class="existingWikiWord" href="/nlab/show/spectral+triple">spectral triple</a> is to take the view of world-line <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> as basic and <em>characterize</em> the spin Riemannian geometry of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> entirely by this algebraic data. In particular the <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</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> is encoded in the <a class="existingWikiWord" href="/nlab/show/operator+spectrum">operator spectrum</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>, which is where the notion “spectral triple” gets its name from.</p> <p>Then with all the ordinary geoemtry re-encoded algebraically this way, in terms of the 1-dimensional <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> that <em>probes</em> this geometry, one can then use the same formulas to interpret spectral triple geometrically that do <em>not</em> come from an ordinary geometry as in the above example.</p> <h4 id="feynman_diagram_in_worldline_formalism_and_monoidal_string_diagrams">Feynman diagram in worldline formalism and Monoidal string diagrams</h4> <p>…<a class="existingWikiWord" href="/nlab/show/worldline+formalism">worldline formalism</a>…</p> <p>…<a class="existingWikiWord" href="/nlab/show/Feynman+diagram">Feynman diagram</a>…</p> <p>…<a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>…</p> <h3 id="QuantumTopologicalString">Quantum topological string</h3> <h4 id="Global2dTQFT">Global 2d TQFT and Frobenius algebra</h4> <p>In view of the above discussion of “topological quantum mechanics”, i.e. of 1-dimensional <a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>, it is immediate to pass to a higher dimensional field theory by using <a class="existingWikiWord" href="/nlab/show/categories+of+cobordisms">categories of cobordisms</a> of higher dimension and consider <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functors">strong monoidal functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Bord</mi> <mi>n</mi> <mo>⊔</mo></msubsup><mo>⟶</mo><msup><mi>Vect</mi> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex"> Z \;\colon\; Bord_n^\sqcup \longrightarrow Vect^\otimes </annotation></semantics></math></div> <p>(<a href="#Atiyah88">Atiyah 88</a>, <a href="#Segal04">Segal 04</a>).</p> <p>for instance 1-dimensional cobordisms with boundary describe a kind <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> with <a class="existingWikiWord" href="/nlab/show/boundary+field+theory">boundary field theory</a>.</p> <p><img src="/nlab/files/frobenius_algebra.jpg" alt="string diagrams for the Frobenius algebra axioms" /></p> <p>As almost immediate from these picture, such 2d TQFTs are equivalent to <a class="existingWikiWord" href="/nlab/show/Frobenius+algebras">Frobenius algebras</a> <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>.</p> <p>In terms of physics: <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> is the <a class="existingWikiWord" href="/nlab/show/space+of+states">space of states</a> of a topological <a class="existingWikiWord" href="/nlab/show/open+string">open string</a> and the algebra and coalgebra structure on it encodes its <a class="existingWikiWord" href="/nlab/show/n-point+function">3-point functions</a>.</p> <p>More generally open and <a class="existingWikiWord" href="/nlab/show/closed+strings">closed strings</a></p> <p><img src="/nlab/files/monoid_laws.jpg" alt="diagrams for the monoid laws in 2Cob" /></p> <p>Now <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> is equivalent to a pair consisting of a Frobenius algebra <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> and a commutative Frobenius algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \to Z(A)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/center">center</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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mn>2</mn> <mo>⊔</mo></msubsup><mo>⟶</mo><msup><mi>Vect</mi> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex"> Bord_2^{\sqcup} \longrightarrow Vect^\otimes </annotation></semantics></math></div> <p>(<a href="#MooreSegal06">Moore-Segal</a> <a href="#Lazaroiu00">Lazaroiu 00</a>, see also <a href="#LaudaPfeiffer05">Lauda-Pfeiffer 05</a>).</p> <p>In physics speak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is the space of states for the topological <em><a class="existingWikiWord" href="/nlab/show/closed+string">closed string</a></em>.</p> <p>Or rather, it is <em>some</em> topological string model, but not the one originally obtained by <a class="existingWikiWord" href="/nlab/show/topological+twist">topological twist</a> from the <a class="existingWikiWord" href="/nlab/show/2d+%282%2C0%29-superconformal+QFT">2d (2,0)-superconformal QFT</a> which is commonly what is understood as the “<a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a>” in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> (<a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>/<a class="existingWikiWord" href="/nlab/show/B-model">B-model</a>).</p> <h4 id="Local2dTQFT">Cohomological 2d TQFT and Calabi-Yau <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>-algebras</h4> <p>Curiously, the above does <em>not</em> capture the original motivating examples for <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> that came from physics, namely it does not capture the “<a class="existingWikiWord" href="/nlab/show/cohomological+quantum+field+theory">cohomological quantum field theory</a>” due to <a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, such as the <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> in its incarnation as the <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a> and <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> and the <a class="existingWikiWord" href="/nlab/show/Landau-Ginzburg+model">Landau-Ginzburg model</a>.</p> <ul> <li> <p>Witten <a class="existingWikiWord" href="/nlab/show/cohomological+field+theory">cohomological field theory</a>: <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a> is <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>, physical quantum states are <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a></p> </li> <li> <p>Kontsevich: <a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a> is equivalence of <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+categories">A-∞ categories</a></p> </li> <li> <p>Aspinwall, Douglas et al: the <a class="existingWikiWord" href="/nlab/show/derived+categories">derived categories</a> here are those of topological <a class="existingWikiWord" href="/nlab/show/A-branes">A-branes</a> (<a class="existingWikiWord" href="/nlab/show/A-branes">A-branes</a>/<a class="existingWikiWord" href="/nlab/show/B-branes">B-branes</a>)</p> </li> </ul> <p>Hence need to regard <a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>/<a class="existingWikiWord" href="/nlab/show/B-model">B-model</a> open <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> as having a <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>. Under string composition this yields not just an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> with <a class="existingWikiWord" href="/nlab/show/trace">trace</a> but an <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a> with suitable trace.</p> <p>Examples come from twisting the <a class="existingWikiWord" href="/nlab/show/2d+%282%2C0%29-CFT">2d (2,0)-CFT</a> induced from a <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a>, hence one speaks of “<a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+algebra">Calabi-Yau A-∞ algebra</a>”.</p> <p>remember space of <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a></p> <ul> <li>Getzler, Segal: “<a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>”</li> </ul> <p>classification by (<a href="#Costello04">Costello 04</a>) sums it up:</p> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+category">Calabi-Yau A-∞ category</a> is equivalent to non-compact open topological string with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>Vect</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(Vect)</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/objects">objects</a> of the category are the <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a>, <a class="existingWikiWord" href="/nlab/show/hom-spaces">hom-spaces</a> are the spaces of quantum states of open strings stretching between these. The closed string <a class="existingWikiWord" href="/nlab/show/bulk+field+theory">bulk field theory</a> sector is given by forming <a class="existingWikiWord" href="/nlab/show/Hochschild+homology">Hochschild homology</a>. Given a <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a>, then the <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a> refinement (see at <em><a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></em>) of its <a class="existingWikiWord" href="/nlab/show/derived+category+of+coherent+sheaves">derived category of coherent sheaves</a> is an example.