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12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> geometry of physics -- smooth sets </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/8077/#Item_7" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" 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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> <h4 id="differential_geometry">Differential geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> </li> </ul> <p><strong>Tangency</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a></p> </li> </ul> <p><strong>The magic algebraic facts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a></p> </li> </ul> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a></p> </li> <li> <p><a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</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">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a></strong></p> <div id="Diagram" 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>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> <p id="models_2"><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>)</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></p> </li> </ul> </div></div> </div> </div> <blockquote> <p>This entry one chapter of <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em>.</p> <p>previous chapters: <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">categories and toposes</a></em></p> <p>next chapters: <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a></em>, <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a></em></p> </blockquote> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>This chapter introduces a generalized kind of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> equipped with <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, to be called <em><a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a></em> or, with an eye towards their generalization to <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a></em>, <em>smooth <a class="existingWikiWord" href="/nlab/show/h-sets">h-sets</a></em> or <em>smooth <a class="existingWikiWord" href="/nlab/show/homotopy+0-types">0-types</a></em><sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup>.</p> <p>The definition (Def. <a class="maruku-ref" href="#SmoothSpace"></a> below) subsumes that of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifolds">Fréchet manifolds</a> and <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> but is both simpler and more powerful: smooth sets are simply <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> on the <a class="existingWikiWord" href="/nlab/show/gros+site">gros site</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+Spaces">Cartesian Spaces</a> (Prop. <a class="maruku-ref" href="#CartSpCategory"></a> below) and as such form a nice <a class="existingWikiWord" href="/nlab/show/category">category</a> – a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> – and this contains as <a class="existingWikiWord" href="/nlab/show/full+subcategories">full subcategories</a> the more “tame” objects such as <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> (Prop. <a class="maruku-ref" href="#InclusionOfSmoothManifoldsIntoSmoothSets"></a> below) and <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> (Prop. <a class="maruku-ref" href="#DiffeologicalSpacesAreTheConcreteSmoothSets"></a> below).</p> <p>In fact smooth sets are an early stage in a long sequence of generalized smooth spaces appearing in <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\;\;\;</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><a class="existingWikiWord" href="/nlab/show/coordinate+systems">coordinate systems</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> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</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><a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}</annotation></semantics></math> (Prop. <a class="maruku-ref" href="#InclusionOfSmoothManifoldsIntoSmoothSets"></a> below) <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</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><a class="existingWikiWord" href="/nlab/show/Hilbert+manifolds">Hilbert manifolds</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> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</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><a class="existingWikiWord" href="/nlab/show/Banach+manifolds">Banach manifolds</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> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</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><a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifolds">Fréchet manifolds</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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}</annotation></semantics></math> (Prop. <a class="maruku-ref" href="#FrechetManifoldsFullyFaithfulInSmoothSets"></a> below) <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</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><a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological 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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}</annotation></semantics></math> (Prop. <a class="maruku-ref" href="#ReflectiveInclusionOfDiffeologicalSpacesinSmoothSets"></a> below) <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</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><a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</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><a class="existingWikiWord" href="/nlab/show/formal+smooth+sets">formal smooth sets</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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</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><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+sets">super formal smooth sets</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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}</annotation></semantics></math> (chapter <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">on supergeometry</a>) <br /> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</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><a class="existingWikiWord" href="/nlab/show/orbifold">smooth orbifolds</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> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</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><a class="existingWikiWord" href="/nlab/show/smooth+groupoids">smooth groupoids</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> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow </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><a class="existingWikiWord" href="/nlab/show/smooth+2-groupoids">smooth 2-groupoids</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> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow </annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>⋯</mi></mrow><annotation encoding="application/x-tex">\cdots</annotation></semantics></math> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow </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><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>AA</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{AA}</annotation></semantics></math> (chapter <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">on smooth ∞-groupoids</a>) <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow </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><a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoids">formal smooth ∞-groupoids</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> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow </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><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoids">super formal smooth ∞-groupoids</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#CoordinateSystemsLayerMod'>Abstract coordinate systems</a></li> <ul> <li><a href='#TheContinuumRealWorldLine'>The continuum real line</a></li> <li><a href='#CartesianSpaces'>Cartesian spaces and smooth functions</a></li> <li><a href='#PropertiesOfSmoothFunctions'>The magic properties of smooth functions</a></li> <li><a href='#CoordinateSystemsLayerSem'>The site of abstract coordinate systems</a></li> </ul> <li><a href='#SmoothSpacesLayerMod'>Smooth sets</a></li> <ul> <li><a href='#PlotsOfSmoothSpacesAndTheirGluing'>Plots of smooth sets and their gluing</a></li> <li><a href='#HomomorphismsOfSmoothSpaces'>Homomorphisms of smooth sets</a></li> <li><a href='#ProductsAndFiberProductsOfSmoothSpaces'>Products and fiber products of smooth sets</a></li> <li><a href='#SmoothMappingSpaces'>Smooth mapping spaces and smooth moduli spaces</a></li> <li><a href='#SmoothSpacesLayerSem'>The cohesive topos of smooth sets</a></li> <li><a href='#concrete_smooth_sets_diffeological_spaces'>Concrete smooth sets: Diffeological spaces</a></li> <li><a href='#DifferentialForms'>Differential forms</a></li> <li><a href='#integration_and_transgression'>Integration and transgression</a></li> </ul> </ul> </div> <h2 id="CoordinateSystemsLayerMod">Abstract coordinate systems</h2> <p>As discussed in the chapter <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">categories and toposes</a></em>, every kind of <em><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></em> is modeled on a collection of <a class="existingWikiWord" href="/nlab/show/generator">archetypical</a> basic <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> and geometric <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> between them. In <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> the archetypical spaces are the abstract standard <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian coordinate systems</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> (in every <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>). The geometric homomorphism between them are <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math>, hence smooth (and possibly degenerate) <a class="existingWikiWord" href="/nlab/show/coordinate+transformations">coordinate transformations</a>.</p> <p>Here we introduce the basic concept, organizing them in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> (Prop. <a class="maruku-ref" href="#CartSpCategory"></a> below.) We highlight three classical theorems about <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> in Prop. <a class="maruku-ref" href="#AlgebraicFactsOfDifferentialGeometry"></a> below, which look innocent but play a decisive role in setting up <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential</a> <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> based on the concept of abstract smooth coordinate systems.</p> <p>At this point these are not yet coordinate systems <em>on</em> some other space. But by applying the general machine of <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">categories and toposes</a> to these, a concept of <a class="existingWikiWord" href="/nlab/show/generalized+spaces">generalized spaces</a> modeled on these abstract coordinate systems is induced. These are the <em><a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a></em> discussed in the next chapter <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">geometry of physics – smooth sets</a></em>.</p> <h3 id="TheContinuumRealWorldLine">The continuum real line</h3> <p>The fundamental premise of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> as a model of <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is the following.</p> <p><strong>Premise.</strong> <em>The abstract <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a> of any <a class="existingWikiWord" href="/nlab/show/particle">particle</a> is modeled by the <a class="existingWikiWord" href="/nlab/show/continuum">continuum</a> <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>.</em></p> <p>This comes down to the following sequence of premises.</p> <ol> <li> <p>There is a <a class="existingWikiWord" href="/nlab/show/linear+ordering">linear ordering</a> of the <a class="existingWikiWord" href="/nlab/show/points">points</a> on a <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a>: in particular if we pick points at some intervals on the worldline we may label these in an order-preserving way by <a class="existingWikiWord" href="/nlab/show/integers">integers</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>.</p> </li> <li> <p>These intervals may each be subdivided into <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> smaller intervals, for each natural number <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>. Hence we may label points on the <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a> in an order-preserving way by the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math>.</p> </li> <li> <p>This labeling is dense: every point on the worldline is the <a class="existingWikiWord" href="/nlab/show/supremum">supremum</a> of an <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a> <a class="existingWikiWord" href="/nlab/show/bounded+subset">bounded subset</a> of such labels. This means that a <a class="existingWikiWord" href="/nlab/show/worldline">worldline</a> <a class="existingWikiWord" href="/nlab/show/equivalence">is</a> the <em><a class="existingWikiWord" href="/nlab/show/real+line">real line</a></em>, the <a class="existingWikiWord" href="/nlab/show/continuum">continuum</a> of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>.</p> </li> </ol> <p>The adjective “real” in “<a class="existingWikiWord" href="/nlab/show/real+number">real number</a>” is a historical shadow of the old idea that real numbers are related to <em>observed reality</em>, hence to <a class="existingWikiWord" href="/nlab/show/physics">physics</a> in this way. The experimental success of this assumption shows that it is valid at least to very good approximation.</p> <h3 id="CartesianSpaces">Cartesian spaces and smooth functions</h3> <div class="num_defn" id="SmoothFunctions"> <h6 id="definition">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>)</strong></p> <p>A <a class="existingWikiWord" href="/nlab/show/function">function</a> 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><mi>f</mi><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">f : \mathbb{R} \to \mathbb{R}</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></strong> if, <a class="existingWikiWord" href="/nlab/show/coinduction">coinductively</a>:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/derivative">derivative</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>d</mi><mi>f</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mo>:</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\frac{d f}{d x} : \mathbb{R} \to \mathbb{R}</annotation></semantics></math> exists;</p> </li> <li> <p>and is itself a smooth function.</p> </li> </ol> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}) \in Set</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/set">set</a> of all smooth functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The superscript “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">{}^\infty</annotation></semantics></math>” in “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R})</annotation></semantics></math>” refers to the order of the <a class="existingWikiWord" href="/nlab/show/derivatives">derivatives</a> that exist for smooth functions. More generally 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 \in \mathbb{N}</annotation></semantics></math> one writes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^k(\mathbb{R})</annotation></semantics></math> for the set of <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>-fold <a class="existingWikiWord" href="/nlab/show/differentiable+functions">differentiable functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>. These will however not play much of a role for our discussion here.</p> </div> <div class="num_defn" id="CartesianSpaceAndHomomorphism"> <h6 id="definition_2">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> and <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, the <strong><a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is the set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>1</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><msup><mi>x</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">|</mo><msup><mi>x</mi> <mi>i</mi></msup><mo>∈</mo><mi>ℝ</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n = \{ (x^1 , \cdots, x^{n}) | x^i \in \mathbb{R} \} </annotation></semantics></math></div> <p>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>-<a class="existingWikiWord" href="/nlab/show/tuples">tuples</a> of <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">1 \leq k \leq n</annotation></semantics></math> write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mi>k</mi></msup><mo>:</mo><mi>ℝ</mi><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> i^k : \mathbb{R} \to \mathbb{R}^n </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/function">function</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>,</mo><mn>0</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i^k(x) = (0, \cdots, 0,x,0,\cdots,0)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/tuple">tuple</a> whose <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>th entry is <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 all whose other entries are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">0 \in \mathbb{R}</annotation></semantics></math>; and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>𝕡</mi> <mi>k</mi></msup><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> \mathbb{p}^k : \mathbb{R}^n \to \mathbb{R} </annotation></semantics></math></div> <p>for the function such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>1</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><msup><mi>x</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">p^k(x^1, \cdots, x^n) = x^k</annotation></semantics></math>.</p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a></strong> of Cartesian spaces is a <em><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></em></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} \,, </annotation></semantics></math></div> <p>hence a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> such that all <a class="existingWikiWord" href="/nlab/show/partial+derivatives">partial derivatives</a> exist and are <a class="existingWikiWord" href="/nlab/show/continuous+map">continuous</a>.</p> </div> <div class="num_example"> <h6 id="example">Example</h6> <p>Regarding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, every <a class="existingWikiWord" href="/nlab/show/linear+function">linear function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> is in particular a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>But a homomorphism of Cartesian spaces in def. <a class="maruku-ref" href="#CartesianSpaceAndHomomorphism"></a> is <em>not</em> required to be a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a>. We do <em>not</em> regard the Cartesian spaces here as <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>.</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a></strong> if there exists another smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_2} \to \mathbb{R}^{n_1}</annotation></semantics></math> such that the underlying functions of sets are <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> to each other</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>g</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex"> f \circ g = id </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>g</mi><mo>∘</mo><mi>f</mi><mo>=</mo><mi>id</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g \circ f = id \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>There exists a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">n_1 = n_2</annotation></semantics></math>.</p> </div> <div class="num_defn" id="CartesianSpacesAndSmoothFunctions"> <h6 id="definition_4">Definition</h6> <p>We will also say equivalently that</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is an <strong><a class="existingWikiWord" href="/nlab/show/coordinate+system">abstract coordinate system</a></strong>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> is an <strong><a class="existingWikiWord" href="/nlab/show/coordinate+transformation">abstract coordinate transformation</a></strong>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mi>k</mi></msup><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">p^k : \mathbb{R}^{n} \to \mathbb{R}</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>th <strong><a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a></strong> of the coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>. We will also write this function as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>k</mi></msup><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">x^k : \mathbb{R}^{n} \to \mathbb{R}</annotation></semantics></math>.</p> </li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">1 \leq k \leq n_2</annotation></semantics></math> we write</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mi>k</mi></msup><mo>≔</mo><msup><mi>p</mi> <mi>k</mi></msup><mo>∘</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">f^k \coloneqq p^k\circ f</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><msup><mi>f</mi> <mn>1</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><msup><mi>f</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≔</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">(f^1, \cdots, f^n) \coloneqq f</annotation></semantics></math>.</p> </li> </ol> </li> </ol> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>It follows with this notation that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msup><mi>x</mi> <mn>1</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><msup><mi>x</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> id_{\mathbb{R}^n} = (x^1, \cdots, x^n) : \mathbb{R}^n \to \mathbb{R}^n \,. </annotation></semantics></math></div> <p>Hence an abstract coordinate transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex"> f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2} </annotation></semantics></math></div> <p>may equivalently be written as the tuple</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msup><mi>f</mi> <mn>1</mn></msup><mrow><mo>(</mo><msup><mi>x</mi> <mn>1</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><msup><mi>x</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>)</mo></mrow><mo>,</mo><mi>⋯</mi><mo>,</mo><msup><mi>f</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mrow><mo>(</mo><msup><mi>x</mi> <mn>1</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><msup><mi>x</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( f^1 \left( x^1, \cdots, x^{n_1} \right) , \cdots, f^{n_2}\left( x^1, \cdots, x^{n_1} \right) \right) \,. </annotation></semantics></math></div></div> <h3 id="PropertiesOfSmoothFunctions">The magic properties of smooth functions</h3> <p>Below we encounter generalizations of ordinary <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> that include explicit “<a class="existingWikiWord" href="/nlab/show/infinitesimals">infinitesimals</a>” in the guise of <em><a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+points">infinitesimally thickened points</a></em>, as well as “super-graded infinitesimals”, in the guise of <em><a class="existingWikiWord" href="/nlab/show/superpoints">superpoints</a></em> (necessary for the description of <a class="existingWikiWord" href="/nlab/show/fermion+fields">fermion fields</a> such as the <a class="existingWikiWord" href="/nlab/show/Dirac+field">Dirac field</a>). As we discuss <a href="#FieldBundles">below</a>, these structures are naturally incorporated into <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> in just the same way as <a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a> introduced them into <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> (in the guise of “<a class="existingWikiWord" href="/nlab/show/formal+schemes">formal schemes</a>”), namely in terms of <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a> <a class="existingWikiWord" href="/nlab/show/rings+of+functions">rings of functions</a> with <a class="existingWikiWord" href="/nlab/show/nilpotent+ideals">nilpotent ideals</a>. That this also works well for <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> rests on the following three basic but important properties, which say that <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> behave “more algebraically” than their definition might superficially suggest:</p> <div class="num_prop" id="AlgebraicFactsOfDifferentialGeometry"> <h6 id="proposition_2">Proposition</h6> <p><strong>(the three magic algebraic properties of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>)</strong></p> <ol> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of Cartesian spaces into formal duals of R-algebras</a></strong></p> <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> and <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> two <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>, the <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</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>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \longrightarrow Y</annotation></semantics></math> between them (def. <a class="maruku-ref" href="#CartesianSpacesAndSmoothFunctionsBetweenThem"></a>) are in <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> with their induced algebra <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X) \overset{f^\ast}{\longrightarrow} C^\infty(Y)</annotation></semantics></math> (example <a class="maruku-ref" href="#AlgebraOfSmoothFunctionsOnCartesianSpaces"></a>), so that one may equivalently handle <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> entirely via their <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-algebras of smooth functions.</p> <p>Stated more <a class="existingWikiWord" href="/nlab/show/category+theory">abstractly</a>, this means equivalently that the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(-)</annotation></semantics></math> that sends a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> (example <a class="maruku-ref" href="#AlgebraOfSmoothFunctionsOnCartesianSpaces"></a>) is a <em><a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a></em>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>SmthMfd</mi><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><mi>ℝ</mi><msup><mi>Alg</mi> <mi>op</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C^\infty(-) \;\colon\; SmthMfd \overset{\phantom{AAAA}}{\hookrightarrow} \mathbb{R} Alg^{op} \,. </annotation></semantics></math></div> <p>(<a href="embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras#KolarSlovakMichor93">Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10</a>)</p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">embedding of smooth vector bundles into formal duals of R-algebra modules</a></strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><mover><mo>→</mo><mrow><msub><mi>vb</mi> <mn>1</mn></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">E_1 \overset{vb_1}{\to} X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub><mover><mo>→</mo><mrow><msub><mi>vb</mi> <mn>2</mn></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">E_2 \overset{vb_2}{\to} X</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> there is then a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> between vector bundle <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> <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>E</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f \colon E_1 \to E_2</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/modules">modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><msub><mi>E</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_\ast \;\colon\; \Gamma_X(E_1) \to \Gamma_X(E_2)</annotation></semantics></math> that these induces between the <a class="existingWikiWord" href="/nlab/show/spaces+of+sections">spaces of sections</a>.