</p> <h4 id="local_2d_tqft_and_2modules">Local 2d TQFT and 2-Modules</h4> <p>Given an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> <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> then its <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math> behaves much like a higher analog of a <a class="existingWikiWord" href="/nlab/show/module">module</a>/<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>.</p> <p>Given an <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>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</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> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mi>N</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mi>Mod</mi><mo>→</mo><mi>B</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">(-)\otimes_A N \colon A Mod \to B Mod</annotation></semantics></math> behaves like a higher dimensional <a class="existingWikiWord" href="/nlab/show/linear+operator">linear operator</a>.</p> <p>This is the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Watts+theorem">Eilenberg-Watts theorem</a>.</p> <p>Hence we speak of a <a class="existingWikiWord" href="/nlab/show/2-module">2-module</a>.</p> <p>Notice that every algebra <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> is canonically an <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>-<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>-bimodule. This way we see that the above construction naturally localizes</p> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/cohomological+QFT">cohomological QFT</a></th><th><a class="existingWikiWord" href="/nlab/show/local+QFT">local QFT</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+string">open string</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">\mapsto</annotation></semantics></math></td><td style="text-align: left;">open string algebra <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></td><td style="text-align: left;">open string <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>A</mi></msub><msub><mi>A</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">{}_{A} A_{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↦</mo></mrow><annotation encoding="application/x-tex">\mapsto</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-module">2-module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math></td></tr> </tbody></table> <p>Hence we regard <span class="newWikiWord">D-Brane<a href="/nlab/new/D-Brane">?</a></span> states as quantum 2-states.</p> <p>We motivate this further <a href="#IntroductionGeneralFormulation">below</a>. First to record the classification results:</p> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> (“<a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>”)</th><th><a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a></th><th><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> structure on <a class="existingWikiWord" href="/nlab/show/space+of+quantum+states">space of quantum states</a></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+string">open</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Vect">Vect</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_k</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a> <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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/folklore">folklore</a>+(<a href="2d+TQFT#Abrams96">Abrams 96</a>)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+string">open</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> with <a class="existingWikiWord" href="/nlab/show/closed+string">closed string</a> <a class="existingWikiWord" href="/nlab/show/bulk+field+theory">bulk theory</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Vect">Vect</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_k</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Frobenius+algebra">Frobenius algebra</a> <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> with <a class="existingWikiWord" href="/nlab/show/trace">trace</a> map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \to Z(A)</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/Cardy+condition">Cardy condition</a></td><td style="text-align: left;">(<a href="#2d+TQFT#Lazaroiu00">Lazaroiu 00</a>, <a href="2d+TQFT#MooreSegal02">Moore-Segal 02</a>)</td></tr> <tr><td style="text-align: left;">non-compact <a class="existingWikiWord" href="/nlab/show/open+string">open</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">Ch(Vect)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+algebra">Calabi-Yau A-∞ algebra</a></td><td style="text-align: left;">(<a href="2d+TQFT#Kontsevich95">Kontsevich 95</a>, <a href="2d+TQFT#Costello04">Costello 04</a>)</td></tr> <tr><td style="text-align: left;">non-compact <a class="existingWikiWord" href="/nlab/show/open+string">open</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> with various <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">Ch(Vect)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+category">Calabi-Yau A-∞ category</a></td><td style="text-align: left;">“</td></tr> <tr><td style="text-align: left;">non-compact <a class="existingWikiWord" href="/nlab/show/open+string">open</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a> with various <a class="existingWikiWord" href="/nlab/show/D-branes">D-branes</a> and with <a class="existingWikiWord" href="/nlab/show/closed+string">closed string</a> <a class="existingWikiWord" href="/nlab/show/bulk+field+theory">bulk</a> sector</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">Ch(Vect)</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+category">Calabi-Yau A-∞ category</a> with <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a></td><td style="text-align: left;">“</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/extended+TQFT">local</a> <a class="existingWikiWord" href="/nlab/show/closed+string">closed</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2Mod">2Mod</a>(<a class="existingWikiWord" href="/nlab/show/Vect">Vect</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_k</annotation></semantics></math>) over <a class="existingWikiWord" href="/nlab/show/field">field</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></td><td style="text-align: left;">separable symmetric <a class="existingWikiWord" href="/nlab/show/Frobenius+algebras">Frobenius algebras</a></td><td style="text-align: left;">(<a href="2d+TQFT#SchommerPries11">SchommerPries 11</a>)</td></tr> <tr><td style="text-align: left;">non-compact <a class="existingWikiWord" href="/nlab/show/extended+TQFT">local</a> <a class="existingWikiWord" href="/nlab/show/closed+string">closed</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2Mod">2Mod</a>(<a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">Ch(Vect)</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+algebra">Calabi-Yau A-∞ algebra</a></td><td style="text-align: left;">(<a href="2d+TQFT#Lurie09">Lurie 09, section 4.2</a>)</td></tr> <tr><td style="text-align: left;">non-compact <a class="existingWikiWord" href="/nlab/show/extended+TQFT">local</a> <a class="existingWikiWord" href="/nlab/show/closed+string">closed</a> <a class="existingWikiWord" href="/nlab/show/topological+string">topological string</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2Mod">2Mod</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>S</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{S})</annotation></semantics></math> for a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+object">Calabi-Yau object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>S</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{S}</annotation></semantics></math></td><td style="text-align: left;">(<a href="2d+TQFT#Lurie09">Lurie 09, section 4.2</a>)</td></tr> </tbody></table> </div> <p>Here the trace operation in the CY conditions corresponds to the cobordism which is the “disappearance of a circle”.