</p> <p>More <a class="existingWikiWord" href="/nlab/show/category+theory">abstractly</a> this means that the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(-)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful 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>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>VectBund</mi> <mi>X</mi></msub><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> \Gamma_X(-) \;\colon\; VectBund_X \overset{\phantom{AAAA}}{\hookrightarrow} C^\infty(X) Mod </annotation></semantics></math></div> <p>(<a href="smooth+Serre-Swan+theorem#Nestruev03">Nestruev 03, theorem 11.29</a>)</p> <p>Moreover, the <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</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> which arise this way as <a class="existingWikiWord" href="/nlab/show/sections">sections</a> of <a class="existingWikiWord" href="/nlab/show/smooth+vector+bundles">smooth vector bundles</a> over a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</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> are precisely the <a class="existingWikiWord" href="/nlab/show/finitely+generated+module">finitely generated</a> <a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math>.</p> <p>(<a href="smooth+Serre-Swan+theorem#Nestruev03">Nestruev 03, theorem 11.32</a>)</p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">vector fields are derivations of smooth functions</a></strong>.</p> <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/Cartesian+space">Cartesian space</a> (Def. <a class="maruku-ref" href="#CartesianSpaceAndHomomorphism"></a>), then any <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo lspace="verythinmathspace">:</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><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">D \colon C^\infty(X) \to C^\infty(X)</annotation></semantics></math> on the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> is given by <a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a> with respect to a uniquely defined smooth <a class="existingWikiWord" href="/nlab/show/tangent+vector+field">tangent vector field</a>: The function that regards <a class="existingWikiWord" href="/nlab/show/tangent+vector+fields">tangent vector fields</a> as <a class="existingWikiWord" href="/nlab/show/derivations">derivations</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>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>T</mi><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mphantom><mi>A</mi></mphantom><mo>≃</mo><mphantom><mi>A</mi></mphantom></mrow></mover></mtd> <mtd><mi>Der</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>v</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>D</mi> <mi>v</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Gamma_X(T X) &\overset{\phantom{A}\simeq\phantom{A}}{\longrightarrow}& Der(C^\infty(X)) \\ v &\mapsto& D_v } </annotation></semantics></math></div> <p>is in fact an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> <p>(This follows directly from the <em><a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a></em>.)</p> </li> </ol> </div> <p>Actually all three statements in prop. <a class="maruku-ref" href="#AlgebraicFactsOfDifferentialGeometry"></a> hold not just for <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>, but generally for <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> (def./prop. <a class="maruku-ref" href="#SmoothManifoldInsideDiffeologicalSpaces"></a> below; if only we generalize in the second statement from <a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a> to <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a>). However for our development here it is useful to first focus on just <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> and then bootstrap the theory of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> and much more from that, which we do <a href="#FieldBundles">below</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="CoordinateSystemsLayerSem">The site of abstract coordinate systems</h3> <p>Much of the above disucssion is usefully summarized by saying that abstract coordinate systems with smooth functions between them form a <em><a class="existingWikiWord" href="/nlab/show/category">category</a></em> (Prop. <a class="maruku-ref" href="#CartSpCategory"></a> below). Equipped with the information of how one abstract coordinate system may be <em>covered</em> by other coordinate systems (Def. <a class="maruku-ref" href="#DifferentiallyGoodOpenCover"></a> below), this becomes a <em><a class="existingWikiWord" href="/nlab/show/site">site</a></em> (Prop. <a class="maruku-ref" href="#TheDifferentiallyGoodOpenCoverCoverage"></a>) below.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_prop" id="CartSpCategory"> <h6 id="propositions">Propositions</h6> <p><strong>(the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> of abstract <a class="existingWikiWord" href="/nlab/show/coordinate+systems">coordinate systems</a>/<a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>)</strong></p> <p>Abstract coordinate systems according to prop. <a class="maruku-ref" href="#CartesianSpacesAndSmoothFunctions"></a> form a <em><a class="existingWikiWord" href="/nlab/show/category">category</a></em> (<a href="geometry+of+physics+--+Categories+and+Toposes#Categories">this def.</a>) – to be denoted <em><a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a></em> – whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/objects">objects</a> are the abstract coordinate systems <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n}</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/class">class</a> of objects is the <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>ℕ</mi></mrow><annotation encoding="application/x-tex">\mathbb{N}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/natural+numbers">natural numbers</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>);</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> are the abstract <a class="existingWikiWord" href="/nlab/show/coordinate+transformations">coordinate transformations</a> = <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>.</p> </li> </ul> <p>Composition of morphisms is given by <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/functions">functions</a>.</p> <p>Under this identification</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/identity+morphisms">identity morphisms</a> are precisely the <a class="existingWikiWord" href="/nlab/show/identity+functions">identity functions</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> are precisely the <a class="existingWikiWord" href="/nlab/show/diffeomorphisms">diffeomorphisms</a>.</p> </li> </ol> </div> <p>We discuss a standard structure of a <em><a class="existingWikiWord" href="/nlab/show/site">site</a></em> on the category <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>. Following <em><a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Johnstone – Sketches of an Elephant</a></em>, it will be useful and convenient to regard a site as a (<a class="existingWikiWord" href="/nlab/show/small+site">small</a>) category equipped with a <em><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a></em>. This generates a genuine <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>, but need not itself already be one.</p> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> the standard <a class="existingWikiWord" href="/nlab/show/open+ball">open n-ball</a> is the subset</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><msubsup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">|</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo><</mo><mn>1</mn><mo stretchy="false">}</mo><mo>↪</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> D^n = \{ (x_i)_{i =1}^n \in \mathbb{R}^n | \sum_{i = 1}^n (x_i)^2 \lt 1 \} \hookrightarrow \mathbb{R}^n \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mo>≃</mo></mover><msup><mi>D</mi> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \stackrel{\simeq}{\to} D^n \,. </annotation></semantics></math></div></div> <div class="num_defn" id="DifferentiallyGoodOpenCover"> <h6 id="definition_6">Definition</h6> <p><strong>(differentially <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a>)</strong></p> <p>A <strong>differentially <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a></strong> of a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>↪</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \hookrightarrow \mathbb{R}^n\}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> inclusions of Cartesian spaces such that these <a class="existingWikiWord" href="/nlab/show/open+cover">cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> and such for each non-empty finite <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> there exists a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mo>∩</mo><mi>⋯</mi><mo>∩</mo><msub><mi>U</mi> <mrow><msub><mi>i</mi> <mi>k</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \stackrel{\simeq}{\to} U_{i_1} \cap \cdots \cap U_{i_k} </annotation></semantics></math></div> <p>that identifies the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-fold intersection with a Cartesian space itself.</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>Differentiably good covers are useful for computations. Their full impact is however on the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> over <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>. This we discuss in the chapter on <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, around this prop.omotopy types#DifferentiablyGoodCoverGivesSPlitHyperCoverOverCartSp).</p> </div> <div class="num_lemma" id="DiffGoodOpenCoversRefineOpenCovers"> <h6 id="lemma">Lemma</h6> <p><strong>(every <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> has refinement by a differentially <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>)</strong></p> <p>Every <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> of a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> has a refinement by a differentially good open cover, according to Def. <a class="maruku-ref" href="#DifferentiallyGoodOpenCover"></a>.</p> </div> <p>For proof see <a href="https://ncatlab.org/schreiber/show/Cech+Cocycles+for+Differential+characteristic+Classes">FSS10, Prop. A1</a>, or see at <em><a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a></em>.</p> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>Lemma <a class="maruku-ref" href="#DiffGoodOpenCoversRefineOpenCovers"></a> is not quite a classical statement. The classical statement is only that every open cover is refined by a <em>topologically</em> <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>. See the comments <em><a href="ball#References">here in the references-section</a></em> at <em><a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a></em> for the situation concerning this statement in the literature.</p> </div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>The <em>good</em> open covers do not yet form a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a> on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>. One of the axioms of a Grothendieck topology is that for every <a class="existingWikiWord" href="/nlab/show/covering">covering</a> family also its <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> along any morphism in the category is a covering family. But while the pullback of every <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> is again an open cover, and hence open covers form a Grothendieck topology on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>, not every pullback of a <em><a class="existingWikiWord" href="/nlab/show/good+open+cover">good</a></em> open cover is again <em>good</em>.</p> </div> <div class="num_example" id="ExampleThatGoodOpenCoversAreNotPullbackStable"> <h6 id="example_2">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mover><mo>↪</mo><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub></mrow></mover><msup><mi>ℝ</mi> <mn>2</mn></msup><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\{\mathbb{R}^2\stackrel{\phi_{i}}{\hookrightarrow}\mathbb{R}^2\}_{i \in \{1,2\}}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> of the <a class="existingWikiWord" href="/nlab/show/plane">plane</a> by an open left half space</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>≃</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo stretchy="false">|</mo><msub><mi>x</mi> <mn>1</mn></msub><mo><</mo><mn>1</mn><mo stretchy="false">}</mo><mover><mo>↪</mo><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub></mrow></mover><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex"> \mathbb{R}^2 \simeq \{ (x_1,x_2) \in \mathbb{R}^2 | x_1 \lt 1 \} \stackrel{\phi_1}{\hookrightarrow} \mathbb{R}^2 </annotation></semantics></math></div> <p>and a right open half space</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>≃</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo stretchy="false">|</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>></mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">}</mo><mover><mo>↪</mo><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub></mrow></mover><msup><mi>ℝ</mi> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^2 \simeq \{ (x_1,x_2) \in \mathbb{R}^2 | x_1 \gt -1 \} \stackrel{\phi_2}{\hookrightarrow} \mathbb{R}^2 \,. </annotation></semantics></math></div> <p>The intersection of the two is the open strip</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>≃</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo stretchy="false">|</mo><mo>−</mo><mn>1</mn><mo><</mo><msub><mi>x</mi> <mn>1</mn></msub><mo><</mo><mn>1</mn><mo stretchy="false">}</mo><mo>↪</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^2 \simeq \{ (x_1, x_2) \in \mathbb{R}^2 | -1 \lt x_1 \lt 1 \} \hookrightarrow \mathbb{R}^2 \,. </annotation></semantics></math></div> <p>So this is a <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math>.</p> <p>But consider then the smooth function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>2</mn><mo stretchy="false">(</mo><mi>cos</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>sin</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>→</mo><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex"> 2(\cos(2 \pi (-)), \sin(2 \pi (-))) \colon \mathbb{R}^1 \to \mathbb{R}^2 </annotation></semantics></math></div> <p>which sends the line to a curve in the plane that periodically goes around the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> of radius 2 in the plane.</p> <p>Then the pullback of the above good open cover on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1</annotation></semantics></math> along this function is an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> by two open subsets, each being a disjoint union of countably many open <a class="existingWikiWord" href="/nlab/show/intervals">intervals</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>. Each of these open intervals is an open 1-ball hence diffeomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1</annotation></semantics></math>, but their <em><a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a></em> is not <a class="existingWikiWord" href="/nlab/show/contractible+topological+space">contractible</a> (it does not contract to the point, but to many points!).</p> <p>So the pullback of the good open cover that we started with is an open cover which is not <em>good</em> anymore. But it has an evident <em>refinement</em> by a good open cover.</p> </div> <p>This is a special case of what the following statement says in generality.</p> <div class="num_prop" id="TheDifferentiallyGoodOpenCoverCoverage"> <h6 id="proposition_4">Proposition</h6> <p><strong>(the <a class="existingWikiWord" href="/nlab/show/site">site</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> with differentially <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a>)</strong></p> <p>The differentially <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a>, Def. <a class="maruku-ref" href="#DifferentiallyGoodOpenCover"></a>, constitute a <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> (<a href="geometry+of+physics+--+categories+and+toposes#Coverage">this Def.</a>) on the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> (from Prop. <a class="maruku-ref" href="#CartSpCategory"></a>).</p> <p>Hence <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> equipped with this coverage is a <a class="existingWikiWord" href="/nlab/show/site">site</a> (<a href="geometry+of+physics+--+categories+and+toposes#Coverage">this def.</a>).</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>By definition of <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> we need to check that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>↪</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{U_i \hookrightarrow \mathbb{R}^n\}_{i \in I}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">f \colon \mathbb{R}^k \to \mathbb{R}^n</annotation></semantics></math> any smooth function, we can find a good open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>K</mi> <mi>j</mi></msub><mo>→</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><msub><mo stretchy="false">}</mo> <mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{K_j \to \mathbb{R}^k\}_{j \in J}</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>→</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">J \to I</annotation></semantics></math> such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">j \in J</annotation></semantics></math> there is a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>K</mi> <mi>j</mi></msub><mo>→</mo><msub><mi>U</mi> <mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\phi \colon K_j \to U_{\rho(j)}</annotation></semantics></math> that makes this <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commute</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>K</mi> <mi>j</mi></msub></mtd> <mtd><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd><msub><mi>U</mi> <mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mi>k</mi></msup></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><msup><mi>ℝ</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ K_j &\stackrel{\phi}{\to}& U_{i(j)} \\ \downarrow && \downarrow \\ \mathbb{R}^k &\stackrel{f}{\to}& \mathbb{R}^n } \,. </annotation></semantics></math></div> <p>To obtain this, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{f^* U_i \to \mathbb{R}^k\}</annotation></semantics></math> be the pullback of the original covering family, in that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><msub><mi>U</mi> <mi>i</mi></msub><mo>≔</mo><mo stretchy="false">{</mo><mi>x</mi><mo>∈</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo><mo>↪</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^* U_i \coloneqq \{ x \in \mathbb{R}^k | f(x) \in U_i \} \hookrightarrow \mathbb{R}^k \,. </annotation></semantics></math></div> <p>This is evidently an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a>, albeit not necessarily a <em><a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a></em>. But by Lemma <a class="maruku-ref" href="#DiffGoodOpenCoversRefineOpenCovers"></a> there does exist a good open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mover><mi>j</mi><mo stretchy="false">˜</mo></mover></msub><mo>↪</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><msub><mo stretchy="false">}</mo> <mrow><mover><mi>j</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo><mover><mi>J</mi><mo stretchy="false">˜</mo></mover></mrow></msub></mrow><annotation encoding="application/x-tex">\{\tilde K_{\tilde j} \hookrightarrow \mathbb{R}^k\}_{\tilde j \in \tilde J}</annotation></semantics></math> <em>refining</em> it, which in turn means that for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>j</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde j</annotation></semantics></math> there 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><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mover><mi>j</mi><mo stretchy="false">˜</mo></mover></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>K</mi> <mrow><mi>j</mi><mo stretchy="false">(</mo><mover><mi>j</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mi>k</mi></msup></mtd> <mtd><mover><mo>→</mo><mo>=</mo></mover></mtd> <mtd><msup><mi>ℝ</mi> <mi>k</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \tilde K_{\tilde j} &\to& K_{j(\tilde j)} \\ \downarrow && \downarrow \\ \mathbb{R}^k &\stackrel{=}{\to}& \mathbb{R}^k } \,. </annotation></semantics></math></div> <p>Therefore then the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> composite of these commuting 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><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mover><mi>j</mi><mo stretchy="false">˜</mo></mover></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>K</mi> <mrow><mi>j</mi><mo stretchy="false">(</mo><mover><mi>j</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo></mrow></msub></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>U</mi> <mrow><mi>i</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">(</mo><mover><mi>j</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></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><msup><mi>ℝ</mi> <mi>k</mi></msup></mtd> <mtd><mover><mo>→</mo><mo>=</mo></mover></mtd> <mtd><msup><mi>ℝ</mi> <mi>k</mi></msup></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><msup><mi>ℝ</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \tilde K_{\tilde j} &\to& K_{j(\tilde j)} &\to& U_{i(j(\tilde j))} \\ \downarrow && \downarrow && \downarrow \\ \mathbb{R}^k &\stackrel{=}{\to}& \mathbb{R}^k &\stackrel{f}{\to}& \mathbb{R}^n } </annotation></semantics></math></div> <p>solves the condition required in the definition of <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>.</p> </div> <p>By example <a class="maruku-ref" href="#ExampleThatGoodOpenCoversAreNotPullbackStable"></a> this <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> is not a <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>. But as any coverage, it uniquely completes to one which has the same <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a>:</p> <div class="num_prop" id="CompletingGoodOpenCoversToAllOpenCovers"> <h6 id="proposition_5">Proposition</h6> <p><strong>(completing <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a> to all <a class="existingWikiWord" href="/nlab/show/open+covers">open covers</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a> induced on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> by the differentially good open cover coverage of def. <a class="maruku-ref" href="#TheDifferentiallyGoodOpenCoverCoverage"></a> has as covering families the ordinary <a class="existingWikiWord" href="/nlab/show/open+covers">open covers</a>.</p> <p>Hence if we explicitly write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CartSp</mi> <mi>good</mi></msub></mrow><annotation encoding="application/x-tex">CartSp_{good}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CartSp</mi> <mi>Groth</mi></msub></mrow><annotation encoding="application/x-tex">CartSp_{Groth}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math> equipped with the coverage of differentially good open covers as and that of all open covers, respectively, then there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> (<a href="geometry+of+physics+--+categories+and+toposes#EquivalenceOfCategories">this Def.</a>) between their <a class="existingWikiWord" href="/nlab/show/categories+of+sheaves">categories of sheaves</a> (<a href="geometry+of+physics+--+categories+and+toposes#Sheaf">this Def.</a>)</p> <div class="maruku-equation" id="eq:SheavesOnGoodOpenCoversCoincideWithThosOnALlOpenCovers"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>CartSp</mi> <mtext>good open cov</mtext></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>CartSp</mi> <mtext>all open cov</mtext></msub><mo stretchy="false">)</mo><mi>\</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sh(CartSp_{\text{good open cov}}) \;\simeq\; Sh(CartSp_{\text{all open cov}}) \. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>Prop. <a class="maruku-ref" href="#CompletingGoodOpenCoversToAllOpenCovers"></a> means that for every sheaf-theoretic construction to follow we may just as well consider the Grothendieck topology of open covers on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math>, and hence we may and will suppress the subscripts in <a class="maruku-eqref" href="#eq:SheavesOnGoodOpenCoversCoincideWithThosOnALlOpenCovers">(1)</a>.</p> <p>While the sheaves of the open cover topology are the same as those of the good open cover coverage. But the latter is (more) useful for several computational purposes in the following. It is the <em>good</em> open cover coverage that makes manifest, below, that sheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> and in consequence then a <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a>. This kind of argument becomes all the more pronounced as we pass <a href="#SmoothnGroupoids">further below</a> to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a> on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>. This will be discussed in <em><a href="#InfinityConnectednessOfSmoothInfinityGrpd">Smooth n-groupoids – Semantic Layer – Local Infinity-Connectedness</a> below.</em></p> </div> <p>There are further <a class="existingWikiWord" href="/nlab/show/sites">sites</a> in use, which induce the same categories of sheaves:</p> <div class="num_defn" id="SiteOfSmoothManifolds"> <h6 id="definition_7">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/sites">sites</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> and <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of <a class="existingWikiWord" href="/nlab/show/Euclidean+spaces">Euclidean spaces</a>)</strong></p> <p>We write</p> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/SmthMfd">SmthMfd</a></em> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> of any finite <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>, with <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> between them;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>EuclOp</mi></mrow><annotation encoding="application/x-tex">EuclOp</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of <a class="existingWikiWord" href="/nlab/show/Euclidean+spaces">Euclidean spaces</a> of any finite <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>, with <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> between them.</p> </li> </ul> <p>Both of these carry the respective <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> of <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a> and as such become <a class="existingWikiWord" href="/nlab/show/sites">sites</a> (<a href="geometry+of+physics+--+categories+and+toposes#Coverage">this Def.</a>)</p> </div> <h2 id="SmoothSpacesLayerMod">Smooth sets</h2> <p>In the section <em><a href="#CoordinateSystems">Coordinate systems</a></em> we have set up the archetypical <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>. Here we now define in terms of these the most general <em><a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a></em> that differential geometry can deal with.</p> <h3 id="PlotsOfSmoothSpacesAndTheirGluing">Plots of smooth sets and their gluing</h3> <p>The general kind of “<a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>” that we want to consider is a <a class="existingWikiWord" href="/nlab/show/type">something</a> that can be <em>probed</em> by laying out <a class="existingWikiWord" href="/nlab/show/coordinate+systems">coordinate systems</a> inside it, as in <a href="geometry+of+physics+--+coordinate+systems#CartesianSpaces">this definition</a>, and which may be reconstructed by <em>gluing</em> all the possible coordinate systems in it together.</p> <p>At this point we want to impose no further conditions on a “space” than this. In particular we do not assume that we know beforehand a <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/points">points</a> underlying <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>. Instead, we define <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</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> (Def. <a class="maruku-ref" href="#SmoothSpace"></a>, below) entirely <em>operationally</em> as something about which we may ask “Which ways are there to lay out <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> inside <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 such that there is a self-consistent answer to this question.</p> <p>By the discussion in the chapter <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">categories and toposes</a></em>, this means that we should define a <em><a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a></em> to be a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on a <a class="existingWikiWord" href="/nlab/show/site">site</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>. The following definitions spell this out.</p> <p>The idea of the following definitions may be summarized like this:</p> <ol> <li> <p>a generalized <em>smooth set</em> is something that may be probed by laying out coordinate systems into it, in a way that respects transformation of coordinate patches and gluing of coordinate patches;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> (<a href="geometry+of+physics+--+categories+and+toposes#YonedaLemma">this prop</a>) says that this is consistent in that coordinate systems themselves as well as <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> may naturally be regarded as generalized smooth sets themselves and that under this identification “laying out a coordinate system” in a smooth set means having a map of smooth sets from the coordinate system to the smooth set.</p> </li> </ol> <p>The first set of consistency conditions on plots of a space is that they respect <em>coordinate transformations</em>. This is what the following definition formalises.</p> <div class="num_defn" id="SmoothPreSpace"> <h6 id="definition_8">Definition</h6> <p><strong>(pre-smooth set)</strong></p> <p>A <strong>pre-smooth set</strong> <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</p> <ol> <li> <p>a collection of sets: for each <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> (hence for each <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</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>) a <a class="existingWikiWord" href="/nlab/show/set">set</a></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><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> X(\mathbb{R}^n) \in Set </annotation></semantics></math></div> <p>– to be thought of as the <em>set of ways of laying out <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> inside <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></em>, also called the set of <em>plots</em> 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>, for short;</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> (to be thought of as an abstract coordinate transformation) a <a class="existingWikiWord" href="/nlab/show/function">function</a> between the corresponding sets of plots</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><mi>f</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mo>⟶</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X(f) \;\colon\; X(\mathbb{R}^{n_2}) \longrightarrow X(\mathbb{R}^{n_1}) </annotation></semantics></math></div> <p>– to be thought of as the function that sends a <em>plot</em> 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><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_2}</annotation></semantics></math> to the correspondingly transformed plot by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_1}</annotation></semantics></math> induced by laying out <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_1}</annotation></semantics></math> inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n_2}</annotation></semantics></math>.</p> </li> </ol> <p>such that this is compatible with coordinate transformations:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> coordinate transformation does not change the plots:</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><mi>id</mi> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>id</mi> <mrow><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X(id_{\mathbb{R}^n}) = id_{X(\mathbb{R}^n)} \,, </annotation></semantics></math></div></li> <li> <p>changing plots along two consecutive coordinate transformations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f_1 \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>3</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f_2 \colon \mathbb{R}^{n_2} \to \mathbb{R}^{n_3}</annotation></semantics></math> is the same as changing them along the <a class="existingWikiWord" href="/nlab/show/composition">composite</a> coordinate transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f_2 \circ f_1</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>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>∘</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X(f_1) \circ X(f_2) = X(f_2 \circ f_1) \,. </annotation></semantics></math></div></li> </ol> </div> <p>But there is one more consistency condition for a collection of plots to really be probes of some space: it must be true that if we glue small coordinate systems to larger ones, then the plots by the larger ones are the same as the plots by the collection of smaller ones that agree where they overlap. We first formalize this idea of “plots that agree where their coordinate systems overlap.”</p> <div class="num_prop" id="DifferentiallyGoodOpenCover"> <h6 id="definition_9">Definition</h6> <p><strong>(differentially <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>, then</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</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><msub><mi>U</mi> <mi>i</mi></msub><mo>↪</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \{U_i \hookrightarrow X\}_{i \in I} </annotation></semantics></math></div> <p>is a set of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</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>, such that their <a class="existingWikiWord" href="/nlab/show/union">union</a> is <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>; hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo>∪</mo><mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></munder><msub><mi>U</mi> <mi>i</mi></msub><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\underset{i \in I}{\cup} U_i = X</annotation></semantics></math>;</p> </li> <li> <p>this is a <em><a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a></em> if all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> as well as all their <a class="existingWikiWord" href="/nlab/show/inhabited+set">non-empty</a> finite <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a> are <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a>, hence to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> itself;</p> </li> <li> <p>this is a <em>differentially good open cover</em> if all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> as well as all their <a class="existingWikiWord" href="/nlab/show/inhabited+set">non-empty</a> finite <a class="existingWikiWord" href="/nlab/show/intersections">intersections</a> are <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphic</a> to <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a>, hence to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> itself.</p> </li> </ol> </div> <div class="num_defn" id="MatchingFamiliesOfPlots"> <h6 id="definition_10">Definition</h6> <p><strong>(glued plots)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a pre-smooth set, def. <a class="maruku-ref" href="#SmoothPreSpace"></a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{U_i \to \mathbb{R}^n\}_{i \in I}</annotation></semantics></math> a differentially <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> (def. <a class="maruku-ref" href="#DifferentiallyGoodOpenCover"></a>) let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GluedPlots</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">}</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> GluedPlots(\{U_i \to \mathbb{R}^n\}, X) \in Set </annotation></semantics></math></div> <p>be the set of <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>-<a class="existingWikiWord" href="/nlab/show/tuples">tuples</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math>-plots 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> which coincide on all double <a class="existingWikiWord" href="/nlab/show/intersections">intersections</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><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ι</mi> <mi>i</mi></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>ι</mi> <mi>j</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>U</mi> <mi>i</mi></msub></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mi>U</mi> <mi>j</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>ℝ</mi> <mi>n</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && U_i \cap U_j \\ & {}^{\mathllap{\iota_i}}\swarrow && \searrow^{\mathrlap{\iota_j}} \\ U_i &&&& U_j \\ & \searrow && \swarrow \\ && \mathbb{R}^n } </annotation></semantics></math></div> <p>(also called the <em><a class="existingWikiWord" href="/nlab/show/matching+families">matching families</a></em> 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> over the given cover):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>GluedPlots</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">}</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mspace width="thickmathspace"></mspace><msub><mrow><mo>(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>)</mo></mrow> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mo>∀</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mi>I</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> GluedPlots(\{U_i \to \mathbb{R}^n\}, X) \;\;\coloneqq\;\; \left\{ \; \left(p_i \in X(U_i)\right)_{i \in I} \;|\;\; \forall_{i,j \in I} \;:\; X(\iota_i)(p_i) = X(\iota_j)(p_j) \; \right\} \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_8">Remark</h6> <p><strong>(interpretation of the gluing condition)</strong></p> <p>In def. <a class="maruku-ref" href="#MatchingFamiliesOfPlots"></a> the <a class="existingWikiWord" href="/nlab/show/equation">equation</a></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><mi>ι</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>ι</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X(\iota_i)(p_i) = X(\iota_j)(p_j) </annotation></semantics></math></div> <p>says in words:</p> <blockquote> <p>The plot <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">p_i</annotation></semantics></math> 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 the coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> inside the bigger coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> coincides with the plot <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">p_j</annotation></semantics></math> 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 the other coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_j</annotation></semantics></math> inside <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> when both are restricted to the <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_i \cap U_j</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_j</annotation></semantics></math> inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>.</p> </blockquote> </div> <div class="num_remark" id="NaiveDescentMorphism"> <h6 id="remark_9">Remark</h6> <p><strong>(comparing global plots to glued plots)</strong></p> <p>For each differentially good open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{U_i \to \mathbb{R}^n\}_{i \in I}</annotation></semantics></math> (def. <a class="maruku-ref" href="#DifferentiallyGoodOpenCover"></a>) and each pre-smooth set <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>, def. <a class="maruku-ref" href="#SmoothPreSpace"></a>, there is a canonical <a class="existingWikiWord" href="/nlab/show/function">function</a></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><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⟶</mo><mi>GluedPlots</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">}</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X(\mathbb{R}^n) \longrightarrow GluedPlots(\{U_i \to \mathbb{R}^n\}, X) </annotation></semantics></math></div> <p>from the set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plots 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> to the set of tuples of glued plots, which sends a plot <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p \in X(\mathbb{R}^n)</annotation></semantics></math> to its restriction to all the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>↪</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\phi_i \colon U_i \hookrightarrow \mathbb{R}^n</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><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> p \mapsto (X(\phi_i)(p))_{i \in I} \,. </annotation></semantics></math></div></div> <p>If <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 supposed to be consistently probable by coordinate systems, then it must be true that the set of ways of laying out a coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> inside it coincides with the set of ways of laying out tuples of glued coordinate systems inside it, for each good cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to \mathbb{R}^n\}</annotation></semantics></math> as above. Therefore:</p> <div class="num_defn" id="SmoothSpace"> <h6 id="definition_11">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>)</strong></p> <p>A pre-smooth set <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>, def. <a class="maruku-ref" href="#SmoothPreSpace"></a> is a <strong><a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a></strong> if for all differentially good open covers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to \mathbb{R}^n\}</annotation></semantics></math> (def. <a class="maruku-ref" href="#DifferentiallyGoodOpenCover"></a>) the canonical comparison function of remark <a class="maruku-ref" href="#NaiveDescentMorphism"></a> from plots to glued plots is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a></p> <div class="maruku-equation" id="eq:ConditionSmoothSet"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>GluedPlots</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></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(\mathbb{R}^n) \stackrel{\simeq}{\longrightarrow} GluedPlots(\{U_i \to \mathbb{R}^n\}, X) \,. </annotation></semantics></math></div></div> <div class="num_remark" id="OnTheNotionOfSmoothSpaces"> <h6 id="remark_10">Remark</h6> <p>We may think of a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> as being a kind of <a class="existingWikiWord" href="/nlab/show/space">space</a> whose <em>local models</em> (in the general sense discussed at <em><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></em>) are <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>:</p> <p id="AmalgamatingSmoothSetsFromCoordinateCharts"> While definition <a class="maruku-ref" href="#SmoothSpace"></a> explicitly says that a smooth set is something that is <em>consistently probeable</em> by such local models, a <a class="existingWikiWord" href="/nlab/show/category+theory">general abstract</a> fact sometimes called the <em><a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a></em> – which we discuss in more detail below in <em><a href="#SmoothSpacesLayerSem">smooth sets - Semantic Layer</a></em> – implies that smooth sets are also precisely the objects that are obtained by <em><a class="existingWikiWord" href="/nlab/show/amalgamation">amalgamating</a> coordinate systems</em>, in some sense.</p> <p>For instance, we will see that two open 2-balls <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>≃</mo><msup><mi>D</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2 \simeq D^2</annotation></semantics></math> along a common rim yields the smooth set version of the <a class="existingWikiWord" href="/nlab/show/sphere">sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math>, a basic example of a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>. But before we examine such explicit constructions, we discuss here for the moment more general properties of smooth sets. The reader instead wishing to see more of these concrete examples at this point should jump ahead to <em><a href="#SmoothSpacesOutlook">smooth sets - Outlook</a></em>.</p> </div> <p>But the following most basic example we consider right now:</p> <div class="num_example" id="CartesianSpaceAsSmoothSpace"> <h6 id="example_3">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> and <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> as <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">n \in \mathbb{R}^n</annotation></semantics></math>, there is a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>, def. <a class="maruku-ref" href="#SmoothSpace"></a>, whose set of plots over the abstract coordinate systems <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> is the set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> CartSp(\mathbb{R}^k, \mathbb{R}^n) \in Set </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>.</p> <p>Clearly this is the rule for plots that characterize <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> itself as a smooth set, and so we will just denote this smooth set by the same symbols “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>”:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>↦</mo><mi>CartSp</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \colon \mathbb{R}^k \mapsto CartSp(\mathbb{R}^k, \mathbb{R}^n) \,. </annotation></semantics></math></div> <p>In particular the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> is this way itself a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>.</p> <p>More generally, if the reader already knows what a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is; these become smooth sets by taking their plots to be the ordinary <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> between smooth manifolds, from Cartesian spaces:</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><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>Hom</mi> <mi>SmthMfd</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X(\mathbb{R}^n) \coloneqq Hom_{SmthMfd}(\mathbb{R}^n, X) \,. </annotation></semantics></math></div></div> <p>Some smooth sets are far from being like smooth manifolds:</p> <div class="num_example" id="SmoothSetOfDifferentialForms"> <h6 id="example_4">Example</h6> <p><strong>(smooth <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a>)</strong></p> <p>Let <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 \in \mathbb{N}</annotation></semantics></math>. Then there is a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> (def. <a class="maruku-ref" href="#CartesianSpaceAsSmoothSpace"></a>) to be denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^k</annotation></semantics></math> given as follows:</p> <ol> <li> <p>The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>k</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^k(\mathbb{R}^n)</annotation></semantics></math> of plots from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is the set of smooth <a class="existingWikiWord" href="/nlab/show/differential+n-form">differential k-forms</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>k</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≔</mo><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Omega}^k(\mathbb{R}^n) \coloneqq \Omega^k(\mathbb{R}^n) </annotation></semantics></math></div></li> <li> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> a smooth function, then the corresponding change-of-plots function is the operation of <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a> along <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mover><msup><mi>Ω</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^{k}(\mathbb{R}^{n_2}) \overset{f^\ast}{\longrightarrow} \Omega^k(\mathbb{R}^{n_1}) \,. </annotation></semantics></math></div></li> </ol> <p>We introduce and discuss this example in detail in more detail <a href="#DifferentialForms">below</a></p> </div> <p>A <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> (def. <a class="maruku-ref" href="#SmoothSpace"></a>) need not have an <em>underlying set</em>, for instance the smooth set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^k</annotation></semantics></math> from example <a class="maruku-ref" href="#SmoothSetOfDifferentialForms"></a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">k \geq 1</annotation></semantics></math> has only a single plot from the point (corresponding to the zero differential form on the point), and yet it is far from being the point. If a smooth set does have an underlying set, then it is called a <em><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></em>, see around Prop. <a class="maruku-ref" href="#DiffeologicalSpacesAreTheConcreteSmoothSets"></a> below.</p> <div class="num_example" id="DiscreteSmoothSpace"> <h6 id="example_5">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/discrete+object">discrete</a> <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">S \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a> a set, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Disc</mi><mi>S</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex"> Disc S \in SmoothSet </annotation></semantics></math></div> <p>for the smooth set whose set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-plots for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">U \in CartSp</annotation></semantics></math> is always <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</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>Disc</mi><mi>S</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>↦</mo><mi>S</mi></mrow><annotation encoding="application/x-tex"> Disc S \colon U \mapsto S </annotation></semantics></math></div> <p>and which sends every coordinate transformation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f \colon \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</annotation></semantics></math> to the identity function on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>.</p> <p>A smooth set of this form we call a <strong><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete smooth set</a></strong>.</p> </div> <p>More examples of smooth sets can be built notably by <a class="existingWikiWord" href="/nlab/show/intersection">intersecting</a> <a class="existingWikiWord" href="/nlab/show/images">images</a> of two smooth sets inside a bigger one. In order to say this we first need a formalization of <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of smooth sets. This we turn to now.</p> <h3 id="HomomorphismsOfSmoothSpaces">Homomorphisms of smooth sets</h3> <p>We discuss “functions” or “maps” between <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>, def. <a class="maruku-ref" href="#SmoothSpace"></a>, which preserve the smooth set <a class="existingWikiWord" href="/nlab/show/structure">structure</a> in a suitable sense. As with any notion of function that preserves structure, we refer to them as <em><a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a></em>.</p> <p>The idea of the following definition is to say that whatever 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>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> between two smooth sets is, it has to take the plots 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><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> to a corresponding plot of <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>, such that this respects coordinate transformations.</p> <div class="num_defn" id="HomomorphismOfSmoothSpaces"> <h6 id="definition_12">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> – smooth functions)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <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> be two <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>, def. <a class="maruku-ref" href="#SmoothSpace"></a>. Then a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <em><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></em> <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>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> between them is</p> <ul> <li> <p>for each abstract coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> (hence for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>) a <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mo>:</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_{\mathbb{R}^n} : X(\mathbb{R}^n) \to Y(\mathbb{R}^n)</annotation></semantics></math></p> <p>that sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plots 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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plots of <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> </ul> <p>such that</p> <ul> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\phi : \mathbb{R}^{n_1} \to \mathbb{R}^{n_2}</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><mi>Y</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>f</mi> <mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mrow></msub><mo>=</mo><msub><mi>f</mi> <mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow></msub><mo>∘</mo><mi>X</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Y(\phi) \circ f_{\mathbb{R}^{n_1}} = f_{\mathbb{R}^{n_2}} \circ X(\phi) </annotation></semantics></math></div> <p>hence a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup></mrow></msub></mrow></mover></mtd> <mtd><mi>Y</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>Y</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>f</mi> <mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup></mrow></msub></mrow></mover></mtd> <mtd><mi>Y</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X(\mathbb{R}^{n_1}) &\stackrel{f_{\mathbb{R}^{n_1}}}{\to}& Y(\mathbb{R}^{n_1}) \\ \downarrow^{\mathrlap{X(\phi)}} && \downarrow^{\mathrlap{Y(\phi)}} \\ X(\mathbb{R}^{n_2}) &\stackrel{f_{\mathbb{R}^{n_2}}}{\to}& Y(\mathbb{R}^{n_1}) } \,. </annotation></semantics></math></div></li> </ul> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f_1 : X \to Y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>2</mn></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f_2 : X \to Y</annotation></semantics></math> two homomorphisms of smooth sets, their <a class="existingWikiWord" href="/nlab/show/composition">composition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f_2 \circ f_1 \colon X \to Y</annotation></semantics></math> is defined to be the homomorphism whose component over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is the composite of functions of the components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">f_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">f_2</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mo>≔</mo><msub><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mo>∘</mo><msub><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (f_2\circ f_1)_{\mathbb{R}^n} \coloneqq {f_2}_{\mathbb{R}^n} \circ {f_1}_{\mathbb{R}^n} \,. </annotation></semantics></math></div></div> <div class="num_defn" id="CategoryOfSmoothSets"> <h6 id="definition_13">Definition</h6> <p><strong>(the <a class="existingWikiWord" href="/nlab/show/category">category</a> <em><a class="existingWikiWord" href="/nlab/show/SmoothSet">SmoothSet</a></em> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> with <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>)</strong></p> <p>Write <a class="existingWikiWord" href="/nlab/show/SmoothSet">SmoothSet</a> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">SmoothSet</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> (<a href="geometry+of+physics+--+Categories+and+Toposes#Categories">this def.