</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><mo>⟵</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↖</mo></mtd></mtr> <mtr><mtd><mi>V</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mi>V</mi> <mo>⊗</mo></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⟶</mo></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>*</mo></mrow><annotation encoding="application/x-tex"> \array{ && \longleftarrow \\ & \swarrow && \nwarrow \\ V && && V^\otimes \\ & \searrow && \nearrow \\ && \longrightarrow } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \ast </annotation></semantics></math></div> <p>One may view this as exhibiting “higher order duality”: where the semi-circles exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^\ast</annotation></semantics></math>, this disappearance of a circle exhibits the upper semi-circle as <a class="existingWikiWord" href="/nlab/show/adjoint+morphism">adjoint</a> to the lower semicircle.</p> <p>(…)</p> <h3 id="IntroductionGeneralFormulation">General local TQFT</h3> <h4 id="covariant_quantization_and_directed_homotopy_types">Covariant quantization and Directed homotopy types</h4> <p>One way to understand from the point of view of <a class="existingWikiWord" href="/nlab/show/physics">physics</a> why the 1-functorial description of <a class="existingWikiWord" href="/nlab/show/2d+CFT">2d CFT</a> and <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> <a href="#Global2dTQFT">above</a> is unsatisfactory is that it breaks what is known as “covariance” in physics, in the sense of “<a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a>” (reflected also in the term “<a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>”): implicit in the concept of a <a class="existingWikiWord" href="/nlab/show/category+of+cobordisms">category of cobordisms</a> is a splitting of a <a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a>/<a class="existingWikiWord" href="/nlab/show/worldvolumes">worldvolumes</a> into spatial slices (the <a class="existingWikiWord" href="/nlab/show/objects">objects</a>) of the category and <a class="existingWikiWord" href="/nlab/show/trajectories">trajectories</a> between these.</p> <p>The standard <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a>-data (“<a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum field theory</a>”) from which <a class="existingWikiWord" href="/nlab/show/topological+quantum+field+theories">topological quantum field theories</a> are supposed to arise under <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a> do not enforce such a splitting as indeed they are <a class="existingWikiWord" href="/nlab/show/general+covariance">generally covariant</a>. Accordingly, a <a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a> should, after <a class="existingWikiWord" href="/nlab/show/quantization">quantization</a>, give rise to a <a class="existingWikiWord" href="/nlab/show/local+quantum+field+theory">local quantum field theory</a> that is still “generally covariant” in that it does not require or depend on such a splitting. In physics this plays a crucial role for instance in considerations related to <a class="existingWikiWord" href="/nlab/show/quantum+gravity">quantum gravity</a>.</p> <p>We saw <a href="#CorrespondencesOfGroupoids">above</a> how 1-dimensional (<a class="existingWikiWord" href="/nlab/show/prequantum+field+theory">prequantum</a>) field theory is encoded by <a class="existingWikiWord" href="/nlab/show/correspondences">correspondences</a> of <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>. For instance the process of a particle and its antiparticle appearing out of the vacuum is given by</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><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>Δ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>×</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{Fields} \\ & \swarrow && \searrow^{\mathrlap{\Delta}} \\ \ast && && \mathbf{Fields} \times \mathbf{Fields} } </annotation></semantics></math></div> <p>and the reverse process of them disappearing is given by</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><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>Δ</mi></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>×</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \mathbf{Fields} \\ & {}^{\mathllap{\Delta}}\swarrow && \searrow \\ \mathbf{Fields} \times \mathbf{Fields} && && \ast } </annotation></semantics></math></div> <p>A particle tracing out a circle is equivalently the composition (via <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a>) of these two process.</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></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>×</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && && [\Pi(S^1), \mathbf{Fields}] \\ && & \swarrow && \searrow \\ && \mathbf{Fields} && && \mathbf{Fields} \\ & \swarrow && \searrow && \swarrow && \searrow \\ \ast && && \mathbf{Fields}\times \mathbf{Fields} && && \ast } </annotation></semantics></math></div> <p>Then we saw <a href="#Local2dTQFT">above</a> for <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> that in higher dimensional general such a circle in turn may appear</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><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↖</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && [\Pi(S^1), \mathbf{Fields}] \\ & \swarrow & \uparrow & \searrow \\ \ast && \mathbf{Fields} && \ast \\ & \nwarrow &\downarrow& \nearrow \\ && \ast } </annotation></semantics></math></div> <p>and disappear</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><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>Fields</mi></mstyle></mtd> <mtd></mtd> <mtd><mo>*</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↖</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && \ast \\ & \swarrow & \uparrow & \searrow \\ \ast && \mathbf{Fields} && \ast \\ & \nwarrow &\downarrow& \nearrow \\ && [\Pi(S^1), \mathbf{Fields}] } </annotation></semantics></math></div> <p>The 2-dimensional composition of such processes, again by <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</a> yields values on all higher spheres</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>Fields</mi></mstyle><munder><mo>×</mo><mrow><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow></munder><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo>≃</mo><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Fields} \underset{[\Pi(S^n)]}{\times} \mathbf{Fields} \simeq [\Pi(S^{n+1}), \mathbf{Fields}] </annotation></semantics></math></div> <p>and in fact all <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>. For instance the <a class="existingWikiWord" href="/nlab/show/trinion">trinion</a> process is represented by this correspondence-of-correspondences:</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><mo>*</mo></mtd> <mtd><mo>⟵</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></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> <mtd></mtd> <mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟵</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>Fig</mi><mn>8</mn><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></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> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mo>⟵</mo></mtd> <mtd><mo stretchy="false">[</mo><mi>Π</mi><mo stretchy="false">(</mo><msup><mi>S</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>Fields</mi></mstyle><mo stretchy="false">]</mo></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \ast &\longleftarrow& [\Pi(S^1 \coprod S^1), \mathbf{Fields}] &\longrightarrow& \ast \\ \uparrow && \uparrow && \uparrow \\ \ast &\longleftarrow& [\Pi(Fig8), \mathbf{Fields}] &\longrightarrow& \ast \\ \downarrow && \downarrow && \downarrow \\ \ast &\longleftarrow& [\Pi(S^1), \mathbf{Fields}] &\longrightarrow& \ast } </annotation></semantics></math></div> <p>To describe local propagation in higher dimensional field theory this way, evidently we need a higher dimensional calculus that deals both with the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> (<a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>) involves as well as with the directionality of these processes.