</a>) whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>, def. <a class="maruku-ref" href="#SmoothSpace"></a>, and whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are homomorphisms of smooth sets according to def. <a class="maruku-ref" href="#HomomorphismOfSmoothSpaces"></a>.</p> </div> <p>At this point it may seem that we have now <em>two different</em> notions for how to lay out a coordinate system in a smooth set <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>: on the hand, <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> comes by definition with a rule for what the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X(\mathbb{R}^n)</annotation></semantics></math> of its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plots is. On the other hand, we can now regard the abstract coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> itself as a smooth set, by example <a class="maruku-ref" href="#CartesianSpaceAsSmoothSpace"></a>, and then say that an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plot 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> should be a homomorphism of smooth sets of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \to X</annotation></semantics></math>.</p> <p>The following proposition says that these two superficially different notions actually naturally coincide.</p> <div class="num_prop" id="YonedaForSmoothSpaces"> <h6 id="proposition_6">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be any <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>, def. <a class="maruku-ref" href="#SmoothSpace"></a>, and regard the abstract coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> as a smooth set, by example <a class="maruku-ref" href="#CartesianSpaceAsSmoothSpace"></a>. There is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a></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><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X(\mathbb{R}^n) \simeq Hom_{SmoothSet}(\mathbb{R}^n, X) </annotation></semantics></math></div> <p>between the <em>postulated</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plots 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 the <em>actual</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plots given by homomorphism of smooth sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \to X</annotation></semantics></math>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is a special case of the <em><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></em> (<a href="geometry+of+physics+--+categories+and+toposes#YonedaLemma">this prop.</a>), as will be made more explicit below in <em><a href="#ToposOfSmoothSpaces">The topos of smooth sets</a></em>. The reader unfamiliar with this should write out the simple proof explicitly: use the defining <a class="existingWikiWord" href="/nlab/show/commuting+diagrams">commuting diagrams</a> in def. <a class="maruku-ref" href="#HomomorphismOfSmoothSpaces"></a> to deduce that a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f : \mathbb{R}^n \to X</annotation></semantics></math> is uniquely fixed by the image of the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≔</mo><mi>CartSp</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R}^n(\mathbb{R}^n) \coloneqq CartSp(\mathbb{R}^n, \mathbb{R}^n)</annotation></semantics></math> under the component function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mo>:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_{\mathbb{R}^n} : \mathbb{R}^n(\mathbb{R}^n) \to X(\mathbb{R}^n)</annotation></semantics></math>.</p> </div> <div class="num_example" id="SmoothFunctionOnSmoothSpace"> <h6 id="example_6">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">\mathbb{R} \in SmoothSet</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, regarded as a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> by example <a class="maruku-ref" href="#CartesianSpaceAsSmoothSpace"></a>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SmoothSet</annotation></semantics></math> any smooth set, a homomorphism of smooth sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> f \colon X \to \mathbb{R} </annotation></semantics></math></div> <p>is a <em><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</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></em>. Prop. <a class="maruku-ref" href="#YonedaForSmoothSpaces"></a> says here that when <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> happens to be an abstract coordinate system regarded as a smooth set by def. <a class="maruku-ref" href="#CartesianSpaceAsSmoothSpace"></a>, then this general notion of smooth functions between smooth sets reproduces the basic notion of this def.ystems#CartesianSpaceAndHomomorphism)</p> </div> <div class="num_defn" id="PointsOfASmoothSpace"> <h6 id="definition_14">Definition</h6> <p>The 0-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> abstract coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^0</annotation></semantics></math> we also call the <strong><a class="existingWikiWord" href="/nlab/show/point">point</a></strong> and regarded as a smooth set we will often write it as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>∈</mo><mi>SmoothSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> * \in SmoothSet \,. </annotation></semantics></math></div> <p>For any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SmoothSet</annotation></semantics></math>, we say that a homomorphism</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 lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> x \colon * \to X </annotation></semantics></math></div> <p>is a <strong>point 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></strong>.</p> </div> <div class="num_remark"> <h6 id="remark_11">Remark</h6> <p>By prop. <a class="maruku-ref" href="#YonedaForSmoothSpaces"></a> the points of a smooth set <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> are naturally identified with its 0-dimensional plots, hence with the “ways of laying out a 0-dimensional coordinate system” 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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mn>0</mn></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom(*, X) \simeq X(\mathbb{R}^0) \,. </annotation></semantics></math></div></div> <h3 id="ProductsAndFiberProductsOfSmoothSpaces">Products and fiber products of smooth sets</h3> <div class="num_defn" id="ProductOfSmoothSpaces"> <h6 id="definition_15">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X, Y \in SmoothSet</annotation></semantics></math> by two smooth sets. Their <strong><a class="existingWikiWord" href="/nlab/show/product">product</a></strong> is the smooth set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X \times Y \in SmoothSet</annotation></semantics></math> whose plots are pairs of plots 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 <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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≔</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>×</mo><mi>Y</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mi>Set</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X\times Y (\mathbb{R}^n) \coloneqq X(\mathbb{R}^n) \times Y(\mathbb{R}^n) \;\; \in Set \,. </annotation></semantics></math></div> <p>The <strong>projection on the first factor</strong> is the homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> p_1 \colon X \times Y \to X </annotation></semantics></math></div> <p>which sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plots of <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 \times Y</annotation></semantics></math> to those 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 forming the projection of the <a class="existingWikiWord" href="/nlab/show/cartesian+product">cartesian product</a> of sets:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>×</mo><mi>Y</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> {p_1}_{\mathbb{R}^n} \colon X(\mathbb{R}^n) \times Y(\mathbb{R}^n) \stackrel{p_1}{\to} X(\mathbb{R}^n) \,. </annotation></semantics></math></div> <p>Analogously for the <strong>projection to the second factor</strong></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> p_2 \colon X \times Y \to Y \,. </annotation></semantics></math></div></div> <div class="num_prop" id="ProductOfSmoothSpaceWithThePoint"> <h6 id="proposition_7">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>=</mo><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">* = \mathbb{R}^0</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/point">point</a>, regarded as a smooth set, def. <a class="maruku-ref" href="#PointsOfASmoothSpace"></a>. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SmoothSet</annotation></semantics></math> any smooth set the canonical projection homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \times * \to X </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <div class="num_defn"> <h6 id="definition_16">Definition</h6> <p>Let <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>Z</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Z</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">g \colon Y \to Z</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of smooth sets, def. <a class="maruku-ref" href="#HomomorphismOfSmoothSpaces"></a>. There is then a new smooth set to be denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>Z</mi></msub><mi>Y</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex"> X \times_Z Y \in SmoothSet </annotation></semantics></math></div> <p>(with <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> and <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> understood), called the <a class="existingWikiWord" href="/nlab/show/fiber+product">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> and <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> along <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> and <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 defined as follows:</p> <p>the set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plots of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><msub><mo>×</mo> <mi>Z</mi></msub><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times_Z Y</annotation></semantics></math> is the set of pairs of plots 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 <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> which become the same plot of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math> under <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> and <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>, respectively:</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><msub><mo>×</mo> <mi>Z</mi></msub><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>X</mi></msub><mo>∈</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo>∈</mo><mi>Y</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><msub><mi>f</mi> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></msub><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>Y</mi></msub><mo stretchy="false">)</mo><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (X \times_Z Y)(\mathbb{R}^n) = \left\{ (p_X \in X(\mathbb{R}^n), p_Y \in Y(\mathbb{R}^n)) \; |\; f_{\mathbb{R}^n}(p_X) = g_{\mathbb{R}^n}(p_Y) \right\} \,. </annotation></semantics></math></div></div> <h3 id="SmoothMappingSpaces">Smooth mapping spaces and smooth moduli spaces</h3> <div class="num_defn" id="SmoothFunctionSpace"> <h6 id="definition_17">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/smooth+set">smooth</a> <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>,</mo><mi>X</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">\Sigma, X \in SmoothSet</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>, def. <a class="maruku-ref" href="#SmoothSpace"></a>. Then the <strong>smooth <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></strong></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>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex"> [\Sigma,X] \in SmoothSet </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> defined by saying that its set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plots 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>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>≔</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Sigma, X](\mathbb{R}^n) \coloneqq Hom(\Sigma \times \mathbb{R}^n, X) \,. </annotation></semantics></math></div></div> <p>Here in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Sigma \times \mathbb{R}^n</annotation></semantics></math> we first regard the abstract coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> as a smooth set by example <a class="maruku-ref" href="#CartesianSpaceAsSmoothSpace"></a> and then we form the <a class="existingWikiWord" href="/nlab/show/product">product</a> smooth set by def. <a class="maruku-ref" href="#ProductOfSmoothSpaces"></a>.</p> <div class="num_remark" id="NatureOfPlotsOfMappingSpace"> <h6 id="remark_12">Remark</h6> <p>This means in words that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-plot of the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math> is a smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>-parameterized <em>family</em> of homomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math>.</p> </div> <div class="num_prop" id="UniversalPropertyOfMappingSpace"> <h6 id="proposition_8">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>K</mi><mo>×</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom(K, [\Sigma, X]) \simeq Hom(K \times \Sigma, X) </annotation></semantics></math></div> <p>for every smooth set <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>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>With a bit of work this is straightforward to check explicitly by unwinding the definitions. It follows however from <a class="existingWikiWord" href="/nlab/show/category+theory">general abstract</a> results once we realize that <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>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math> is of course the <em><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a></em> of smooth sets. This we come to below in <em><a href="#SmoothSpacesLayerSem">Smooth sets - Semantic Layer</a></em>.</p> </div> <div class="num_remark" id="MappingSpaceAsModuliSpace"> <h6 id="remark_13">Remark</h6> <p>This says in words that a smooth function from any <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> into the mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math> is equivalently a smooth function from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>×</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">K \times \Sigma</annotation></semantics></math> to <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>. The latter we may regard as a <em><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>-parameterized smooth family</em> of smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math>. Therefore in view of the previous remark <a class="maruku-ref" href="#NatureOfPlotsOfMappingSpace"></a> this says that smooth mapping spaces have a <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> not just over abstract coordinate systems, but over all smooth sets.</p> <p>We will therefore also say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math> is the <strong>smooth <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a></strong> of smooth functions from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math>, because it is such that smooth maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">K \to [\Sigma,X]</annotation></semantics></math> into it <em>modulate</em>, as we move around on <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>, a family of smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma\to X</annotation></semantics></math>, depending on <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>.</p> </div> <p>First interesting examples of such smooth moduli spaces are discussed in <em><a href="#DifferentialFormsLayerMod">Differential forms – Model Layer</a></em> below. Many more interesting examples follow once we pass from smooth 0-types to smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-types below in <em><a href="#SmoothnGroupoids">Smooth n-groupoids</a></em>.</p> <p>We will see many more examples of smooth moduli spaces, starting below in <em><a href="#DifferentialFormsLayerMod">Differential forms - Model Layer</a></em>.</p> <div class="num_prop" id="UnderlyingSetOfSmoothMappingSpace"> <h6 id="proposition_9">Proposition</h6> <p>The set of points, def. <a class="maruku-ref" href="#PointsOfASmoothSpace"></a>, of a smooth mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math> is the bare set of homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math>: there is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom(*, [\Sigma, X]) \simeq Hom(\Sigma, X) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>Combine prop. <a class="maruku-ref" href="#UniversalPropertyOfMappingSpace"></a> with prop. <a class="maruku-ref" href="#ProductOfSmoothSpaceWithThePoint"></a>.</p> </div> <div class="num_example" id="SmoothPathSpace"> <h6 id="example_7">Example</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SmoothSet</annotation></semantics></math>, its smooth <strong><a class="existingWikiWord" href="/nlab/show/path+space">path space</a></strong> is the smooth mapping space</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>P</mi></mstyle><mi>X</mi><mo>≔</mo><mo stretchy="false">[</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{P}X \coloneqq [\mathbb{R}^1, X] \,. </annotation></semantics></math></div> <p>By prop. <a class="maruku-ref" href="#UnderlyingSetOfSmoothMappingSpace"></a> the points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">P X</annotation></semantics></math> are indeed precisely the smooth trajectories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^1 \to X</annotation></semantics></math>. But <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">P X</annotation></semantics></math> also knows how to <em>smoothly vary</em> such smooth trajectories.</p> <p>This is central for <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> which determines <a class="existingWikiWord" href="/nlab/show/equations+of+motion">equations of motion</a> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. This we turn to below in <em><a href="#VariationalCalculus">Variational calculus</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_14">Remark</h6> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a model for <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">P X</annotation></semantics></math> may notably be interpreted as the smooth set of <a class="existingWikiWord" href="/nlab/show/worldlines">worldlines</a> <em>in</em> <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>, hence the smooth set of paths or <em>trajectories</em> of a <a class="existingWikiWord" href="/nlab/show/particle">particle</a> 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>.</p> </div> <div class="num_example" id="SmoothLoopSpace"> <h6 id="example_8">Example</h6> <p>If in the above example <a class="maruku-ref" href="#SmoothPathSpace"></a> the path is constraind to be a <a class="existingWikiWord" href="/nlab/show/loop">loop</a> 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>, one obtains the <em><a class="existingWikiWord" href="/nlab/show/smooth+loop+space">smooth loop space</a></em></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>L</mi></mstyle><mi>X</mi><mo>≔</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{L}X \coloneqq [S^1, X] \,. </annotation></semantics></math></div></div> <p>In example <a class="maruku-ref" href="#SmoothFunctionOnSmoothSpace"></a> we saw that a smooth function on a general <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</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> is a homomorphism of smooth sets, def. <a class="maruku-ref" href="#HomomorphismOfSmoothSpaces"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \colon X \to \mathbb{R} \,. </annotation></semantics></math></div> <p>The collection of these forms the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{SmoothSet}(X, \mathbb{R})</annotation></semantics></math>. But by the discussion in <em><a href="#SmoothMappingSpaces">Smooth mapping spaces</a></em> such hom-sets are naturally refined to smooth sets themselves.</p> <div class="num_defn" id="SmoothSpaceOfSmoothFunctions"> <h6 id="definition_18">Definition</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>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SmoothSet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>, we say that the <strong>moduli space of smooth functions</strong> 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 the smooth mapping space (def. <a class="maruku-ref" href="#SmoothFunctionSpace"></a>), 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> into the standard <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math></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>ℝ</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>SmoothSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X, \mathbb{R}] \in SmoothSet \,. </annotation></semantics></math></div> <p>We will also denote this by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>C</mi></mstyle> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{C}^\infty(X) \coloneqq [X, \mathbb{R}] \,, </annotation></semantics></math></div> <p>since in the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> this is the smooth refinement of the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_15">Remark</h6> <p>We call this a <em><a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a></em> because by prop. <a class="maruku-ref" href="#UniversalPropertyOfMappingSpace"></a> above and in the sense of remark <a class="maruku-ref" href="#MappingSpaceAsModuliSpace"></a> it is such that smooth functions into it <em>modulate</em> smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">X \to \mathbb{R}</annotation></semantics></math>.</p> <p>By prop. <a class="maruku-ref" href="#UnderlyingSetOfSmoothMappingSpace"></a> a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">* \to [X,\mathbb{R}^1]</annotation></semantics></math> of the moduli space is equivalently a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">X \to \mathbb{R}^1</annotation></semantics></math>.</p> </div> <h3 id="SmoothSpacesLayerSem">The cohesive topos of smooth sets</h3> <p>In the language of <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">categories and toposes</a>, we may summarize the concept of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> by saying that they form the <em><a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a></em> over the <em><a class="existingWikiWord" href="/nlab/show/site">site</a></em> of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> (Prop. <a class="maruku-ref" href="#SmoothSetsAreSheavesOnCartSp"></a> below).</p> <p>This perspective allows to see good abstract properties enjoyed by the <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>. The key such property is that the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> which they form is a <em><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></em> (Prop. <a class="maruku-ref" href="#SmoothSetsFormACohesiveTopos"></a> below).</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_prop" id="SmoothSetsAreSheavesOnCartSp"> <h6 id="proposition_10">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> between <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> and <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>)</strong></p> <p>There is a canonical <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> (<a href="geometry+of+physics+--+Categories+and+Toposes#EquivalenceOfCategories">this def.</a>) between the <a class="existingWikiWord" href="/nlab/show/category">category</a> <em><a class="existingWikiWord" href="/nlab/show/SmoothSet">SmoothSet</a></em> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> from def. <a class="maruku-ref" href="#CategoryOfSmoothSets"></a>, and the <a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a> (<a href="geometry+of+physics+--+Categories+and+Toposes#Sheaf">this def.</a>) on the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> (<a href="geometry+of+physics+--+coordinate+systems#CartSpCategory">this def.</a>) equipped with the <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> (<a href="geometry+of+physics+--+Categories+and+Toposes#Coverage">this def.</a>) of differentiably <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a> (def. <a class="maruku-ref" href="#DifferentiallyGoodOpenCover"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> SmoothSet \;\simeq\; Sh(CartSp) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>This is a straightforward matter of matching definitions. We spell it out:</p> <ul> <li> <p>A <em>pre-smooth set</em>, def. <a class="maruku-ref" href="#SmoothPreSpace"></a> is equivalently a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> (<a href="geometry+of+physics+--+Categories+and+Toposes#CategoryOfPresheaves">this Example</a>) on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> (<a href="geometry+of+physics+--+coordinate+systems#CartSpCategory">this prop.</a>), hence a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> (<a href="geometry+of+physics+--+Categories+and+Toposes#Functors">this def.</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><msup><mi>CartSp</mi> <mi>op</mi></msup><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">X : CartSp^{op} \to Set</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> to the <a class="existingWikiWord" href="/nlab/show/category+of+sets">category of sets</a> (<a href="geometry+of+physics+--+Categories+and+Toposes#CategoryOfSets">this Example</a>);</p> </li> <li> <p>a <em><a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a></em>, def. <a class="maruku-ref" href="#SmoothSpace"></a>, is equivalently a presheaf on <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> (<a href="geometry+of+physics+--+coordinate+systems#CartSpCategory">this prop.</a>) which is a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> (<a href="geometry+of+physics+--+Categories+and+Toposes#Sheaf">this def.</a>) with respect to the <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> (<a href="geometry+of+physics+--+Categories+and+Toposes#Coverage">this def.</a>) of differentially <a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> (def. <a class="maruku-ref" href="#DifferentiallyGoodOpenCover"></a>):</p> <ul> <li> <p>the set of “glued plots” (def. <a class="maruku-ref" href="#MatchingFamiliesOfPlots"></a>) is the set of <a class="existingWikiWord" href="/nlab/show/matching+families">matching families</a> (<a href="geometry+of+physics+--+Categories+and+Toposes#CompatibleElements">this def.</a>)</p> </li> <li> <p>the comparison morphism from global plots to glued plots of remark <a class="maruku-ref" href="#NaiveDescentMorphism"></a> is the comparison map from to <a class="existingWikiWord" href="/nlab/show/matching+families">matching families</a> (<a href="geometry+of+physics+--+categories+and+toposes#eq:SheafComparison">here</a>);</p> </li> <li> <p>the condition <a class="maruku-eqref" href="#eq:ConditionSmoothSet">(2)</a> that this be a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> is the <em><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf condition</a></em> (<a href="geometry+of+physics+--+Categories+and+Toposes#eq:SheafCondition">here</a>).