</p> <p>We already saw the first hint of how this works: <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> above appeared in two different guises, on the one hand as <a class="existingWikiWord" href="/nlab/show/homotopy+1-types">homotopy 1-types</a>, on the other as special kinds of <a class="existingWikiWord" href="/nlab/show/categories">categories</a> with directed morphisms.</p> <p>Now <a class="existingWikiWord" href="/nlab/show/homotopy+1-types">homotopy 1-types</a> have a classical generalization to general <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a>, traditionally taken to be represented by <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> regarded up to <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>.</p> <p>A crucial fact is that one may pair this full homotopy-theoretic aspect with the category-theoretic aspect to get <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">∞-categories</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></mtd> <mtd></mtd> <mtd><mi>groupoids</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>categories</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mn>∞</mn><mtext>-</mtext><mi>groupoids</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mn>∞</mn><mtext>-</mtext><mi>categories</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && groupoids \\ & \swarrow && \searrow \\ categories && && \infty\text{-}groupoids \\ & \searrow && \swarrow \\ && \infty\text{-}categories } </annotation></semantics></math></div> <p><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>-fold <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>-categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⟶</mo></mrow><annotation encoding="application/x-tex">\longrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-categories">(∞,n)-categories</a></p> <p><a class="existingWikiWord" href="/nlab/show/k-morphisms">k-morphisms</a> for all <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>, such that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \gt n</annotation></semantics></math> they are invertible</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable rowalign="center"><mtr><mtd><mi>O</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mo>=</mo></mtd> <mtd><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo></mtd></mtr> <mtr><mtd><mi>O</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>1</mn></msup><mo stretchy="false">)</mo><mo>=</mo></mtd> <mtd><mrow><mo>{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo>}</mo></mrow></mtd></mtr> <mtr><mtd><mi>O</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>2</mn></msup><mo stretchy="false">)</mo><mo>=</mo></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><semantics><annotation-xml encoding="SVG1.1"> [[!include oriental > Delta2]] </annotation-xml></semantics></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mtd></mtr> <mtr><mtd><mi>O</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>3</mn></msup><mo stretchy="false">)</mo><mo>=</mo></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><semantics><annotation-xml encoding="SVG1.1"> [[!include oriental > Delta3]] </annotation-xml></semantics></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mtd></mtr> <mtr><mtd><mi>O</mi><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mn>4</mn></msup><mo stretchy="false">)</mo><mo>=</mo></mtd> <mtd><mrow><mo>{</mo><mrow><mtable><mtr><mtd><semantics><annotation-xml encoding="SVG1.1"> [[!include oriental > Delta4]] </annotation-xml></semantics></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{\arrayopts{\rowalign{center}} O(\Delta^0) = & \{ 0\} \\ O(\Delta^1) = & \left\{ 0 \to 1\right\} \\ O(\Delta^2) = & \left\{ \array{\begin{svg} [[!include oriental > Delta2]] \end{svg}} \right\}\\ O(\Delta^3) = & \left\{ \array{\begin{svg} [[!include oriental > Delta3]] \end{svg}}\right\}\\ O(\Delta^4) = & \left\{ \array{\begin{svg} [[!include oriental > Delta4]] \end{svg}} \right\} } </annotation></semantics></math></div> <p>in particular</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bord</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Bord_n</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+correspondences">(∞,n)-category of correspondences</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Corr</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Corr_n</annotation></semantics></math></p> </li> <li> <p><span class="newWikiWord">(∞,n)-category of (∞,n)-modules<a href="/nlab/new/%28%E2%88%9E%2Cn%29-category+of+%28%E2%88%9E%2Cn%29-modules">?</a></span><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">Mod_n</annotation></semantics></math></p> </li> </ul> <h4 id="spaces_of_states_and_the_cobordism_hypothesis"><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>-Spaces of states and the Cobordism hypothesis</h4> <p>These <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-categories">(∞,n)-categories</a> are <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2Cn%29-categories">symmetric monoidal (∞,n)-categories</a> in the same way that their 1-categorical shadows are, only that everything is lifted up to homotopy.</p> <p>hence one may consider</p> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2Cn%29-functors">monoidal (∞,n)-functors</a></p> <p><a class="existingWikiWord" href="/schreiber/show/Local+prequantum+field+theory">Local prequantum field theory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mi>n</mi> <mo>⊔</mo></msubsup><mo>⟶</mo><msub><mi>Corr</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>Phases</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><msub><mo>⊗</mo> <mi>phased</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/local+quantum+field+theory">local quantum field theory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Bord</mi> <mi>n</mi> <mo>⊔</mo></msubsup><mo>⟶</mo><msubsup><mi>Mod</mi> <mi>n</mi> <mo>⊗</mo></msubsup></mrow><annotation encoding="application/x-tex"> Bord_n^\sqcup \longrightarrow Mod_n^\otimes </annotation></semantics></math></div> <p>The classification theory of these, the <em><a class="existingWikiWord" href="/nlab/show/cobordism+theorem">cobordism theorem</a></em> says roughly that such local topological field theories assign <a class="existingWikiWord" href="/nlab/show/fully+dualizable+objects">fully dualizable objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> to the point and are entirely determined by this assignment in that every higher dimensional manifold is sent to the <a class="existingWikiWord" href="/nlab/show/higher+dimensional+trace">higher dimensional trace</a> on the identity on that object, i.e. the higher codimension analogs of the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>↦</mo><msub><mi>tr</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>V</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Sigma \mapsto tr_\Sigma(id_V) \,. </annotation></semantics></math></div> <p>(…)</p> <h3 id="Global3dTQFT">3d TQFT</h3> <h4 id="chernsimons_theory">Chern-Simons theory</h4> <p>…<a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a>…</p> <p>…<a class="existingWikiWord" href="/nlab/show/Turaev-Viro+construction">Turaev-Viro construction</a>…</p> <p>…<a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a>…</p> <p>…<a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+construction">Reshetikhin-Turaev construction</a>…</p> <p>…<a class="existingWikiWord" href="/nlab/show/quantization+of+3d+Chern-Simons+theory">quantization of 3d Chern-Simons theory</a>…</p> <h4 id="Local3dTQFT">Modular functor</h4> <p>… <a class="existingWikiWord" href="/nlab/show/modular+functor">modular functor</a> …</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a></p> <ul> <li> <p><strong>FQFT</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+functorial+field+theory">unitary functorial field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/0-dimensional+TQFT">0-dimensional TQFT</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a>, <a class="existingWikiWord" href="/nlab/show/3d+TQFT">3d TQFT</a> <a class="existingWikiWord" href="/nlab/show/4d+TQFT">4d TQFT</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/HQFT">HQFT</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net+of+observables">local net of observables</a></p> </li> </ul> </li> </ul> </li> </ul> <h2 id="references">References</h2> <h3 id="terminology">Terminology</h3> <p>The idea of functorial field theory originates with the unpublished precursor note of <a href="#Segal04">Segal (2004)</a> and became popular (in the special case of <a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological field theory</a>) with <a href="#Atiyah88">Atiyah (1988)</a>.</p> <p>The term <em>functorial quantum field theory</em> appears to originate around June 2008 with <a href="#Schreiber09">Schreiber (2009)</a>.</p> <p>At some point later, the adjective “quantum” was dropped because the formalism also encodes classical and prequantum field theories.</p> <h3 id="formalization_of_sewing_and_locality_in_terms_of_functoriality">Formalization of sewing and locality in terms of functoriality</h3> <p>It was in</p> <ul> <li id="Atiyah88"><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <em>Topological quantum field theory</em>, Publications Mathématiques de l’IHÉS, 68 (1988), p. 175-186</li> </ul> <p>that it was realized that</p> <ul> <li> <p>this means that this property can be taken as the <em>defining</em> property of the path integral, thereby circumventing the problem of constructing it as an actual integral;</p> </li> <li> <p>this property can be conveniently axiomatized by saying that the path integral is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from a suitable category whose morphisms are <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s to a category of vector spaces.</p> </li> </ul> <p>(Strictly speaking, Atiyah’s original article mentions this functor slightly indirectly only.)</p> <p>All this was originally formalized in the context of <a class="existingWikiWord" href="/nlab/show/TQFT">topological quantum field theory</a> only. This is the easiest case that already exhibits all the functoriality that is implied by “FQFT” but by far not the only case (see below).</p> <p>A pedagogical exposition of how the physicist’s way of thinking about the path integral leads to its definition as a functor is given in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kevin+Walker">Kevin Walker</a>, <em>TQFTs</em> (<a href="http://canyon23.net/math/tc.pdf">pdf</a>)</li> </ul> <p>A pedagogical exposition of the notion of quantum field theory as a functor on <a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a> is in</p> <ul> <li id="Baez04"><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em>Quantum quandaries: a Category-Theoretic perspective</em> (<a href="http://arxiv.org/abs/quant-ph/0404040">arXiv</a>)</li> </ul> <p>and a review of much of the existing material in the literature is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bruce+Bartlett">Bruce Bartlett</a>, <em>Categorical Aspects of Topological Quantum Field Theories</em> (<a href="http://arxiv.org/abs/math/0512103">arXiv</a>).</li> </ul> <p>See also:</p> <ul> <li id="Stolz14"><a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stephan Stolz</a>, <em>Topology and Field Theories</em>, Contemporary Mathematics 613, American Mathematical Society 2014 (<a href="https://bookstore.ams.org/conm-613">ams:conm-613</a>)</li> </ul> <p>A survey of some further developments and conjectures (especially as related to the <a class="existingWikiWord" href="/nlab/show/AGT+correspondence">AGT correspondence</a>) is in</p> <ul> <li id="Tachikawa17"><a class="existingWikiWord" href="/nlab/show/Yuji+Tachikawa">Yuji Tachikawa</a>, <em>On ‘categories’ of quantum field theories</em> (<a href="https://arxiv.org/abs/1712.09456">arXiv:1712.09456</a>)</li> </ul> <p>The discussion of the open-closed case of 2d TQFT goes back to</p> <ul> <li id="MooreSegal06"> <p><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>D-branes and K-theory in 2D topological field theory</em> (<a href="http://arxiv.org/abs/hep-th/0609042">arXiv:hep-th/0609042</a>)</p> </li> <li id="Lazaroiu00"> <p><a class="existingWikiWord" href="/nlab/show/Calin+Lazaroiu">Calin Lazaroiu</a>, <em>On the structure of open-closed topological field theory in two dimensions</em>, Nuclear Phys. B 603(3), 497–530 (2001), (<a href="http://arxiv.org/abs/hep-th/0010269">arXiv:hep-th/0010269</a>)</p> </li> </ul> <p>A picture-rich discussion is in</p> <ul> <li id="LaudaPfeiffer05"><a class="existingWikiWord" href="/nlab/show/Aaron+Lauda">Aaron Lauda</a>, <a class="existingWikiWord" href="/nlab/show/Hendryk+Pfeiffer">Hendryk Pfeiffer</a>, <em>Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras</em>, Topology Appl. 155 (2008) 623-666. (<a href="http://arxiv.org/abs/math.AT/0510664">arXiv:math.AT/0510664</a>)</li> </ul> <div> <h3 id="ReferencesRelationBetweenAQFTAndFQFT">Relation between algebraic and functorial field theory</h3> <p>On the relation between <a class="existingWikiWord" href="/nlab/show/functorial+quantum+field+theory">functorial quantum field theory</a> (axiomatizing the <a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a> of <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>) and <a class="existingWikiWord" href="/nlab/show/algebraic+quantum+field+theory">algebraic quantum field theory</a> (axiomatizing the <a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a>):</p> <ul> <li id="Schreiber09"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/files/AQFTfromFQFT.pdf" title="AQFT from n-Functorial QFT">AQFT from n-Functorial QFT</a></em>, Comm. Math. Phys. <strong>291</strong> 2 (2009) 357-401 &lbrack;<a href="https://arxiv.org/abs/0806.1079">arXiv:0806.1079</a>, <a href="https://doi.org/10.1007/s00220-009-0840-2">doi:10.1007/s00220-009-0840-2</a>&rbrack;</p> </li> <li id="Johnson-Freyd21"> <p><a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, <em>Heisenberg-picture quantum field theory</em>, in <em>Representation Theory, Mathematical Physics, and Integrable Systems</em>, Progress in Mathematics <strong>340</strong> (2021) &lbrack;<a href="https://arxiv.org/abs/1508.05908">arXiv:1508.05908</a>, <a href="https://doi.org/10.1007/978-3-030-78148-4_13">doi:10.1007/978-3-030-78148-4_13</a>&rbrack;</p> </li> <li id="BunkMacManusSchenkel23"> <p><a class="existingWikiWord" href="/nlab/show/Severin+Bunk">Severin Bunk</a>, <a class="existingWikiWord" href="/nlab/show/James+MacManus">James MacManus</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, <em>Lorentzian bordisms in algebraic quantum field theory</em> &lbrack;<a href="https://arxiv.org/abs/2308.01026">arXiv:2308.01026</a>&rbrack;</p> </li> </ul> </div> <h3 id="nontopological_fqfts_especially_conformal">Non-topological FQFTs (especially conformal)</h3> <p>This mostly concentrates on <a class="existingWikiWord" href="/nlab/show/TQFT">topological quantum field theories</a>, those where the path integral depends only on the diffeomorphism class of the domain it is evaluated on. This is the simplest and by far best understood case. But the idea of functorial FQFT is not restricted to this case.</p> <p>This was realized in</p> <ul> <li id="Segal04"><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>The definition of conformal field theory</em>, in: <em>Topology, Geometry and quantum field theory</em>, London. Math. Soc. LNS 308, edited by <a class="existingWikiWord" href="/nlab/show/Ulrike+Tillmann">Ulrike Tillmann</a>, Cambridge Univ. Press 2004, 247-343</li> </ul> <p>There the notion of 2-dimensional <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> is axiomatized as a functor on a category of 2-dimensional cobordisms with conformal structure.</p> <p>(Apparently a similar definition has been given by Kontsevich, but never published.) The details of the category of conformal cobordisms can get a bit technical and slight variations of Segal’s original definition may be necessary. The work by Huang and Kong can be regarded as a further refinement and maybe completion of Segal’s program</p> <ul> <li> <p>Yi-Zhi Huang, <em>Geometric interpretation of vertex operator algebras</em>, Proc. Natl. Acad. Sci. USA, Vol 88. (1991) pp. 9964-9968</p> </li> <li> <p>Liang Kong, <em>Open-closed field algebras</em> Commun. Math. Physics. 280, 207-261 (2008) (<a href="http://arxiv.org/abs/math/0610293">arXiv</a>).</p> </li> </ul> <p>A very concrete construction of functorial CFTs (for the special case of <em>rational</em> CFTs) is provided by the <a class="existingWikiWord" href="/nlab/show/FFRS-formalism">FFRS-formalism</a>.