</p> </li> </ul> </li> </ul> </div> <div class="num_prop" id="EquivalentSitesForCartSp"> <h6 id="proposition_11">Proposition</h6> <p><strong>(equivalent <a class="existingWikiWord" href="/nlab/show/sites">sites</a> for <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>)</strong></p> <p>Consider the canonical <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a>-inclusion <a class="existingWikiWord" href="/nlab/show/functors">functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi><mover><mo>↪</mo><mphantom><mi>AAA</mi></mphantom></mover><mi>EuclOp</mi><mover><mo>↪</mo><mphantom><mi>AAA</mi></mphantom></mover><mi>SmthMfd</mi></mrow><annotation encoding="application/x-tex"> CartSp \overset{\phantom{AAA}}{\hookrightarrow} EuclOp \overset{\phantom{AAA}}{\hookrightarrow} SmthMfd </annotation></semantics></math></div> <p>which regard, in turn, a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> (Def. <a class="maruku-ref" href="#CartSpCategory"></a>) as an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> of itself, and regard every <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> of <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> (Def. <a class="maruku-ref" href="#SiteOfSmoothManifolds"></a>) as a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> (<a href="differentiable+manifold#DifferentiableManifoldCartesianSpace">this Example</a>[this Example] and (differentiable+manifold#OpenSubsetsOfDifferentiableManifoldsAreDifferentiableManifolds)).</p> <p>Then the induced pre-composition functors induce <a class="existingWikiWord" href="/nlab/show/equivalences+of+categories">equivalences of categories</a> (<a href="geometry+of+physics+--+categories+and+toposes#EquivalenceOfCategories">this Def.</a>) between the corresponding <a class="existingWikiWord" href="/nlab/show/categories+of+sheaves">categories of sheaves</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Sh</mi><mo stretchy="false">(</mo><mi>EuclOp</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Sh</mi><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> SmoothSet \;\simeq\; Sh(CartSp) \;\simeq\; Sh(EuclOp) \;\simeq\; Sh(SmthMfd) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>By Prop. <a class="maruku-ref" href="#CompletingGoodOpenCoversToAllOpenCovers"></a> we may identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>CartSp</mi> <mtext>good open cov</mtext></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(CartSp) = Sh(CartSp_{\text{good open cov}})</annotation></semantics></math>. With that, both inclusions are evidently <a class="existingWikiWord" href="/nlab/show/dense+subsite">dense subsite</a>-inclusions (<a href="geometry+of+physics+--+categories+and+toposes#DenseSubsite">this Def.</a>). Therefore the statement follows by the <a class="existingWikiWord" href="/nlab/show/comparison+lemma">comparison lemma</a> (<a href="geometry+of+physics+--+categories+and+toposes#ComparisonLemma">this prop.</a>).</p> </div> <p>As a corollary we obtain:</p> <div class="num_prop" id="InclusionOfSmoothManifoldsIntoSmoothSets"> <h6 id="proposition_12">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a> inside <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>)</strong></p> <p>Regarding <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> as <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> via Example <a class="maruku-ref" href="#CartesianSpaceAsSmoothSpace"></a> yields a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a>-inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothManifold</mi><mover><mo>↪</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>ι</mi><mphantom><mi>AA</mi></mphantom></mrow></mover><mi>SmoothSet</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> SmoothManifold \overset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow} SmoothSet \,, </annotation></semantics></math></div> <p>meaning that 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><mo>∈</mo><mi>SmoothManifold</mi></mrow><annotation encoding="application/x-tex">X, Y \in SmoothManifold</annotation></semantics></math> any two <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>, this functor induces a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> between the <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <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> regarded in the sense of smooth manifolds, and regarded in the sense of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> (Def. <a class="maruku-ref" href="#HomomorphismOfSmoothSpaces"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>SmoothManifold</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><mi>ι</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ι</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{SmoothManifold}(X,Y) \;\simeq\; Hom_{SmoothSet}(\iota(X), \iota(Y)) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>By Prop. <a class="maruku-ref" href="#EquivalentSitesForCartSp"></a> we have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Sh</mi><mo stretchy="false">(</mo><mi>SmoothManifold</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> SmoothSet \;\simeq\; Sh(SmoothManifold) \,. </annotation></semantics></math></div> <p>With this the statement is given by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> (<a href="geometry+of+physics+--+categories+and+toposes#YonedaLemma">this prop.</a>).</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_prop" id="SmoothSetsFormACohesiveTopos"> <h6 id="proposition_13">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> form a <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/site">site</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> (Prop. <a class="maruku-ref" href="#TheDifferentiallyGoodOpenCoverCoverage"></a>) is a <a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a> (<a href="geometry+of+physics+--+categories+and+toposes#OneCohesiveSite">this Def.</a>), hence its <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a> is a <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> (by <a href="geometry+of+physics+--+categories+and+toposes#CategoriesOfSheavesOnCohesiveSiteIsCohesive">this Prop.</a>). Under the identification of Prop. <a class="maruku-ref" href="#SmoothSetsAreSheavesOnCartSp"></a>, this means that:</p> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/SmoothSet">SmoothSet</a> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> (Def. <a class="maruku-ref" href="#CategoryOfSmoothSets"></a>) is a <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> (<a href="geometry+of+physics+--+categories+and+toposes#CohesiveTopos">this Def.</a>):</p> <div class="maruku-equation" id="eq:SheafToposAdjointQuadruple"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mrow><mtable><mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AAA</mi></mphantom><msub><mi>Π</mi> <mn>0</mn></msub><mphantom><mi>AAA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>Disc</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AAA</mi></mphantom><mi>Γ</mi><mphantom><mi>AAA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>coDisc</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr></mtable></mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex"> SmoothSet \array{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\longleftarrow} } Set </annotation></semantics></math></div> <p>Moreover, this cohesive topos satisfies the following equivalent conditions (from <a href="geometry+of+physics+-+cohesive+toposes#PiecesHavePoints">this Prop.</a>):</p> <ol> <li> <p><em><a class="existingWikiWord" href="/nlab/show/pieces+have+points">pieces have points</a></em>,</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/discrete+objects+are+concrete">discrete objects are concrete</a></em>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math> clearly has <a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a>: The <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> is the <a class="existingWikiWord" href="/nlab/show/point">point</a>, given by the 0-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>=</mo><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex"> \ast = \mathbb{R}^0 </annotation></semantics></math></div> <p>and the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of two <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> is the Cartesian space whose <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> is the <a class="existingWikiWord" href="/nlab/show/sum">sum</a> of the two separate dimensions:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \;\simeq\; \mathbb{R}^{ n_1 + n_2 } \,. </annotation></semantics></math></div> <p>This establishes the first clause in the definition of <a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a> (<a href="geometry+of+physics+--+categories+and+toposes#OneCohesiveSite">this def.</a>)</p> <p>For the second clause, consider a differentiably-<a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow></mrow></mover><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \overset{}{\to} \mathbb{R}^n\}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#DifferentiallyGoodOpenCover"></a>). This being a <a class="existingWikiWord" href="/nlab/show/good+cover">good cover</a> implies that its <a class="existingWikiWord" href="/nlab/show/Cech+groupoid">Cech groupoid</a> is, as an <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> (via <a href="geometry+of+physics+--+categories+and+toposes#PresheavesOfGroupoidsAsInternalGroupoidsInPresheaves">this remark</a>), of the form</p> <div class="maruku-equation" id="eq:CechGroupoidForCartSp"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><munder><mo>∩</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C(\{U_i\}_i) \;\simeq\; \left( \array{ \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} y(U_i) } \right) \,. </annotation></semantics></math></div> <p>where we used the defining property of <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a> to identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>X</mi></msub><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo>∩</mo> <mi>X</mi></msub><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(U_i) \times_X y(U_j) \simeq y( U_i \cap_X U_j )</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <a class="maruku-eqref" href="#eq:CechGroupoidForCartSp">(4)</a>, regarded just as a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> of <a class="existingWikiWord" href="/nlab/show/reflexive+graph">reflexive</a> <a class="existingWikiWord" href="/nlab/show/directed+graph">directed</a> <a class="existingWikiWord" href="/nlab/show/graphs">graphs</a> (hence ignoring <a class="existingWikiWord" href="/nlab/show/composition">composition</a> for the moment), is readily seen to be the <a class="existingWikiWord" href="/nlab/show/graph">graph</a> of the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of the components (the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> follows immediately from that of the component colimits):</p> <div class="maruku-equation" id="eq:ColimitOfCechGroupoidOverCartSp"><span class="maruku-eq-number">(5)</span><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><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><munder><mo>∩</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><munder><mo>∩</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mo>*</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><mo>*</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \array{ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i) } \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} \ast \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \ast } \right) \end{aligned} \,. </annotation></semantics></math></div> <p>Here we first used that <a class="existingWikiWord" href="/nlab/show/colimits+commute+with+colimits">colimits commute with colimits</a>, hence in particular with <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> (<a href="geometry+of+physics+--+categories+and+toposes#LimitsCommuteWithLimits">this prop.</a>) and then that the colimit of a <a class="existingWikiWord" href="/nlab/show/representable+presheaf">representable presheaf</a> is <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a> set (<a href="geometry+of+physics+--+categories+and+toposes#ColimitOfRepresentableIsSingleton">this Lemma</a>).</p> <p>This colimiting <a class="existingWikiWord" href="/nlab/show/graph">graph</a> carries a unique <a class="existingWikiWord" href="/nlab/show/composition">composition</a> structure making it a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, since there is at most one morphism between any two objects, and every object carries a morphism from itself to itself. This implies that this groupoid is actually the colimiting groupoid of the Cech groupoid: hence the groupoid obtained from replacing each representable summand in the Cech groupoid by a point.</p> <p>Precisely this operation on <a class="existingWikiWord" href="/nlab/show/Cech+groupoids">Cech groupoids</a> of <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> is what <em><a class="existingWikiWord" href="/nlab/show/Borsuk%27s+nerve+theorem">Borsuk's nerve theorem</a></em> is about, a classical result in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>. This theorem implies directly that the set of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> of the groupoid <a class="maruku-eqref" href="#eq:ColimitOfCechGroupoidOverCartSp">(6)</a> is in <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> with the set of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>, regarded as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. But this is evidently a <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected topological space</a>, which finally shows that, indeed</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0 \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; \ast \,. </annotation></semantics></math></div> <p>The second item of the second clause in Def. <a class="maruku-ref" href="#OneCohesiveSite"></a> follows similarly, but more easily: The <a class="existingWikiWord" href="/nlab/show/limit">limit</a> of the <a class="existingWikiWord" href="/nlab/show/Cech+groupoid">Cech groupoid</a> is readily seen to be, as before, the unique groupoid structure on the limiting underlying graph of presheaves. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>=</mo><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\ast = \mathbb{R}^0</annotation></semantics></math>, which is hence an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CartSp^{op}</annotation></semantics></math>, limits over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CartSp^{op}</annotation></semantics></math> yield simply the evaluation on that object:</p> <div class="maruku-equation" id="eq:ColimitOfCechGroupoidOverCartSp"><span class="maruku-eq-number">(6)</span><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><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><munder><mo>∩</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>Hom</mi> <mi>CartSp</mi></msub><mrow><mo>(</mo><mo>*</mo><mo>,</mo><msub><mi>U</mi> <mi>i</mi></msub><munder><mo>∩</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><msub><mi>Hom</mi> <mi>CartSp</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \array{ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \phantom{A} \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} Hom_{CartSp}\left( \ast, U_i \underset{\mathbb{R}^n}{\cap} U_j \right) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} Hom_{CartSp}( \ast, U_i ) } \right) \end{aligned} \,. </annotation></semantics></math></div> <p>Here we used that <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> (here <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a>) of <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> are computed objectwise, and then the definition of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> <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> <p>But the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> induced by this graph on its set of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><msub><mi>Hom</mi> <mi>CartSp</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{i}{\coprod} Hom_{CartSp}( \ast, U_i )</annotation></semantics></math> precisely identifies pairs of points, one in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> the other in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_j</annotation></semantics></math>, that are actually the same point of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> being covered. Hence the set of <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> is the set of points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>, which is just what remained to be shown:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>CartSp</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0 \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; Hom_{CartSp}(\ast, \mathbb{R}^n) \,. </annotation></semantics></math></div> <p>Finally to see that <em><a class="existingWikiWord" href="/nlab/show/pieces+have+points">pieces have points</a></em> and <em><a class="existingWikiWord" href="/nlab/show/discrete+objects+are+concrete">discrete objects are concrete</a></em> is satisfied, it is sufficient to observe, by <a href="geometry+of+physics+-+cohesive+toposes#CohesiveSiteSuchThatDiscreteObjectsAreConcrete">this prop.</a> that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/hom+set">hom set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>CartSp</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{CartSp}(\ast, \mathbb{R}^n)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/inhabited+set">non-empty</a>.</p> </div> <p>The following statement is a <a class="existingWikiWord" href="/nlab/show/0-truncated+object">0-truncated</a> shadow of the <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometric</a> analog of the concept of <em><a class="existingWikiWord" href="/nlab/show/A1-homotopy+theory">A1-localization</a></em>:</p> <div class="num_prop" id="ShapeModalityOnSmoothSetsIsR1Localization"> <h6 id="proposition_14">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> on <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/reflective+localization">localization</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a>-inclusion from Prop. <a class="maruku-ref" href="#SmoothSetsFormACohesiveTopos"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Set</mi><munderover><mo>⊥</mo><munder><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><mi>Disc</mi><mphantom><mi>A</mi></mphantom></mrow></munder><mover><mo>⟵</mo><mi>Π</mi></mover></munderover><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex"> Set \underoverset {\underset{ \phantom{A}Disc\phantom{A} }{\hookrightarrow}} {\overset{\Pi}{\longleftarrow}} {\bot} SmoothSet </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> as the <a class="existingWikiWord" href="/nlab/show/discrete+objects">discrete</a> <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>, exhibits the <em><a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflection</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/local+objects">local objects</a></em> (<a href="geometry+of+physics+--+categories+and+toposes#LocalizationAtACollectionOfMorphisms">this Def.</a>) for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/real+line">real line</a>, in the sense of <a class="existingWikiWord" href="/nlab/show/localization+at+an+object">localization at an object</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>L</mi> <mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Pi \;\simeq\; L_{\mathbb{R}^1} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>Since we already know that the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</annotation></semantics></math> exists, we need to show that the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/discrete+object">discrete</a> <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mover><mo>↪</mo><mi>Disc</mi></mover><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">Set \overset{Disc}{\hookrightarrow} SmoothSet</annotation></semantics></math> is equivalently that of <a class="existingWikiWord" href="/nlab/show/local+objects">local objects</a> (<a href="geometry+of+physics+–+basic+notions+of+category+theory#LocalObjects">this Def.</a>) with respect to the class of morphisms 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>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \times \mathbb{R}^1 \overset{ p_1 }{\longrightarrow} X </annotation></semantics></math></div> <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> any <a class="existingWikiWord" href="/nlab/show/object">object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">SmoothSet</annotation></semantics></math>.</p> <p>We claim that a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</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> is a <a class="existingWikiWord" href="/nlab/show/local+object">local object</a> with respect to this class, already if it is a local object with respect to just the <a class="existingWikiWord" href="/nlab/show/small+set">small set</a></p> <div class="maruku-equation" id="eq:RnTimesR1Projections"><span class="maruku-eq-number">(7)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>}</mo></mrow> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \left\{ \mathbb{R}^n \times \mathbb{R}^1 \overset{p_1}{\longrightarrow} \mathbb{R}^n \right\}_{n \in \mathbb{N}} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X = \mathbb{R}^n</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>.</p> <p>To see this, we identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SmoothSet \simeq Sh(CartSp)</annotation></semantics></math> via Prop. <a class="maruku-ref" href="#EquivalentSitesForCartSp"></a>, and then use the <a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a> (<a href="geometry+of+physics+--+categories+and+toposes#TopologicalCoYonedaLemma">this Prop.</a>) to express any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SmoothSet</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>≃</mo><msup><mo>∫</mo> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∈</mo><mi>Cartsp</mi></mrow></msup><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>⋅</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \simeq \int^{\mathbb{R}^n \in Cartsp} \mathbb{R}^n \cdot X(\mathbb{R}^n)</annotation></semantics></math>.</p> <p>Assuming then that <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 local with respect to the <a class="existingWikiWord" href="/nlab/show/small+set">small set</a> <a class="maruku-eqref" href="#eq:RnTimesR1Projections">(7)</a> we obtain</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><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><mi>X</mi><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mo>≃</mo><msub><mo>∫</mo> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>∈</mo><mi>CartSp</mi></mrow></msub><msub><mi>Hom</mi> <mi>Set</mi></msub><mrow><mo>(</mo><mi>X</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><munder><munder><mrow><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>×</mo><msup><mi>ℝ</mi> <mn>1</mn></msup><mover><mo>→</mo><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover><msup><mi>ℝ</mi> <mi>n</mi></msup><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><mo>⏟</mo></munder><mtext>bijection</mtext></munder><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} Hom_{SmoothSet}\big( X \times \mathbb{R}^1 \overset{p_1}{\to} X \;\,,\; A \big) & \simeq \int_{\mathbb{R}^n \in CartSp} Hom_{Set}\left( X(\mathbb{R}^n), \, \underset{\text{bijection}}{ \underbrace{ Hom_{SmoothSet}\big( \mathbb{R}^n \times \mathbb{R}^1 \overset{p_1}{\to} \mathbb{R}^n \;\,,\; A \big) }} \right) \end{aligned} </annotation></semantics></math></div> <p>where we used that</p> <ol> <li> <p>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><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">(-) \times \mathbb{R}^1</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a>, since it is a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> (<a href="geometry+of+physics+--+categories+and+toposes#PropertiesOfSheafToposes">this Prop.</a>) and since <a class="existingWikiWord" href="/nlab/show/left+adjoints+preserve+colimits">left adjoints preserve colimits</a> (<a href="geometry+of+physics+--+categories+and+toposes#AdjointsPreserveCoLimits">this Prop.</a>),</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/hom-functor+preserves+limits">hom-functor preserves limits</a> (<a href="geometry+of+physics+--+categories+and+toposes#HomFunctorPreservesLimits">this Prop.</a>), hence sends colimits in the first argument to limits.</p> </li> </ol> <p>But on the right this is now a morphisms of <a class="existingWikiWord" href="/nlab/show/limits">limits</a> induced by morphism of <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> which is objectwise an <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>, as shown, and hence is a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>) itself.</p> <p>Hence we are reduced to showing that the <a class="existingWikiWord" href="/nlab/show/local+objects">local objects</a> with respect to <a class="maruku-eqref" href="#eq:RnTimesR1Projections">(7)</a> are the discrete smooth sets. But by <a class="existingWikiWord" href="/nlab/show/induction">induction</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, locality with respect to <a class="maruku-eqref" href="#eq:RnTimesR1Projections">(7)</a> means equivalently locality with respect to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mover><mo>→</mo><mrow><mo>∃</mo><mo>!</mo></mrow></mover><msup><mi>ℝ</mi> <mn>0</mn></msup><mo>=</mo><mo>*</mo><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ \mathbb{R}^n \overset{\exists !}{\to} \mathbb{R}^0 = \ast \right\} </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/local+objects">local objects</a> with respect to this set are manifestly exactly the <a class="existingWikiWord" href="/nlab/show/constant+presheaves">constant presheaves</a>. But by the proof of Prop. <a class="maruku-ref" href="#SmoothSetsFormACohesiveTopos"></a> (proceeding via the proof of <a href="geometry+of+physics+--+categories+and+toposes#CategoriesOfSheavesOnCohesiveSiteIsCohesive">this Prop.</a>) these are indeed exactly the objects in the image of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi><mover><mo>↪</mo><mi>Disc</mi></mover><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">Set \overset{Disc}{\hookrightarrow} SmoothSet</annotation></semantics></math>.</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="concrete_smooth_sets_diffeological_spaces">Concrete smooth sets: Diffeological spaces</h3> <p>The cohesiveness of smooth sets (Prop. <a class="maruku-ref" href="#SmoothSetsFormACohesiveTopos"></a>) implies in particular that there is a concept of <em><a class="existingWikiWord" href="/nlab/show/concrete+objects">concrete objects</a></em> among the smooth sets (<a href="geometry+of+physics+--+categories+and+toposes#CohesiveModalities">this Def.</a>). We show here (Prop. <a class="maruku-ref" href="#DiffeologicalSpacesAreTheConcreteSmoothSets"></a> below) that these concrete smooth sets are <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to a kind of <a class="existingWikiWord" href="/nlab/show/generalized+smooth+spaces">generalized smooth spaces</a> that are known as <em><a class="existingWikiWord" href="/nlab/show/Chen+smooth+spaces">Chen smooth spaces</a></em> (<a href="diffeological+space#Chen77">Chen 77</a>) or <em><a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a></em> (<a href="diffeological+space#Souriau79">Souriau 79</a>, <a href="diffeological+space#IglesiasZemmour85">Iglesias-Zemmour 85</a>), which we recall as Def. <a class="maruku-ref" href="#DiffeologicalSpace"></a> below.</p> <p>A comprehensive development of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> in terms of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> is spelled out in (<a href="diffeological+space#PIZ">Iglesias-Zemmour 13</a>).</p> <p>Since the <a class="existingWikiWord" href="/nlab/show/concrete+objects">concrete objects</a> in any <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a>, and hence the <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> among all <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>, form a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> (<a href="geometry+of+physics+--+categories+and+toposes#QuasitoposOfConcreteObjects">This prop.