</p> <h3 id="Extended">Extended (multi-tiered) FQFT</h3> <p>But one notices that the formalization of quantum field theory as a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> on <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s encodes only a small aspect of the full sewing law imagined to be satisfied by the path integral: In a 1-<a class="existingWikiWord" href="/nlab/show/category">category</a> 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>-dimensional <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a>s these are glued along <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>-dimensional boundaries. One could imagine more generally a formalization where a given <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> is allowed to be chopped into arbitrary parts of arbitrary co-dimension such that the path integral can still consistently be evaluated on each of these parts.</p> <p>This leads to the notion of <em>extended quantum field theory</em>, which is taken to be 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>-functor on an <a class="existingWikiWord" href="/nlab/show/higher+category+theory">infinity category</a> of <a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a>s. Early ideas about a formalization of this approach were given in</p> <ul> <li id="BaezDolan95"><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/James+Dolan">James Dolan</a>, <em>Higher-dimensional algebra and Topological Quantum Field Theory</em> (<a href="http://arxiv.org/abs/q-alg/9503002">arXiv</a>) .</li> </ul> <p>Making this precise involves giving a precise definition of 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>-category of cobordisms. Several approaches exist, such as</p> <ul> <li>Eugenia Cheng and Nick Gurski, <em>Towards 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>-category of cobordisms</em>, Theory and Applications of Categories, Vol. 18, 2007, No. 10, pp 274-302. (<a href="http://www.tac.mta.ca/tac/volumes/18/10/18-10abs.html">tac</a>)</li> </ul> <p>or</p> <ul> <li>Marco Grandis, <em>Collared cospans, cohomotopy and TQFT (Cospans in Algebraic Topology, II)</em>, Dip. Mat. Univ. Genova, Preprint 555 (2007). (<a href="http://www.dima.unige.it/~grandis/wCub2.pdf">pdf</a>)</li> </ul> <p>There is a long-term project by Stephan Stolz and Peter Teichner which originally tried to refine Segal’s 1-functorial formulation of conformal field theory to a 2-functorial extended FQFT, as indicated in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stephan Stolz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em>What is an elliptic object?</em> (<a href="http://math.berkeley.edu/~teichner/Papers/Oxford.pdf">pdf</a>).</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stephan Stolz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em>Supersymmetric field theories and generalized cohomology</em>, in: <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">H. Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">U. Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em>, Proceedings of Symposia in Pure Mathematics 83 (2011), 279–340 (<a href="https://arxiv.org/abs/1108.0189">arXiv:1108.0189</a>, <a href="https://doi.org/10.1090/pspum/083">doi:10.1090/pspum/083/2742432</a>)</p> </li> </ul> <p>In 2008, Mike Hopkins and Jacob Lurie claimed (<a href="http://golem.ph.utexas.edu/category/2008/05/hopkinslurie_on_baezdolan.html">Hopkins-Lurie on Baez-Dolan</a>) to have found a complete coherent formalization of topological extended FQFT in the context of <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,n)-categories</a> using an <a class="existingWikiWord" href="/nlab/show/%28infinity%2Cn%29-category+of+cobordisms">(infinity,n)-category of cobordisms</a>. This is described in</p> <ul> <li id="Luire09"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/On+the+Classification+of+Topological+Field+Theories">On the Classification of Topological Field Theories</a></em> (<a href="http://arxiv.org/abs/0905.0465">arXiv:0905.0465</a>)</li> </ul> <p>An explicit account of this for the 2-dimensional case is presented in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Chris+Schommer-Pries">Chris Schommer-Pries</a>, <em>The Classification of Two-Dimensional Extended Topological Field Theories</em>, PhD thesis, Berkeley, 2009. <p><a href="https://arxiv.org/abs/1112.1000">arXiv:1112.1000</a></p> </li> </ul> <p>see also</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Damien+Calaque">Damien Calaque</a>, <a class="existingWikiWord" href="/nlab/show/Claudia+Scheimbauer">Claudia Scheimbauer</a>, <em>A note on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,n)</annotation></semantics></math>-category of cobordisms</em>, Algebraic & Geometric Topology 19:2 (2019), 533–655 (<a href="https://arxiv.org/abs/1509.08906">arXiv:1509.08906</a>, <a href="https://doi.org/10.2140/agt.2019.19.533">doi:10.2140/agt.2019.19.533</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Christopher+Schommer-Pries">Christopher Schommer-Pries</a>, <em>Invertible topological field theories</em> (<a href="https://arxiv.org/abs/1712.08029">arXiv:1712.08029</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Pavlov">Dmitri Pavlov</a>, <em>Extended field theories are local</em> (<a href="https://arxiv.org/abs/2011.01208">arXiv:2011.01208</a>)</p> </li> </ul> <p>Claim of classification of geometric FQFTs (including <a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a> etc.) via a geometric <a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>:</p> <ul> <li id="GradyPavlov21"><a class="existingWikiWord" href="/nlab/show/Daniel+Grady">Daniel Grady</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Pavlov">Dmitri Pavlov</a>, <em>The geometric cobordism hypothesis</em> (<a href="https://arxiv.org/abs/2111.01095">arXiv:2111.01095</a>)</li> </ul> <p>For the case of one-dimensional smooth field theories, see</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>Parallel Transport and Functors</em>, (<a href="https://arxiv.org/abs/0705.0452">arXiv:0705.0452</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Berwick-Evans">Daniel Berwick-Evans</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Pavlov">Dmitri Pavlov</a>, <em>Smooth one-dimensional topological field theories are vector bundles with connection</em>( <a href="https://arxiv.org/abs/1501.00967">arXiv:1501.00967</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Matthias+Ludewig">Matthias Ludewig</a>, Augusto Stoffel, <em>A framework for geometric field theories and their classification in dimension one</em> (<a href="https://arxiv.org/abs/2001.05721">arXiv:2001.05721</a>)</p> </li> </ul> <p>And for the analogous discussion in <a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a> see also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marco+Benini">Marco Benini</a>, <a class="existingWikiWord" href="/nlab/show/Marco+Perin">Marco Perin</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Schenkel">Alexander Schenkel</a>, <em>Smooth 1-dimensional algebraic quantum field theories</em> [<a href="https://arxiv.org/abs/2010.13808">arXiv:2010.13808</a>]</li> </ul> <p>For a functorial construction of two-dimensional smooth field theories from 2-form U(1)-connections and D-branes:</p> <ul> <li id="BunkWaldorf21a"> <p><a class="existingWikiWord" href="/nlab/show/Severin+Bunk">Severin Bunk</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>Transgression of D-branes</em>, Adv. Theor. Math. Phys. <strong>25</strong> 5 (2021) 1095-1198 [<a href="https://arxiv.org/abs/arXiv:1808.04894">arXiv:1808.04894</a>, <a href="https://dx.doi.org/10.4310/ATMP.2021.v25.n5.a1">doi:10.4310/ATMP.2021.v25.n5.a1</a>]</p> </li> <li id="BunkWaldorf21b"> <p><a class="existingWikiWord" href="/nlab/show/Severin+Bunk">Severin Bunk</a>, <a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>Smooth functorial field theories from B-fields and D-branes</em>, J. Homot. Rel. Struc. <strong>16</strong> 1 (2021) 75-153 [<a href="https://doi.org/10.1007/s40062-020-00272-2">doi:10.1007/s40062-020-00272-2</a>, <a href="https://arxiv.org/abs/arXiv:1911.09990">arXiv:1911.09990</a>]</p> </li> </ul> <h3 id="extended_fqft_from_background_fields_models">(extended) FQFT from background fields: <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>-models</h3> <p>In this context Dan Freed is picking up again his old work on higher algebraic structures in quantum field theory, as described in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <em>Quantum Groups from Path Integrals</em> (<a href="https://arxiv.org/abs/q-alg/9501025">arXiv</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <em>Higher Algebraic Structures and Quantization</em> (<a href="https://arxiv.org/abs/hep-th/9212115">arXiv</a>)</p> </li> </ul> <p>where he argued that and how the path integral should assign <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>-categorical objects to domains of codimension <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 is re-expressing this in 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>-functorial context. (Freed speaks of <em>multi-tiered</em> QFT instead of extended QFT.)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <em>Remarks on Chern-Simons Theory</em> (<a href="https://arxiv.org/abs/0808.2507">arXiv</a>).</li> </ul> <p>Freed’s ideas on how an extended or multi-tiered QFT arises from a path integral coming from a given background field were further formalized in the context of “finite” QFTs in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Simon+Willerton">Simon Willerton</a>, <em>The twisted Drinfeld double of a finite group via gerbes and finite groupoids</em> (<a href="https://arxiv.org/abs/math.QA/0503266">arXiv</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bruce+Bartlett">Bruce Bartlett</a>, <em>On unitary 2-representations of finite groups and topological quantum field theory</em>, PhD thesis, Sheffield (2008) (<a href="https://arxiv.org/abs/0901.3975">arXiv</a>)</p> </li> </ul> <p>There are indications that a complete picture of this involves <a class="existingWikiWord" href="/nlab/show/groupoidification">groupoidification</a></p> <ul> <li>Jeffrey Morton, <em>Extended TQFTs and Quantum Gravity</em> (<a href="https://arxiv.org/abs/0710.0032">arXiv</a>)</li> </ul> <p>and, more generally <a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a>:</p> <p>a big advancement in the understanding of extended <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>-model QFTs is the discussion in</p> <ul> <li>David Ben-Zvi, John Francis, David Nadler, <em>Integral Transforms and Drinfeld Centers in Derived Geometry</em> (<a href="https://arxiv.org/abs/0805.0157">arXiv</a>)</li> </ul> <p>which realizes <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>-models by homming <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a> <a class="existingWikiWord" href="/nlab/show/cospan">cospan</a>s into the total spaces (realized as an <a class="existingWikiWord" href="/nlab/show/infinity-stack">infinity-stack</a>) of background fields and regarding the resulting <a class="existingWikiWord" href="/nlab/show/span">span</a>s as pull-push operators on suitable <a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric functions</a>.</p> <p>A similar approach to bring the old work by Dan Freed mentioned above in contact with the picture of extended functorial QFT and the Baez-Dolan-Lurie structure theorem is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dan+Freed">Dan Freed</a>, <a class="existingWikiWord" href="/nlab/show/Mike+Hopkins">Mike Hopkins</a>, <a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>, <em><a class="existingWikiWord" href="/nlab/show/Topological+Quantum+Field+Theories+from+Compact+Lie+Groups">Topological Quantum Field Theories from Compact Lie Groups</a></em></li> </ul> <h3 id="super_qft">Super QFT</h3> <p>See</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theories+and+tmf">(2,1)-dimensional Euclidean field theories and tmf</a></p> </li> </ul> <h3 id="homological_2d_fqft_and_tcft">homological 2d FQFT (and TCFT)</h3> <p>As usual, the problem of constructing FQFT becomes much more tractable when linear approximations are applied. In homological FQFT and in <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> the Hom-spaces of the cobordism category (the moduli spaces of cobordisms with given punctures/boundaries) are approximated by complexes of chains on them. This leads to formalization 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>-functorial QFT in the context of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a></p> <p>The concept is essentially a formalization of what used to be called <a class="existingWikiWord" href="/nlab/show/cohomological+field+theory">cohomological field theory</a> in</p> <ul> <li id="Witten91"><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, <em>Introduction to cohomological field theory</em>, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 (<a class="existingWikiWord" href="/nlab/files/WittenCQFT.pdf" title="pdf">pdf</a>)</li> </ul> <p>The definition of <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a> was given independently by</p> <ul> <li id="Getzler92"><a class="existingWikiWord" href="/nlab/show/Ezra+Getzler">Ezra Getzler</a>, <em>Batalin-Vilkovisky algebras and two-dimensional topological field theories</em>, Comm. Math. Phys. 159(2), 265–285 (1994) (<a href="http://arxiv.org/abs/hep-th/9212043">arXiv:hep-th/9212043</a>)</li> </ul> <p>and</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Topological field theory</em>, (1999), Notes of lectures at Stanford university. (<a href="http://www.cgtp.duke.edu/ITP99/segal/">web</a>). See in particular <a href="http://www.cgtp.duke.edu/ITP99/segal/stanford/lect5.pdf">lecture 5</a> (“topological field theory with cochain values”).</li> </ul> <p>The classification of <a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a>s by <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+categories">Calabi-Yau A-∞ categories</a> was discussed in</p> <ul> <li id="Costello04"> <p><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em>Topological conformal field theories and Calabi-Yau categories</em> Advances in Mathematics, Volume 210, Issue 1, (2007), (<a href="http://arxiv.org/abs/math/0412149">arXiv:math/0412149</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kevin+Costello">Kevin Costello</a>, <em>The Gromov-Witten potential associated to a TCFT</em> (<a href="http://arxiv.org/abs/math/0509264">arXiv:math/0509264</a>)</p> </li> </ul> <p>following conjectures by <a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, e.g.</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <em>Homological algebra of mirror symmetry</em>, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), pages 120–139, Basel, 1995, Birkhäuser.</li> </ul> <div> <h3 id="2dCFTAsFunctorialQFTReferences"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">D=2</annotation></semantics></math> CFT as functorial field theory</h3> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/D%3D2+conformal+field+theory">D=2 conformal field theory</a> as a <a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a>, namely as a <a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from a 2d <a class="existingWikiWord" href="/nlab/show/conformal+cobordism+category">conformal cobordism category</a> to <a class="existingWikiWord" href="/nlab/show/Hilbert+spaces">Hilbert spaces</a>:</p> <ul> <li id="Segal88"><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>The definition of conformal field theory</em>, in: K. Bleuler, M. Werner (eds.), <em>Differential geometrical methods in theoretical physics</em> (Proceedings of Research Workshop, Como 1987), NATO Adv. Sci. Inst., Ser. C: Math. Phys. Sci. <strong>250</strong> Kluwer Acad. Publ., Dordrecht (1988) 165-171 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1007/978-94-015-7809-7">doi:10.1007/978-94-015-7809-7</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>and including discussion of <a class="existingWikiWord" href="/nlab/show/modular+functors">modular functors</a>:</p> <ul> <li id="Segal89"> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>Two-dimensional conformal field theories and modular functors</em>, in: <em>Proceedings of the IXth International Congress on Mathematical Physics</em>, Swansea, 1988, Hilger, Bristol (1989) 22-37.</p> </li> <li id="Segal04"> <p><a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>The definition of conformal field theory</em>, in: <a class="existingWikiWord" href="/nlab/show/Ulrike+Tillmann">Ulrike Tillmann</a> (ed.), <em>Topology, geometry and quantum field theory</em> , London Math. Soc. Lect. Note Ser. <strong>308</strong>, Cambridge University Press (2004) 421-577 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1017/CBO9780511526398.019">doi:10.1017/CBO9780511526398.019</a>, <a href="https://people.maths.ox.ac.uk/segalg/0521540496txt.