</a>), every smooth set has a <em><a class="existingWikiWord" href="/nlab/show/concretification">concretification</a></em> to a concrete smooth set, hence to a diffeological space. An important example of this construction are <a class="existingWikiWord" href="/nlab/show/moduli+spaces">moduli spaces</a> of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a>, this we turn to in Def. <a class="maruku-ref" href="#SmoothSpaceOfFormsOnSmoothSpace"></a> below.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_defn" id="DiffeologicalSpace"> <h6 id="definition_19">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a> (e.g. <a href="diffeological+space#IglesiasZemmour18">Iglesias Zemmour 18, def. 2 and 5</a>))</strong></p> <p>A <em><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math> is</p> <ol> <li> <p>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>X</mi><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">X \in Set</annotation></semantics></math>, called the <em>underlying set</em> of the diffeological space;</p> </li> <li> <p>for each <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><msup><mi>E</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U \subset E^n</annotation></semantics></math> of some <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">E^n</annotation></semantics></math>, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/subset">subset</a></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>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>⊂</mo><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{X}(U) \subset Hom_{Set}(U, X) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/function+set">function set</a> of <a class="existingWikiWord" href="/nlab/show/functions">functions</a> of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> to <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>, called the set of <em>plots</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>;</p> </li> </ol> <p>such that for every open subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> as above the following conditions hold:</p> <ol> <li> <p>the set of plots <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{X}(U)</annotation></semantics></math> contains all the <a class="existingWikiWord" href="/nlab/show/constant+functions">constant functions</a></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><msub><mi>const</mi> <mi>U</mi></msub></mrow></mover><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> X \overset{const_U}{\hookrightarrow} \mathbf{X}(U) \hookrightarrow Hom_{Set}(U,X) \,, </annotation></semantics></math></div></li> <li> <p>for every <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \;\colon\; U \to X</annotation></semantics></math> and for every <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow><msub><mi>ι</mi> <mi>i</mi></msub></mrow></mover><mi>U</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \overset{\iota_i}{\to} U\}</annotation></semantics></math>, if each <a class="existingWikiWord" href="/nlab/show/restriction">restriction</a> is a plot, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow></msub><mo>∈</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi\vert_{U_i} \in \mathbf{X}(U_i)</annotation></semantics></math>, then <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> itself is a plot: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi \in \mathbf{X}(U)</annotation></semantics></math>;</p> </li> <li> <p>for all plots <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi \in \mathbf{X}(U)</annotation></semantics></math>, all open subsets <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> of any <a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a>, and all <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mover><mo>→</mo><mi>f</mi></mover><mi>U</mi></mrow><annotation encoding="application/x-tex">V \overset{f}{\to} U</annotation></semantics></math>, we have, the composition is again a plot</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∘</mo><mi>f</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi \circ f \;\in\; \mathbf{X}(V) \,. </annotation></semantics></math></div></li> </ol> <p>Moreover, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Y</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Y}</annotation></semantics></math> two <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>, as above, then a <em>smooth map</em> between them</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>X</mi></mstyle><mover><mo>⟶</mo><mi>f</mi></mover><mstyle mathvariant="bold"><mi>Y</mi></mstyle></mrow><annotation encoding="application/x-tex"> \mathbf{X} \overset{f}{\longrightarrow} \mathbf{Y} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/function">function</a> of underlying sets</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><mphantom><mi>A</mi></mphantom><mi>f</mi><mphantom><mi>A</mi></mphantom></mrow></mover><mi>Y</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>Set</mi></mrow><annotation encoding="application/x-tex"> X \overset{\phantom{A} f \phantom{A}}{\longrightarrow} Y \;\;\in\;\; Set </annotation></semantics></math></div> <p>such that for each plot <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi \in \mathbf{X}(U)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> with that function is a plot of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Y</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Y}</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>f</mi><mo>∘</mo><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \circ \phi \;\in\; \mathbf{X}(U) \,. </annotation></semantics></math></div> <p>This defines a <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><mi>DiffeologicalSpace</mi></mrow><annotation encoding="application/x-tex">DiffeologicalSpace</annotation></semantics></math> (<a href="geometry+of+physics+--+categories+and+toposes#Categories">this def.</a>) whose <a class="existingWikiWord" href="/nlab/show/objects">objects</a> are the diffeological spaces, whose <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> are the smooth maps between them, with <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of morphisms the ordinary composition of functions of underlying sets.</p> </div> <div class="num_prop" id="DiffeologicalSpacesAreTheConcreteSmoothSets"> <h6 id="proposition_15">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> are the <a class="existingWikiWord" href="/nlab/show/concrete+object">concret</a> <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>)</strong></p> <p>The category of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> (Def. <a class="maruku-ref" href="#DiffeologicalSpace"></a>) is a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the category of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> (Def. <a class="maruku-ref" href="#CategoryOfSmoothSets"></a>).</p> <p>Moreover, in terms of the <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive</a> structure on the category of smooth sets from Prop. <a class="maruku-ref" href="#SmoothSetsFormACohesiveTopos"></a>, the diffeological spaces are precisely the <em><a class="existingWikiWord" href="/nlab/show/concrete+objects">concrete objects</a></em> (<a href="geometry+of+physics+--+categories+and+toposes#CohesiveModalities">this def.</a>) among the <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DiffeologicalSpace</mi><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>SmoothSet</mi> <mi>conc</mi></msub><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><mi>SmoothSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> DiffeologicalSpace \;\simeq\; SmoothSet_{conc} \overset{\phantom{AAAA}}{\hookrightarrow} SmoothSet \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>First observe that the assignment of sets of plots</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>↦</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> U \mapsto \mathbf{X}(U) </annotation></semantics></math></div> <p>of a diffeological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math>, according to Def. <a class="maruku-ref" href="#DiffeologicalSpace"></a> constitutes a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> (<a href="geometry+of+physics+--+categories+and+toposes#Sheaf">this Def.</a>) on the <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>EuclOp</mi></mrow><annotation encoding="application/x-tex">EuclOp</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> of <a class="existingWikiWord" href="/nlab/show/Euclidean+spaces">Euclidean spaces</a>, by the same unwinding of Definitions as in Prop. <a class="maruku-ref" href="#SmoothSetsAreSheavesOnCartSp"></a>:</p> <ul> <li> <p>the third clause in the list of properties in Def. <a class="maruku-ref" href="#DiffeologicalSpace"></a> says that the assignment of sets of plots is a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a>,</p> </li> <li> <p>the second clause in the list of properties says that this presheaf satisfies the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf condition</a>,</p> </li> <li> <p>while the first clause in the list of properties is an extra condition, singling out diffeological spaces among all sheaves.</p> </li> </ul> <p>Under this identification, the definition of a smooth map of diffeological spaces in Def. <a class="maruku-ref" href="#DiffeologicalSpace"></a> says that it is equivalently a morphism of presheaves of sets of plots between sheaves of sets of plots, and hence a morphism of sheaves. This establishe a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a>-inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DiffeologicalSpace</mi><mo>↪</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>EuclOp</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> DiffeologicalSpace \hookrightarrow Sh(EuclOp) \,. </annotation></semantics></math></div> <p>But by Prop. <a class="maruku-ref" href="#EquivalentSitesForCartSp"></a> the restriction from the site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>EuclOp</mi></mrow><annotation encoding="application/x-tex">EuclOp</annotation></semantics></math> of all open subsets of Euclidean spaces to that of just the site <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> between the corresponding <a class="existingWikiWord" href="/nlab/show/sheaf+toposes">sheaf toposes</a>. This yields the full subcategory inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DiffeologicalSpace</mi><mo>↪</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>EuclOp</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>SmoothSet</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> DiffeologicalSpace \hookrightarrow Sh(EuclOp) \simeq Sh(CartSp) \simeq SmoothSet \,, </annotation></semantics></math></div> <p>where the last equivalence is Prop. <a class="maruku-ref" href="#SmoothSetsAreSheavesOnCartSp"></a>.</p> <p>It remains to see that under this inclusion, the diffeological spaces are identified with the <a class="existingWikiWord" href="/nlab/show/concrete+objects">concrete objects</a> among the smooth set.</p> <p>By definition (<a href="geometry+of+physics+--+categories+and+toposes#CohesiveModalities">this Def.</a>), a smooth set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">\mathbf{X} \in SmoothSet</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete</a>, precisely if its <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp</a>-unit is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a></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>X</mi></mstyle><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><msubsup><mi>η</mi> <mi>X</mi> <mo>♯</mo></msubsup><mphantom><mi>A</mi></mphantom></mrow></mover><mo>♯</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{X} \overset{\phantom{A} \eta_X^\sharp \phantom{A}}{\hookrightarrow} \sharp \mathbf{X} \,, </annotation></semantics></math></div> <p>which is the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> (<a href="geometry+of+physics+--+categories+and+toposes#AdjunctionUnitFromHomIsomorphism">this Def.</a>) of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Γ</mi><mo>⊣</mo><mi>coDisc</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Gamma \dashv coDisc)</annotation></semantics></math>-adjunction</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>X</mi></mstyle><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><msub><mi>η</mi> <mi>X</mi></msub><mphantom><mi>A</mi></mphantom></mrow></mover><mi>coDisc</mi><mi>Γ</mi><mstyle mathvariant="bold"><mi>X</mi></mstyle><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{X} \overset{\phantom{A} \eta_X \phantom{A}}{\hookrightarrow} coDisc \Gamma \mathbf{X} \,. </annotation></semantics></math></div> <p>Now a morphism of sheaves is a monomorphism, precisely if for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">U \in CartSp</annotation></semantics></math> in the site, its component function</p> <div class="maruku-equation" id="eq:DecohesAsGammacoDiscUnit"><span class="maruku-eq-number">(8)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><msubsup><mi>η</mi> <mi>X</mi> <mo>♯</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow></mover><mo stretchy="false">(</mo><mi>coDisc</mi><mi>Γ</mi><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{X}(U) \overset{\phantom{A} \eta_X^\sharp(U) \phantom{A}}{\hookrightarrow} (coDisc \Gamma \mathbf{X})(U) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/injective+function">injective function</a> (<a href="geometry+of+physics+--+categories+and+toposes#RecognitionOfEpimorphisms">this Prop.</a>). Unde the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, this function may be re-identified as follows:</p> <div class="maruku-equation" id="eq:CodiscretePlotIdentidification"><span class="maruku-eq-number">(9)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>η</mi> <mi>X</mi> <mo>♯</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><msubsup><mi>η</mi> <mi>X</mi> <mo>♯</mo></msubsup><mo stretchy="false">)</mo></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mover><mrow><msubsup><mi>η</mi> <mi>X</mi> <mo>♯</mo></msubsup><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover></mpadded></msup></mtd> <mtd></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>coDisc</mi><mi>Γ</mi><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>coDisc</mi><mi>Γ</mi><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>Γ</mi><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Γ</mi><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>ϕ</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mover><mi>ϕ</mi><mo>˜</mo></mover><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{X}(U) &\simeq& Hom_{SmoothSet}(y(U), \mathbf{X}) \\ {}^{\mathllap{ \eta_X^\sharp(U) }}\big\downarrow && {}^{\mathllap{ Hom_{SmoothSet}(y(U), \eta_X^\sharp ) }}\big\downarrow &\searrow^{\mathrlap{ \widetilde{\eta^\sharp_X \circ (-)} }}& \\ (coDisc \Gamma \mathbf{X}(U)) &\simeq& Hom_{SmoothSet}( y(U), coDisc \Gamma \mathbf{X} ) &\simeq& Hom_{Set}( \Gamma y(U), \Gamma \mathbf{X} ) &\simeq& Hom_{Set}( U, X ) \\ && \phi &\mapsto& \widetilde \phi \,, } </annotation></semantics></math></div> <p>where we first used the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> (<a href="geometry+of+physics+--+categories+and+toposes#YonedaLemma">this Prop.</a>), then the adjunction isomorphism (<a href="geometry+of+physics+--+categories+and+toposes#eq:HomIsomorphismForAdjointFunctors">here</a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Γ</mi><mo>⊣</mo><mi>coDisc</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Gamma \dashv coDisc)</annotation></semantics></math>. In the final step we used that the cohesive structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">SmoothSet</annotation></semantics></math> comes from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math> being a <a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a> (Prop. <a class="maruku-ref" href="#SmoothSetsFormACohesiveTopos"></a>) and that in this case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is given by evaluation on the point (<a href="geometry+of+physics+--+categories+and+toposes#CohesiveGlobalSectionsGivenByPointEvaluation">here</a>), and we wrote</p> <div class="maruku-equation" id="eq:UnderlyingSetOfDiffeologicalSpaceAsSmoothSet"><span class="maruku-eq-number">(10)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>Γ</mi><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo>=</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X \;\coloneqq\; \Gamma \mathbf{X} = \mathbf{X}(\ast) </annotation></semantics></math></div> <p>for the set of points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math>. Notice that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math> is indeed a <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, then this set is indeed its underlying set, by the first clause in the list of conditions on a diffeological space in Def. <a class="maruku-ref" href="#DiffeologicalSpace"></a>.</p> <p>This shows that <a class="maruku-eqref" href="#eq:DecohesAsGammacoDiscUnit">(8)</a> being an injection means equivalently that we have an injection of the form</p> <div class="maruku-equation" id="eq:RearrangedDecohesAsGammacoDiscUnit"><span class="maruku-eq-number">(11)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><msubsup><mi>η</mi> <mi>X</mi> <mo>♯</mo></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mphantom><mi>A</mi></mphantom></mrow></mover><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{X}(U) \overset{\phantom{A} \eta_X^\sharp(U) \phantom{A}}{\hookrightarrow} Hom_{Set}(U, X) \,. </annotation></semantics></math></div> <p>Hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{X}(U)</annotation></semantics></math> is always a subset of the <a class="existingWikiWord" href="/nlab/show/function+set">function set</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> to the set <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>, as in the second clause in Def. <a class="maruku-ref" href="#DiffeologicalSpace"></a>.</p> <p>This shows that every concrete smooth set is a diffeological space. For the converse, it remains to check that if we start with a diffeological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{X}</annotation></semantics></math> with prescribed inclusion function</p> <div class="maruku-equation" id="eq:DiffeologicalInclusionFunction"><span class="maruku-eq-number">(12)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{X}(U) \hookrightarrow Hom_{Set}(U,X) </annotation></semantics></math></div> <p>then <a class="maruku-eqref" href="#eq:RearrangedDecohesAsGammacoDiscUnit">(11)</a> indeed reproduces this inclusion.</p> <p>To see this, first notice that, by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> (<a href="geometry+of+physics+--+categories+and+toposes#YonedaLemma">this prop.</a>) and the definition of smooth maps between diffeological spaces, the inclusion function <a class="maruku-eqref" href="#eq:DiffeologicalInclusionFunction">(12)</a> equals the component function of the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>SmoothSet</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">\Gamma \;\colon\; SmoothSet \to Set</annotation></semantics></math>, that acts by point evaluation:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mrow><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>Γ</mi><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Γ</mi><mstyle mathvariant="bold"><mi>X</mi></mstyle><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mi>f</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>Γ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \Gamma_{y(U),\mathbf{X}} \;\colon\; \array{ Hom_{SmoothSet}( y(U), \mathbf{X} ) &\hookrightarrow& Hom_{Set}( \Gamma y(U), \Gamma \mathbf{X} ) \\ f &\mapsto& \Gamma(f) } </annotation></semantics></math></div> <p>Hence, by <a class="maruku-eqref" href="#eq:DecohesAsGammacoDiscUnit">(8)</a>, we need to show that</p> <div class="maruku-equation" id="eq:PostcompositionWithEtaAsGamma"><span class="maruku-eq-number">(13)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mrow><mi>y</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>,</mo><mstyle mathvariant="bold"><mi>X</mi></mstyle></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mover><mrow><msub><mi>η</mi> <mstyle mathvariant="bold"><mi>X</mi></mstyle></msub><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mo>˜</mo></mover><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma_{y(U), \mathbf{X}} \;=\; \widetilde{\eta_{\mathbf{X}} \circ (-)} \,. </annotation></semantics></math></div> <p>But this holds as a general fact about <a class="existingWikiWord" href="/nlab/show/adjunctions">adjunctions</a> (a special case of <a href="geometry+of+physics+–+basic+notions+of+category+theory#ReExpressingMiddleFunctorInAdjointTriple">this Example</a>).</p> </div> <p>By <a href="geometry+of+physics+--+categories+and+toposes#QuasitoposOfConcreteObjects">this prop.</a> it follows that</p> <div class="num_prop" id="ReflectiveInclusionOfDiffeologicalSpacesinSmoothSets"> <h6 id="proposition_16">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflection</a> of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> in <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> (Def. <a class="maruku-ref" href="#DiffeologicalSpace"></a>) is “in between” the <a class="existingWikiWord" href="/nlab/show/category+of+sets">category of sets</a> (<a href="geometry+of+physics+--+categories+and+toposes#CategoryOfSets">this Example</a>) and the category of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> (Def. <a class="maruku-ref" href="#CategoryOfSmoothSets"></a>) as exhibited by the following system of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</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></mtr> <mtr><mtd><mphantom><mi>a</mi></mphantom></mtd></mtr> <mtr><mtd><mphantom><mi>A</mi></mphantom></mtd></mtr> <mtr><mtd><mi>Γ</mi><mspace width="thickmathspace"></mspace><mo>⊣</mo><mspace width="thickmathspace"></mspace><mi>coDisc</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>SmoothSet</mi><mrow><mtable><mtr><mtd><mphantom><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mn>0</mn></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mphantom></mtd></mtr> <mtr><mtd><mphantom><mover><mo>↩</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>Disc</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mphantom></mtd></mtr> <mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>conc</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>↩</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>ι</mi> <mi>conc</mi></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr></mtable></mrow><mi>DiffeologicalSpace</mi><mrow><mtable><mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><msub><mi>Π</mi> <mn>0</mn></msub><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>↩</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>Disc</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>Γ</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>↩</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>coDisc</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr></mtable></mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex"> \array{ \\ \phantom{a} \\ \phantom{A} \\ \Gamma \;\dashv\; coDisc } \;\;\colon\;\; SmoothSet \array{ \phantom{\overset{ \phantom{AA} \Pi_0 \phantom{AA} }{\longrightarrow}} \\ \phantom{\overset{ \phantom{AA} Disc \phantom{AA} }{\hookleftarrow}} \\ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } DiffeologicalSpace \array{ \overset{ \phantom{AA} \Pi_0 \phantom{AA} }{\longrightarrow} \\ \overset{ \phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AA}\Gamma \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}coDisc\phantom{AA}}{\hookleftarrow} } Set </annotation></semantics></math></div> <p>where on the left we have a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> with reflector being <em><a class="existingWikiWord" href="/nlab/show/concretification">concretification</a></em> (<a href="geometry+of+physics+--+categories+and+toposes#QuasitoposOfConcreteObjects">this prop.</a>), and on the right we have the <a class="existingWikiWord" href="/nlab/show/corestriction">co</a><a class="existingWikiWord" href="/nlab/show/restriction">restriction</a> of the <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a> of <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a> from <a class="maruku-eqref" href="#eq:SheafToposAdjointQuadruple">(3)</a>.</p> </div> <div class="num_prop" id="FrechetManifoldsFullyFaithfulInSmoothSets"> <h6 id="proposition_17">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifolds">Fréchet manifolds</a> <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a> among <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>)</strong></p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>FrechetMfd</mi></mrow><annotation encoding="application/x-tex">FrechetMfd</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifolds">Fréchet manifolds</a> and <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> between these, which generalizes smooth manifolds to possibly <a class="existingWikiWord" href="/nlab/show/infinite-dimensional+manifold">infinite-dimensional</a> smooth manifolds</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><mi>SmoothMfd</mi><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><mi>FrechetMfd</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CartSp \overset{\phantom{AAAA}}{\hookrightarrow} SmoothMfd \overset{\phantom{AAAA}}{\hookrightarrow} FrechetMfd \,. </annotation></semantics></math></div> <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 <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifolds">Fréchet manifolds</a>, write again <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X,Y)</annotation></semantics></math> for the set of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> between them. Then the same kind of construction as for <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>, sending a Fréchet manifold to the <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>↦</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^n \mapsto C^\infty(\mathbb{R}^n, X) </annotation></semantics></math></div> <p>defines a <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a> (<a href="geometry+of+physics+--+categories+and+toposes#FullyFaithfulFunctor">this Example</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>FrechetMfd</mi><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><mi>SmoothSet</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> FrechetMfd \overset{\phantom{AAAA}}{\hookrightarrow} SmoothSet \,, </annotation></semantics></math></div> <p>hence a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> inclusion.</p> </div> <div class="proof"> <h6 id="proof_11">Proof</h6> <p>The construction clearly factors through <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> (Def. <a class="maruku-ref" href="#DiffeologicalSpace"></a>), identified as a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> via Prop. <a class="maruku-ref" href="#ReflectiveInclusionOfDiffeologicalSpacesinSmoothSets"></a>.