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/SegalDefinitionCFT.pdf" title="pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> </li> </ul> <p>General construction for the case of <a class="existingWikiWord" href="/nlab/show/rational+2d+conformal+field+theory">rational 2d conformal field theory</a> is given by the</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/FRS-theorem+on+rational+2d+CFT">FRS-theorem on rational 2d CFT</a></em></li> </ul> <p>See also:</p> <ul> <li id="MooreSegal06"> <p><a class="existingWikiWord" href="/nlab/show/Greg+Moore">Greg Moore</a>, <a class="existingWikiWord" href="/nlab/show/Graeme+Segal">Graeme Segal</a>, <em>D-branes and K-theory in 2D topological field theory</em> (<a href="http://arxiv.org/abs/hep-th/0609042">arXiv:hep-th/0609042</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Richard+Blute">Richard Blute</a>, <a class="existingWikiWord" href="/nlab/show/Prakash+Panangaden">Prakash Panangaden</a>, <a class="existingWikiWord" href="/nlab/show/Dorette+Pronk">Dorette Pronk</a>, <em>Conformal field theory as a nuclear functor</em>, Electronic Notes in Theoretical Computer Science Volume 172, 1 April 2007, Pages 101-132 GDP Festschrift (<a href="http://aix1.uottawa.ca/~rblute/conf.pdf">pdf</a>, <a href="https://doi.org/10.1016/j.entcs.2007.02.005">doi:10.1016/j.entcs.2007.02.005</a>)</p> </li> </ul> <p>A different but closely analogous development for chiral 2d CFT (<a class="existingWikiWord" href="/nlab/show/vertex+operator+algebras">vertex operator algebras</a>, see <a href="vertex+operator+algebra#AsOperadAlgebras">there</a> for more):</p> <ul> <li id="Huang91"><a class="existingWikiWord" href="/nlab/show/Yi-Zhi+Huang">Yi-Zhi Huang</a>, <em>Geometric interpretation of vertex operator algebras</em>, Proc. Natl. Acad. Sci. USA <strong>88</strong> (1991) pp. 9964-9968 (<a href="https://doi.org/10.1073/pnas.88.22.9964">doi:10.1073/pnas.88.22.9964</a>)</li> </ul> <p>Discussion of the case of <a class="existingWikiWord" href="/nlab/show/Liouville+theory">Liouville theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Colin+Guillarmou">Colin Guillarmou</a>, <a class="existingWikiWord" href="/nlab/show/Antti+Kupiainen">Antti Kupiainen</a>, <a class="existingWikiWord" href="/nlab/show/R%C3%A9mi+Rhodes">Rémi Rhodes</a>, <a class="existingWikiWord" href="/nlab/show/Vincent+Vargas">Vincent Vargas</a>, <em>Segal’s axioms and bootstrap for Liouville Theory</em> &lbrack;<a href="https://arxiv.org/abs/2112.14859">arXiv:2112.14859</a>&rbrack;</li> </ul> <p>Early suggestions to refine this to an <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended</a> <a class="existingWikiWord" href="/nlab/show/2-functor">2-functorial</a> construction:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stefan+Stolz">Stefan Stolz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em><a class="existingWikiWord" href="/nlab/show/What+is+an+elliptic+object%3F">What is an elliptic object?</a></em></li> </ul> <p>A step towards generalization to <a class="existingWikiWord" href="/nlab/show/2d+super-conformal+field+theory">2d super-conformal field theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stephan+Stolz">Stephan Stolz</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Teichner">Peter Teichner</a>, <em>Supersymmetric field theories and generalized cohomology</em>, in: <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">H. Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">U. Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Mathematical+Foundations+of+Quantum+Field+and+Perturbative+String+Theory">Mathematical Foundations of Quantum Field and Perturbative String Theory</a></em>, Proceedings of Symposia in Pure Mathematics 83 (2011), 279–340 (<a href="https://arxiv.org/abs/1108.0189">arXiv:1108.0189</a>, <a href="https://doi.org/10.1090/pspum/083">doi:10.1090/pspum/083/2742432</a>)</li> </ul> <p>Discussion of 2-functorial chiral 2d CFT:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Henriques">André Henriques</a>, <em>The complex cobordism 2-category</em>, 2021 (<a href="http://andreghenriques.com/ComplexCob2CatandCentralExt.mp4">video</a>)</li> </ul> </div><div> <h3 id="TheoryXAsAnFQFTReferences">The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><mn>6</mn></mrow><annotation encoding="application/x-tex">D=6</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒩</mi><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{N}=(2,0)</annotation></semantics></math> SCFT as an extended functorial field theory</h3> <p>On the (conjectural) suggestion to view at least some aspects of the <a class="existingWikiWord" href="/nlab/show/D%3D6+N%3D%282%2C0%29+SCFT">D=6 N=(2,0) SCFT</a> (such as its <a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a> or its image as a <a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a> under the <a class="existingWikiWord" href="/nlab/show/AGT+correspondence">AGT correspondence</a>) as a <a class="existingWikiWord" href="/nlab/show/functorial+field+theory">functorial field theory</a> given by a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> on a suitable <a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a>, or rather as an <a class="existingWikiWord" href="/nlab/show/extended+TQFT">extended</a> such <a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a>, given by an <a class="existingWikiWord" href="/nlab/show/n-functor">n-functor</a> (at least a <a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> on a <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category+of+cobordisms">2-category of cobordisms</a>):</p> <ul> <li id="Witten09"> <p><a class="existingWikiWord" href="/nlab/show/Edward+Witten">Edward Witten</a>, Section 1 of: <em>Geometric Langlands From Six Dimensions</em>, in Peter Kotiuga (ed.) <em>A Celebration of the Mathematical Legacy of Raoul Bott</em>, CRM Proceedings & Lecture Notes Volume: 50, AMS 2010 (<a href="http://arxiv.org/abs/0905.2720">arXiv:0905.2720</a>, <a href="https://bookstore.ams.org/crmp-50">ISBN:978-0-8218-4777-0</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <em><a class="existingWikiWord" href="/nlab/show/4-3-2+8-7-6">4-3-2 8-7-6</a></em>, talk at <em><a href="https://people.maths.ox.ac.uk/tillmann/ASPECTS.html">ASPECTS of Topology</a></em> Dec 2012 (<a href="https://people.maths.ox.ac.uk/tillmann/ASPECTSfreed.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Freed432876.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, p. 32 of: <em>The cobordism hypothesis</em>, Bulletin of the American Mathematical Society 50 (2013), pp. 57-92, (<a href="http://arxiv.org/abs/1210.5100">arXiv:1210.5100</a>, <a href="https://doi.org/10.1090/S0273-0979-2012-01393-9">doi:10.1090/S0273-0979-2012-01393-9</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Freed">Daniel Freed</a>, <a class="existingWikiWord" href="/nlab/show/Constantin+Teleman">Constantin Teleman</a>: <em>Relative quantum field theory</em>, Commun. Math. Phys. <strong>326</strong> (2014) 459–476 [<a href="https://doi.org/10.1007/s00220-013-1880-1">doi:10.1007/s00220-013-1880-1</a>, <a href="https://arxiv.org/abs/1212.1692">arXiv:1212.1692</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Ben-Zvi">David Ben-Zvi</a>: <em>Theory <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">\mathcal{X}</annotation></semantics></math> and Geometric Representation Theory</em>, talks at <em><a href="http://www.ihes.fr/~celliott/workshop/">Mathematical Aspects of Six-Dimensional Quantum Field Theories</a></em> IHES 2014, notes by <a class="existingWikiWord" href="/nlab/show/Qiaochu+Yuan">Qiaochu Yuan</a> (<a class="existingWikiWord" href="/nlab/files/BenZvi-TheoryX-I.pdf" title="pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BenZvi-TheoryX-II.pdf" title="pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/BenZvi-TheoryX-III.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/David+Ben-Zvi">David Ben-Zvi</a>, <em>Algebraic geometry of topological field theories</em>, talk at <em><a href="https://www.msri.org/workshops/689">Reimagining the Foundations of Algebraic Topology April 07, 2014 - April 11, 2014</a></em> (<a href="https://www.msri.org/workshops/689/schedules/18216">web video</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lukas+M%C3%BCller">Lukas Müller</a>, <em>Extended Functorial Field Theories and Anomalies in Quantum Field Theories</em> (<a href="https://arxiv.org/abs/2003.08217">arXiv:2003.08217</a>)</p> </li> </ul> </div></body></html> </div> <div class="revisedby"> <p> Last revised on April 24, 2024 at 09:43:58. 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