</p> <p>With this it is now sufficient to see that <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifolds">Fréchet manifolds</a> are fully faithful among <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>. This is due to (<a href="diffeological+space#Losik">Losik 94, theorem 3.1.1</a>),</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="DifferentialForms">Differential forms</h3> <p>We have seen above in <em><a href="#TheContinuumRealWorldLine">The continuum real line</a></em> that that <a class="existingWikiWord" href="/nlab/show/real+line">real line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math> is the basic <a class="existingWikiWord" href="/nlab/show/kinematics">kinematical structure</a> in the <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> of <a class="existingWikiWord" href="/nlab/show/physics">physics</a>. Notably the smooth <a class="existingWikiWord" href="/nlab/show/path+spaces">path spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>ℝ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\mathbb{R}, X]</annotation></semantics></math> from example <a class="maruku-ref" href="#SmoothPathSpace"></a> are to be thought of as the smooth spaces of <em>trajectories</em> (for instance of some <a class="existingWikiWord" href="/nlab/show/particle">particle</a>) in a <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</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>, hence of smooth maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbb{R} \to X</annotation></semantics></math>.</p> <p>But moreover, <em><a class="existingWikiWord" href="/nlab/show/dynamics">dynamics</a></em> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is encoded by <em><a class="existingWikiWord" href="/nlab/show/linear+functionals">functionals</a> on such trajectories</em>: by “<a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a>”. In the simplest case these are for instance homomorphisms of smooth spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo lspace="verythinmathspace">:</mo><mrow><mo>[</mo><mi>I</mi><mo>,</mo><mi>X</mi><mo>]</mo></mrow><mo>→</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> S \colon \left[I, X\right] \to \mathbb{R} \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>↪</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">I \hookrightarrow \mathbb{R}</annotation></semantics></math> is the standard unit <a class="existingWikiWord" href="/nlab/show/interval">interval</a>.</p> <p>Such <a class="existingWikiWord" href="/nlab/show/action+functionals">action functionals</a> we discuss in their own right in <em><a href="#VariationalCalculus">Variational calculus</a></em> below. Here we first examine in detail a fundamental property they all have: they are supposed to be <em><a class="existingWikiWord" href="/nlab/show/local+action+functional">local</a></em>.</p> <p>Foremost this means that the value associated to a trajectory is <em>built up incrementally</em> from small contributions associated to small sub-trajectories: if a trajectory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> is decomposed as a trajectory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\gamma_1</annotation></semantics></math> followed by a trajectory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>γ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\gamma_2</annotation></semantics></math>, then the action functional is additive</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>γ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>S</mi><mo stretchy="false">(</mo><msub><mi>γ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S(\gamma) = S(\gamma_1) + S(\gamma_2) \,. </annotation></semantics></math></div> <p>As one takes this property to the limit of iterative subdivision, one finds that action functionals are entirely determined by their value on <em><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal</a> displacements</em> along the worldline. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><mi>ℝ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma \colon \mathbb{R} \to X</annotation></semantics></math> denotes a path and “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>γ</mi><mo>˙</mo></mover><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\dot \gamma(x)</annotation></semantics></math>” denotes the corresponding “infinitesimal path” at worldline parameter <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>, then the value of the action functional on such an infinitesimal path is traditionally written as</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>d</mi></mstyle><mi>S</mi><mo stretchy="false">(</mo><mover><mi>γ</mi><mo>˙</mo></mover><msub><mo stretchy="false">)</mo> <mi>x</mi></msub><mo>∈</mo><mi>ℝ</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{d}S(\dot \gamma)_x \in \mathbb{R} \,, </annotation></semantics></math></div> <p>to be read as “the small change <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>S</mi></mrow><annotation encoding="application/x-tex">\mathbf{d}S</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> along the infinitesimal path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>γ</mi><mo>˙</mo></mover> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">\dot \gamma_x</annotation></semantics></math>”.</p> <p>This function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><mi>S</mi></mrow><annotation encoding="application/x-tex">\mathbf{d}S</annotation></semantics></math> that assigns numbers to infinitesimal paths is called a <em><a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a></em>. Etymologically this originates in the use of “form” as in <em><a class="existingWikiWord" href="/nlab/show/bilinear+form">bilinear form</a></em>: something that is evaluated. Here it is evaluated on <em>infinitesimal differences</em>, referred to as <em>differentials</em>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <div class="num_defn" id="Differential1FormsOnCartesianSpaces"> <h6 id="definition_20">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> a <strong><a class="existingWikiWord" href="/nlab/show/smooth+differential+1-form">smooth differential 1-form</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> on a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> is 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>-<a class="existingWikiWord" href="/nlab/show/tuple">tuple</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mrow><mo>(</mo><msub><mi>ω</mi> <mi>i</mi></msub><mo>∈</mo><mi>CartSp</mi><mrow><mo>(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>,</mo><mi>ℝ</mi><mo>)</mo></mrow><mo>)</mo></mrow> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex"> \left(\omega_i \in CartSp\left(\mathbb{R}^n,\mathbb{R}\right)\right)_{i = 1}^n </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>, which we think of equivalently as the <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> of a <a class="existingWikiWord" href="/nlab/show/formal+linear+combination">formal linear combination</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></munderover><msub><mi>f</mi> <mi>i</mi></msub><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex"> \omega = \sum_{i = 1}^n f_i \mathbf{d}x^i </annotation></semantics></math></div> <p>on 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><mo stretchy="false">{</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mn>1</mn></msup><mo>,</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mn>2</mn></msup><mo>,</mo><mi>⋯</mi><mo>,</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{\mathbf{d}x^1, \mathbf{d}x^2, \cdots, \mathbf{d}x^n\}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/cardinality">cardinality</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>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>CartSp</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>,</mo><mi>ℝ</mi><msup><mo stretchy="false">)</mo> <mrow><mo>×</mo><mi>k</mi></mrow></msup><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> \Omega^1(\mathbb{R}^k) \simeq CartSp(\mathbb{R}^k, \mathbb{R})^{\times k}\in Set </annotation></semantics></math></div> <p>for the set of smooth <a class="existingWikiWord" href="/nlab/show/differential+1-forms">differential 1-forms</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_16">Remark</h6> <p>We think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{d} x^i</annotation></semantics></math> as a measure for <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal</a> displacements along the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">x^i</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a> of a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>. This idea is made precise below in <em><a href="#1FormsAsSmoothFunctors">Differential 1-forms are smooth increnemental path measures</a></em>.</p> </div> <p>If we have a measure of infintesimal displacement on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> and a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mover><mi>n</mi><mo stretchy="false">˜</mo></mover></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">f \colon \mathbb{R}^{\tilde n} \to \mathbb{R}^n</annotation></semantics></math>, then this induces a measure for infinitesimal displacement on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mover><mi>n</mi><mo stretchy="false">˜</mo></mover></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{\tilde n}</annotation></semantics></math> by sending whatever happens there first with <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> and then applying the given measure there. This is captured by the following definition.</p> <div class="num_defn" id="PullbackOfDifferential1FormsOnCartesianSpaces"> <h6 id="definition_21">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mover><mi>k</mi><mo stretchy="false">˜</mo></mover></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\phi \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>, the <strong><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential 1-forms</a></strong> along <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> is the <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mo lspace="verythinmathspace">:</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mover><mi>k</mi><mo stretchy="false">˜</mo></mover></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi^* \colon \Omega^1(\mathbb{R}^{k}) \to \Omega^1(\mathbb{R}^{\tilde k}) </annotation></semantics></math></div> <p>between sets of differential 1-forms, def. <a class="maruku-ref" href="#Differential1FormsOnCartesianSpaces"></a>, which is defined on <a class="existingWikiWord" href="/nlab/show/basis">basis</a>-elements by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup><mo>≔</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow> <mover><mi>k</mi><mo stretchy="false">˜</mo></mover></munderover><mfrac><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>i</mi></msup></mrow><mrow><mo>∂</mo><msup><mover><mi>x</mi><mo stretchy="false">˜</mo></mover> <mi>j</mi></msup></mrow></mfrac><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mover><mi>x</mi><mo stretchy="false">˜</mo></mover> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex"> \phi^* \mathbf{d} x^i \coloneqq \sum_{j = 1}^{\tilde k} \frac{\partial \phi^i}{\partial \tilde x^j} \mathbf{d}\tilde x^j </annotation></semantics></math></div> <p>and then extended linearly by</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><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>ω</mi></mtd> <mtd><mo>=</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mrow><mo>(</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><msub><mi>ω</mi> <mi>i</mi></msub><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≔</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mrow><mo>(</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>ω</mi><mo>)</mo></mrow> <mi>i</mi></msub><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow> <mover><mi>k</mi><mo stretchy="false">˜</mo></mover></munderover><mfrac><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>i</mi></msup></mrow><mrow><mo>∂</mo><msup><mover><mi>x</mi><mo stretchy="false">˜</mo></mover> <mi>j</mi></msup></mrow></mfrac><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mover><mi>x</mi><mo stretchy="false">˜</mo></mover> <mi>j</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow> <mover><mi>k</mi><mo stretchy="false">˜</mo></mover></munderover><mo stretchy="false">(</mo><msub><mi>ω</mi> <mi>i</mi></msub><mo>∘</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>⋅</mo><mfrac><mrow><mo>∂</mo><msup><mi>ϕ</mi> <mi>i</mi></msup></mrow><mrow><mo>∂</mo><msup><mover><mi>x</mi><mo stretchy="false">˜</mo></mover> <mi>j</mi></msup></mrow></mfrac><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mover><mi>x</mi><mo stretchy="false">˜</mo></mover> <mi>j</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \phi^* \omega & = \phi^* \left( \sum_{i} \omega_i \mathbf{d}x^i \right) \\ & \coloneqq \sum_{i = 1}^k \left(\phi^* \omega\right)_i \sum_{j = 1}^{\tilde k} \frac{\partial \phi^i }{\partial \tilde x^j} \mathbf{d} \tilde x^j \\ & = \sum_{i = 1}^k \sum_{j = 1}^{\tilde k} (\omega_i \circ \phi) \cdot \frac{\partial \phi^i }{\partial \tilde x^j} \mathbf{d} \tilde x^j \end{aligned} \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_17">Remark</h6> <p>The term “pullback” in <em><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a></em> is not really related, certainly not historically, to the term <em><a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></em> in <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a>. One can relate the pullback of differential forms to categorical pullbacks, but this is not really essential here. The most immediate property that both concepts share is that they take a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> going in one direction to a map between structures over domain and codomain of that morphism which goes in the other direction, and in this sense one is “pulling back structure along a morphism” in both cases.</p> </div> <p>Even if in the above definition we speak only about the <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><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^1(\mathbb{R}^k)</annotation></semantics></math> of differential 1-forms, this set naturally carries further <a class="existingWikiWord" href="/nlab/show/structure">structure</a>.</p> <div class="num_defn" id="ModuleStructureOn1FormsOnRk"> <h6 id="definition_22">Definition</h6> <ol> <li> <p>The set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^1(\mathbb{R}^k)</annotation></semantics></math> is naturally an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> with addition given by componentwise addition</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><mi>ω</mi><mo>+</mo><mi>λ</mi></mtd> <mtd><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mi>ω</mi> <mi>i</mi></msub><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><msub><mi>λ</mi> <mi>j</mi></msub><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>j</mi></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><mo stretchy="false">(</mo><msub><mi>ω</mi> <mi>i</mi></msub><mo>+</mo><msub><mi>λ</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \omega + \lambda & = \sum_{i = 1}^k \omega_i \mathbf{d}x^i + \sum_{j = 1}^k \lambda_j \mathbf{d}x^j \\ & = \sum_{i = 1}^k(\omega_i + \lambda_i) \mathbf{d}x^i \end{aligned} \,, </annotation></semantics></math></div></li> <li> <p>The abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^1(\mathbb{R}^k)</annotation></semantics></math> is naturally equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/module">module</a> over the <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>CartSp</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}^k,\mathbb{R}) = CartSp(\mathbb{R}^k, \mathbb{R})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>, where the <a class="existingWikiWord" href="/nlab/show/action">action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo><mo>×</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}^k,\mathbb{R}) \times\Omega^1(\mathbb{R}^k) \to \Omega^1(\mathbb{R}^k)</annotation></semantics></math> is given by componentwise multiplication</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>ω</mi><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></munderover><mo stretchy="false">(</mo><mi>f</mi><mo>⋅</mo><msub><mi>ω</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \cdot \omega = \sum_{i = 1}^k( f \cdot \omega_i) \mathbf{d}x^i \,. </annotation></semantics></math></div></li> </ol> </div> <div class="num_defn"> <h6 id="remark_18">Remark</h6> <p>More abstractly, this just says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^1(\mathbb{R}^k)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/free+module">free module</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}^k)</annotation></semantics></math> on the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup><msubsup><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">\{\mathbf{d}x^i\}_{i = 1}^k</annotation></semantics></math>.</p> </div> <p>The following definition captures the idea that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{d} x^i</annotation></semantics></math> is a measure for displacement along the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">x^i</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{d}x^j</annotation></semantics></math> a measure for displacement along the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex">x^j</annotation></semantics></math> coordinate, then there should be a way te get a measure, to be called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{d}x^i \wedge \mathbf{d} x^j</annotation></semantics></math>, for infinitesimal <em>surfaces</em> (squares) in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex">x^i</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex">x^j</annotation></semantics></math>-plane. And this should keep track of the <a class="existingWikiWord" href="/nlab/show/orientation">orientation</a> of these squares, with</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>d</mi></mstyle><msup><mi>x</mi> <mi>j</mi></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>i</mi></msup><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mi>j</mi></msup></mrow><annotation encoding="application/x-tex"> \mathbf{d}x^j \wedge \mathbf{d}x^i = - \mathbf{d}x^i \wedge \mathbf{d} x^j </annotation></semantics></math></div> <p>being the same infinitesimal measure with orientation reversed.</p> <div class="num_defn" id="DifferentialnForms"> <h6 id="definition_23">Definition</h6> <p>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><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">k,n \in \mathbb{N}</annotation></semantics></math>, the <strong>smooth <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>≔</mo><msubsup><mo>∧</mo> <mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow> <mo>•</mo></msubsup><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet(\mathbb{R}^k) \coloneqq \wedge^\bullet_{C^\infty(\mathbb{R}^k)} \Omega^1(\mathbb{R}^k) </annotation></semantics></math></div> <p>over the <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}^k)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> of the <a class="existingWikiWord" href="/nlab/show/module">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^1(\mathbb{R}^k)</annotation></semantics></math> of smooth 1-forms, prop. <a class="maruku-ref" href="#ModuleStructureOn1FormsOnRk"></a>.</p> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^n(\mathbb{R}^k)</annotation></semantics></math> for the sub-module of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> and call its elements the <strong>smooth <a class="existingWikiWord" href="/nlab/show/differential+n-forms">differential n-forms</a></strong>.</p> </div> <div class="num_remark"> <h6 id="remark_19">Remark</h6> <p>Explicitly this means that a <a class="existingWikiWord" href="/nlab/show/differential+n-form">differential n-form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^n(\mathbb{R}^k)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/formal+linear+combination">formal linear combination</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}^k)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/basis">basis</a> elements of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbf{d} x^{i_1} \wedge \cdots \wedge \mathbf{d}x^{i_n}</annotation></semantics></math> for <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><msub><mi>i</mi> <mn>2</mn></msub><mo><</mo><mi>⋯</mi><mo><</mo><msub><mi>i</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">i_1 \lt i_2 \lt \cdots \lt i_n</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>ω</mi><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mn>1</mn><mo>≤</mo><msub><mi>i</mi> <mn>1</mn></msub><mo><</mo><msub><mi>i</mi> <mn>2</mn></msub><mo><</mo><mi>⋯</mi><mo><</mo><msub><mi>i</mi> <mi>n</mi></msub><mo><</mo><mi>k</mi></mrow></munder><msub><mi>ω</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>i</mi> <mi>n</mi></msub></mrow></msub><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega = \sum_{1 \leq i_1 \lt i_2 \lt \cdots \lt i_n \lt k} \omega_{i_1, \cdots, i_n} \mathbf{d}x^{i_1} \wedge \cdots \wedge \mathbf{d}x^{i_n} \,. </annotation></semantics></math></div></div> <div class="num_defn" id="PullbackOfDifferentialForms"> <h6 id="definition_24">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential 1-forms</a> of def. <a class="maruku-ref" href="#Differential1FormsOnCartesianSpaces"></a> extends as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(\mathbb{R}^k)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^n(-)</annotation></semantics></math>, given for a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mover><mi>k</mi><mo stretchy="false">˜</mo></mover></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">f \colon \mathbb{R}^{\tilde k} \to \mathbb{R}^k</annotation></semantics></math> on basis elements by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mrow><mo>(</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></msup><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><msup><mi>f</mi> <mo>*</mo></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msup><mo>∧</mo><mi>⋯</mi><mo>∧</mo><msup><mi>f</mi> <mo>*</mo></msup><mstyle mathvariant="bold"><mi>d</mi></mstyle><msup><mi>x</mi> <mrow><msub><mi>i</mi> <mi>n</mi></msub></mrow></msup><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^* \left( \mathbf{d}x^{i_1} \wedge \cdots \wedge \mathbf{d}x^{i_n} \right) = \left(f^* \mathbf{d}x^{i_1} \wedge \cdots \wedge f^* \mathbf{d}x^{i_n} \right) \,. </annotation></semantics></math></div></div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>So far we have defined differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form on abstract coordinate systems. Here we extend this definition to one of differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms on arbitrary <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>. We start by observing that the space of <em>all</em> differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms on cordinate systems themselves naturally is a smooth set.</p> <div class="num_prop" id="SmoothModuliSpaceOfnForms"> <h6 id="proposition_18">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/smooth+set">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a>)</strong></p> <p>The assignment of differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>↦</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^n(-) \colon \mathbb{R}^k \mapsto \Omega^n(\mathbb{R}^k) </annotation></semantics></math></div> <p>of def. <a class="maruku-ref" href="#DifferentialnForms"></a> together with the <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>-functions of def. <a class="maruku-ref" href="#PullbackOfDifferentialForms"></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><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></msup></mtd> <mtd><mo>↦</mo></mtd> <mtd><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></msup><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></msup></mtd> <mtd><mo>↦</mo></mtd> <mtd><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{R}^{k_1} &\mapsto & \Omega^n(\mathbb{R}^{k_1}) \\ \uparrow^{\mathrlap{f}} && \downarrow^{\mathrlap{f^*}} \\ \mathbb{R}^{k_2} &\mapsto& \Omega^n(\mathbb{R}^{k_2}) } </annotation></semantics></math></div> <p>constitutes a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> in the sense of def. <a class="maruku-ref" href="#SmoothSpace"></a>, which we denote by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>SmoothSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Omega}^n(-) \;\in\; SmoothSet \,. </annotation></semantics></math></div> <p>We call this the <strong>universal smooth <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a></strong> of differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms.</p> </div> <p>The reason for this terminology is that homomorphisms of smooth sets into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Omega^1</annotation></semantics></math> <em>modulate</em> differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms on their <a class="existingWikiWord" href="/nlab/show/domain">domain</a>, by prop. <a class="maruku-ref" href="#YonedaForSmoothSpaces"></a> (and hence by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, <a href="geometry+of+physics+--+categories+and+toposes#YonedaLemma">this prop</a>):</p> <div class="num_example"> <h6 id="example_9">Example</h6> <p>For the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> by example <a class="maruku-ref" href="#CartesianSpaceAsSmoothSpace"></a>, there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>,</mo><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^n(\mathbb{R}^k) \simeq Hom(\mathbb{R}^k, \Omega^1) </annotation></semantics></math></div> <p>between the set of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> according to def. <a class="maruku-ref" href="#Differential1FormsOnCartesianSpaces"></a> and the set of homomorphism of smooth set, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>→</mo><msup><mi>Ω</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k \to \Omega^1</annotation></semantics></math>, according to def. <a class="maruku-ref" href="#HomomorphismOfSmoothSpaces"></a>.</p> </div> <p>In view of this we have the following elegant definition of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms on an arbitrary smooth set.</p> <div class="num_defn" id="DifferentialnFormOnSmoothSpace"> <h6 id="definition_25">Definition</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>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SmoothSet</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>, def. <a class="maruku-ref" href="#SmoothSpace"></a>, a <strong><a class="existingWikiWord" href="/nlab/show/differential+n-form">differential n-form</a></strong> 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 a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</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><mi>ω</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega \colon X \to \Omega^n(-) \,. </annotation></semantics></math></div> <p>Accordingly we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>SmoothSet</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Omega^n(X) \coloneqq SmoothSet(X,\Omega^n) </annotation></semantics></math></div> <p>for the set of smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>We may unwind this definition to a very explicit description of differential forms on smooth sets. This we do in a moment in remark <a class="maruku-ref" href="#DifferentialFormOnSmoothSpaceAsSystemOfDiffFormsOnCoordinates"></a>.</p> <p>Notice that differential 0-forms are equivalently smooth <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-valued functions.</p> <div class="num_example" id="SpaceOf0FormsIsRealLine"> <h6 id="examples">Examples</h6> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>0</mn></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℝ</mi></mrow><annotation encoding="application/x-tex"> \Omega^0 \;\simeq\; \mathbb{R} </annotation></semantics></math></div></div> <div class="num_defn" id="PullbackOfDifferentialFormsOnSmoothSpaces"> <h6 id="definition_26">Definition</h6> <p>For <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>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>, def. <a class="maruku-ref" href="#HomomorphismOfSmoothSpaces"></a>, the <strong><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a></strong> along <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/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo lspace="verythinmathspace">:</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> f^* \colon \Omega^n(Y) \to \Omega^n(X) </annotation></semantics></math></div> <p>given by the <a class="existingWikiWord" href="/nlab/show/hom-functor">hom-functor</a> into the smooth set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^n</annotation></semantics></math> of def. <a class="maruku-ref" href="#SmoothModuliSpaceOfnForms"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mo>≔</mo><mi>Hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^* \coloneqq Hom(-, \Omega^n) \,. </annotation></semantics></math></div> <p>This means that it sends 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>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^n(Y)</annotation></semantics></math> which is modulated by a homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>→</mo><msup><mi>Ω</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">Y \to \Omega^n</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f^* \omega \in \Omega^n(X)</annotation></semantics></math> which is modulated by the <a class="existingWikiWord" href="/nlab/show/composition">composite</a> <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><mo>→</mo><msup><mi>Ω</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\to} Y \to \Omega^n</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="proposition_19">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>ℝ</mi> <mover><mi>k</mi><mo stretchy="false">˜</mo></mover></msup></mrow><annotation encoding="application/x-tex">X = \mathbb{R}^{\tilde k}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>=</mo><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">Y = \mathbb{R}^{k}</annotation></semantics></math> definition <a class="maruku-ref" href="#PullbackOfDifferentialFormsOnSmoothSpaces"></a> reproduces def. <a class="maruku-ref" href="#PullbackOfDifferentialForms"></a>.</p> </div> <div class="proof"> <h6 id="proof_12">Proof</h6> <p>Again by the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> (<a href="geometry+of+physics+--+categories+and+toposes#YonedaLemma">this prop.</a>)</p> </div> <div class="num_remark" id="DifferentialFormOnSmoothSpaceAsSystemOfDiffFormsOnCoordinates"> <h6 id="remark_20">Remark</h6> <p>Using def. <a class="maruku-ref" href="#PullbackOfDifferentialFormsOnSmoothSpaces"></a></p> <p>Unwinding def. <a class="maruku-ref" href="#DifferentialnFormOnSmoothSpace"></a> yields the following explicit description:</p> <p>a differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^n(X)</annotation></semantics></math> on a smooth set <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</p> <ol> <li> <p>for each way <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon \mathbb{R}^k \to X</annotation></semantics></math> of laying out a coordinate system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi^* \omega \in \Omega^n(\mathbb{R}^k) </annotation></semantics></math></div> <p>on the abstract coordinate system, as given by def. <a class="maruku-ref" href="#DifferentialnForms"></a>;</p> </li> <li> <p>for each abstract <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></msup><mo>→</mo><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></msup></mrow><annotation encoding="application/x-tex">f \colon \mathbb{R}^{k_2} \to \mathbb{R}^{k_1}</annotation></semantics></math> a corresponding compatibility condition between local differential forms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>1</mn></msub></mrow></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi_1 \colon \mathbb{R}^{k_1} \to X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mrow><msub><mi>k</mi> <mn>2</mn></msub></mrow></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi_2 \colon \mathbb{R}^{k_2} \to X</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><msup><mi>f</mi> <mo>*</mo></msup><msubsup><mi>ϕ</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>ω</mi><mo>=</mo><msubsup><mi>ϕ</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>ω</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f^* \phi_1^* \omega = \phi_2^* \omega \,. </annotation></semantics></math></div></li> </ol> <p>Hence a differential form on a smooth set is simply a collection of differential forms on all its coordinate systems such that these glue along all possible coordinate transformations.</p> </div> <p>The following adds further explanation to the role of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo>∈</mo><mi>Smooth</mi><mn>0</mn><mi>Tye</mi></mrow><annotation encoding="application/x-tex">\Omega^n \in Smooth0Tye</annotation></semantics></math> as a <em><a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a></em>. Notice that since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^n</annotation></semantics></math> is itself a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>, we may speak about differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^n</annotation></semantics></math> itsefl.</p> <div class="num_defn" id="UniversalDifferentialnForm"> <h6 id="definition_27">Definition</h6> <p>The <strong>universal differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form</strong> is the differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>ω</mi> <mi>univ</mi> <mi>n</mi></msubsup><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega^n_{univ} \in \Omega^n(\Omega^n) </annotation></semantics></math></div> <p>which is modulated by the <a class="existingWikiWord" href="/nlab/show/identity">identity</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>id</mi><mo lspace="verythinmathspace">:</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>Ω</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">id \colon \Omega^n \to \Omega^n</annotation></semantics></math>.</p> </div> <p>With this definition we have:</p> <div class="num_prop"> <h6 id="proposition_20">Proposition</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>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SmoothSet</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>, every differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form 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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^n(X)</annotation></semantics></math> is the pullback of differential forms, def. <a class="maruku-ref" href="#PullbackOfDifferentialFormsOnSmoothSpaces"></a>, of the universal differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form, def. <a class="maruku-ref" href="#UniversalDifferentialnForm"></a>, along a homomorphism <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> 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> into the moduli space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^n</annotation></semantics></math> of differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><msup><mi>f</mi> <mo>*</mo></msup><msubsup><mi>ω</mi> <mi>univ</mi> <mi>n</mi></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega = f^* \omega^n_{univ} \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_21">Remark</h6> <p>This statement is of course in a way a big tautology. Nevertheless it is a very useful tautology to make explicit. The whole concept of differential forms on smooth sets here may be thought of as simply a variation of the theme of the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> (<a href="geometry+of+physics+--+categories+and+toposes#YonedaLemma">this prop.</a>).</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <p>We discuss the smooth space of differential forms <em>on a fixed smooth space</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <div class="num_remark"> <h6 id="remark_22">Remark</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/smooth+space">smooth space</a>, the smooth mapping space <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><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">]</mo><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">[X, \Omega^n] \in SmoothSet</annotation></semantics></math> is the smooth space whose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math>-plots are differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms on the <a class="existingWikiWord" href="/nlab/show/product">product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">X \times \mathbb{R}^k</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">]</mo><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>↦</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X, \Omega^n] \colon \mathbb{R}^k \mapsto \Omega^n(X \times \mathbb{R}^k) \,. </annotation></semantics></math></div> <p>This is not <em>quite</em> what one usually wants to regard as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math>-parameterized of differential forms 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>. That is instead usually meant to be a differential form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">X \times \mathbb{R}^k</annotation></semantics></math> which has “no leg along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math>”. Another way to say this is that the family of forms 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> that is represented by some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">X \times \mathbb{R}^k</annotation></semantics></math> is that which over a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo lspace="verythinmathspace">:</mo><mo>*</mo><mo>→</mo><msup><mi>ℝℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">v \colon * \to \mathbb{RR}^k</annotation></semantics></math> has the value <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><mi>v</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi></mrow><annotation encoding="application/x-tex">(id_X,v)^* \omega</annotation></semantics></math>. Under this <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a> any components of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> with “legs along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math>” are identified with the 0 differential form</p> </div> <p>This is captured by the following definition.</p> <div class="num_defn" id="SmoothSpaceOfFormsOnSmoothSpace"> <h6 id="definition_28">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/concrete+object">concrete</a> <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> on a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>)</strong></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>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X \in SmoothSet</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, the <em>smooth space of differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^n(X)</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> is the <a class="existingWikiWord" href="/nlab/show/concretification">concretification</a> (Prop. <a class="maruku-ref" href="#ReflectiveInclusionOfDiffeologicalSpacesinSmoothSets"></a>) of the smooth mapping space <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><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X, \Omega^n]</annotation></semantics></math>, def. <a class="maruku-ref" href="#SmoothFunctionSpace"></a>, into the smooth moduli space of differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms, def. <a class="maruku-ref" href="#SmoothModuliSpaceOfnForms"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>conc</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>Conc</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{\Omega}^n(X)_{conc} \; \coloneqq \; Conc([X, \Omega^n]) \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_21">Proposition</h6> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math>-plots of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^n(\mathbb{R}^k)</annotation></semantics></math> are indeed smooth differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">X \times \mathbb{R}^k</annotation></semantics></math> which are such that their evaluation on vector fields tangent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> vanish.</p> </div> <div class="proof"> <h6 id="proof_sketch">Proof (sketch)</h6> <p>By the proof of <a href="geometry+of+physics+--+categories+and+toposes#QuasitoposOfConcreteObjects">this Prop.</a> spring the set of plots of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^n(X)</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/image">image</a> of the <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>Γ</mi> <mrow><msup><mi>ℝ</mi> <mi>k</mi></msup><mo>,</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">]</mo></mrow></msub></mrow></mover><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mi>Γ</mi><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><msubsup><mi>ℝ</mi> <mi>s</mi> <mi>k</mi></msubsup><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Omega^n(X \times \mathbb{R}^k) \simeq Hom_{SmoothSet}(\mathbb{R}^k, [X,\Omega^n]) \stackrel{\Gamma_{ \mathbb{R}^k, [X,\Omega^n] }}{\to} Hom_{Set}(\Gamma(\mathbb{R}^k), \Gamma [X, \Omega^n]) \simeq Hom_{Set}(\mathbb{R}^k_s, \Omega^n(X)) \,, </annotation></semantics></math></div> <p>where on the right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℝ</mi> <mi>s</mi> <mi>k</mi></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k_s</annotation></semantics></math> denotes, just for emphasis, the underlying set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math>. This function manifestly sends a smooth differential form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega \in \Omega^n(X \times \mathbb{R}^k)</annotation></semantics></math> to the function from points <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> to differential forms 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> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>↦</mo><mrow><mo>(</mo><mi>v</mi><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>X</mi></msub><mo>,</mo><mi>v</mi><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega \mapsto \left(v \mapsto (id_X, v)^* \omega \right) \,. </annotation></semantics></math></div> <p>Under this function all components of differential forms with a “leg along” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math> are sent to the 0-form. Hence the image of this function is the collection of smooth forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">X \times \mathbb{R}^k</annotation></semantics></math> with “no leg along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^k</annotation></semantics></math>”.</p> </div> <div class="num_remark"> <h6 id="remark_23">Remark</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> we have (for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">X\in SmoothSet</annotation></semantics></math>)</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><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mn>0</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mi>Conc</mi><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><msup><mi>Ω</mi> <mn>0</mn></msup><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Conc</mi><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \mathbf{\Omega}^0(X) & \coloneqq Conc [X, \Omega^0] \\ & \simeq Conc [X, \mathbb{R}] \\ & \simeq [X, \mathbb{R}] \end{aligned} \,, </annotation></semantics></math></div> <p>by prop. <a class="maruku-ref" href="#SpaceOf0FormsIsRealLine"></a>.</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="integration_and_transgression">Integration and transgression</h3> <p>The traditional concept of <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> over a <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> applies in smooth families of differential forms and hence constitutes in fact a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> from the <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> on the given manifold, this is Def. <a class="maruku-ref" href="#ParameterizedIntegrationOfDifferentialForms"></a> below.</p> <p>Using this, <a class="existingWikiWord" href="/nlab/show/transgression+of+differential+forms">transgression of differential forms</a> may be defined as the application of the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a>-<a class="existingWikiWord" href="/nlab/show/functor">functor</a> out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/modulating+morphisms">modulating morphisms</a> of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a> and applying <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> to the result (Def. <a class="maruku-ref" href="#TransgressionOfDifferentialFormsToMappingSpaces"></a> below). This simple construction turns out to be equivalent to the traditional definition (Prop. <a class="maruku-ref" href="#EquivalenceOfTransgressionOfDifferentialFormsToMappingSpaces"></a> below).</p> <div class="num_defn" id="ParameterizedIntegrationOfDifferentialForms"> <h6 id="definition_29">Definition</h6> <p><strong>(parameterized <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a>)</strong></p> <p>Let</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, regarded as a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> via Example <a class="maruku-ref" href="#CartesianSpaceAsSmoothSpace"></a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \geq k \in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>;</p> </li> </ol> <p>Consider the <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth</a> <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mi>k</mi></msub><mo>,</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma_k, \mathbf{\Omega}^n]</annotation></semantics></math> (Def. <a class="maruku-ref" href="#SmoothFunctionSpace"></a>) out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_k</annotation></semantics></math> into the universal <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^n</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/differential+n-forms">differential n-forms</a> (Prop. <a class="maruku-ref" href="#SmoothModuliSpaceOfnForms"></a>).</p> <p>Then we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>Σ</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mi>k</mi></msub><mo>,</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">]</mo><mo>⟶</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mrow><annotation encoding="application/x-tex"> \int_{\Sigma} \;\colon\; [\Sigma_k, \mathbf{\Omega}^n] \longrightarrow \mathbf{\Omega}^{n-k} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> (Def. <a class="maruku-ref" href="#HomomorphismOfSmoothSpaces"></a>) which takes a plot <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>→</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>k</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\omega_{(-)} \colon U \to [\Sigma, \mathbf{\Omega}^k]</annotation></semantics></math>, hence equivalently a differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega_{(-)}(-)</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">U \times \Sigma</annotation></semantics></math>, to the result of <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mi>ω</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≔</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mi>ω</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \int_{\Sigma} \omega_{(-)}(-) \coloneqq \int_\Sigma \omega_{(-)} \,. </annotation></semantics></math></div></div> <div class="num_defn" id="TransgressionOfDifferentialFormsToMappingSpaces"> <h6 id="definition_30">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/transgression+of+differential+forms">transgression of differential forms</a> to <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>)</strong></p> <p>Let</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, regarded as a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> via Example <a class="maruku-ref" href="#CartesianSpaceAsSmoothSpace"></a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \geq k \in \mathbb{N}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a>;</p> </li> </ol> <p>Consider the <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth</a> <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mi>k</mi></msub><mo>,</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma_k, \mathbf{\Omega}^n]</annotation></semantics></math> (Def. <a class="maruku-ref" href="#SmoothFunctionSpace"></a>) out of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_k</annotation></semantics></math> into the universal <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth</a> <a class="existingWikiWord" href="/nlab/show/moduli+space">moduli space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbf{\Omega}^n</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/differential+n-forms">differential n-forms</a> (Prop. <a class="maruku-ref" href="#SmoothModuliSpaceOfnForms"></a>).</p> <p>Then the operation of <em><a class="existingWikiWord" href="/nlab/show/transgression+of+differential+n-forms">transgression of differential n-forms</a></em> 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> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_k</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>Σ</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tau_\Sigma \;\coloneqq\; \int_\Sigma [\Sigma,-] \;\colon\; \Omega^n(X) \to \Omega^{n-k}([\Sigma,X]) </annotation></semantics></math></div> <p>from <a class="existingWikiWord" href="/nlab/show/differential+n-forms">differential n-forms</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n-k</annotation></semantics></math>-forms on the <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth</a> <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>Σ</mi> <mi>k</mi></msub><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma_k,X]</annotation></semantics></math> spring which takes the differential form corresponding to the smooth function</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><mover><mo>→</mo><mi>ω</mi></mover><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (X \stackrel{\omega}{\to} \Omega^n) \in \Omega^n(X) </annotation></semantics></math></div> <p>to the differential form corresponding to the following composite smooth function:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>Σ</mi></msub><mi>ω</mi><mo>≔</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">]</mo></mrow></mover><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">]</mo><mover><mo>⟶</mo><mrow><msub><mo>∫</mo> <mi>Σ</mi></msub></mrow></mover><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \tau_\Sigma \omega \coloneqq \int_{\Sigma} [\Sigma,\omega] \;\colon\; [\Sigma, X] \stackrel{[\Sigma, \omega]}{\longrightarrow} [\Sigma, \Omega^n] \stackrel{\int_{\Sigma}}{\longrightarrow} \Omega^{n-k} \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,\omega]</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> on <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">\int_{\Sigma}</annotation></semantics></math> is the parameterized integration of differential forms from def. <a class="maruku-ref" href="#ParameterizedIntegrationOfDifferentialForms"></a>.</p> <p>More explicitly in terms of plots this means equivalently the following</p> <p>A plot of the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>→</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \phi_{(-)} \;\colon\; U \to [\Sigma, X] </annotation></semantics></math></div> <p>is equivalently a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</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><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>U</mi><mo>×</mo><mi>Σ</mi><mo>→</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi_{(-)}(-) \;\colon\; U \times \Sigma \to X \,. </annotation></semantics></math></div> <p>The smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,\omega]</annotation></semantics></math> takes this smooth function to the plot</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>×</mo><mi>Σ</mi><mo>→</mo><mi>X</mi><mover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>X</mi><mover><mo>⟶</mo><mi>ω</mi></mover><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> U \times \Sigma \to X \overset{\phi_{(-)}(-)}{\longrightarrow} X \overset{\omega}{\longrightarrow} \mathbf{\Omega}^{n} </annotation></semantics></math></div> <p>which is equivalently a differential form</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>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><mi>Σ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\phi_{(-)}(-))^\ast \omega \in \Omega^n(U \times \Sigma) \,. </annotation></semantics></math></div> <p>Finally the smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∫</mo> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">\int_\Sigma</annotation></semantics></math> takes this to the result of <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>Σ</mi></msub><mi>ω</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tau_{\Sigma}\omega\vert_{\phi_{(-)}} \;=\; \int_\Sigma (\phi_{(-)}(-))^\ast \omega \;\in\; \Omega^{n-k}(U) \,. </annotation></semantics></math></div></div> <div class="num_defn" id="TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap"> <h6 id="definition_31">Definition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/transgression+of+differential+forms">transgression of differential forms</a> to <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> via <a class="existingWikiWord" href="/nlab/show/evaluation+map">evaluation map</a>)</strong></p> <p>Let</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \geq k \in \mathbb{N}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </li> </ol> <p>Then the operation of <em>transgression of differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms</em> 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> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/function">function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>Σ</mi></msub><mo>≔</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><msup><mi>ev</mi> <mo>*</mo></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>ev</mi> <mo>*</mo></msup></mrow></mover><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>Σ</mi><mo>×</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mo>∫</mo> <mi>Σ</mi></msub></mrow></mover><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tau_\Sigma \coloneqq \int_\Sigma ev^\ast \;\colon\; \Omega^n(X) \overset{ev^\ast}{\longrightarrow} \Omega^n(\Sigma \times [\Sigma, X]) \overset{\int_\Sigma}{\longrightarrow} \Omega^{n-k}([\Sigma,X]) </annotation></semantics></math></div> <p>from differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-forms 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> to differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">n-k</annotation></semantics></math>-forms on the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math> (Def. <a class="maruku-ref" href="#SmoothFunctionSpace"></a>) which is the <a class="existingWikiWord" href="/nlab/show/composition">composite</a> of forming the <a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a> along 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><mi>ev</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">ev \colon [\Sigma, X] \times \Sigma \to X</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>.</p> <p>This construction manifestly extends to the <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a> of <a class="existingWikiWord" href="/nlab/show/differential+forms">differential forms</a></p> </div> <div class="num_prop" id="EquivalenceOfTransgressionOfDifferentialFormsToMappingSpaces"> <h6 id="proposition_22">Proposition</h6> <p>The two definitions of <a class="existingWikiWord" href="/nlab/show/transgression+of+differential+forms">transgression of differential forms</a> to mapping spaces from def. <a class="maruku-ref" href="#TransgressionOfDifferentialFormsToMappingSpaces"></a> and def. <a class="maruku-ref" href="#TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap"></a> are equivalent.</p> </div> <div class="proof"> <h6 id="proof_13">Proof</h6> <p>We need to check that for all plots <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>→</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\gamma \colon U \to [\Sigma, X]</annotation></semantics></math> the pullbacks of the two forms to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> coincide.</p> <p>For def. <a class="maruku-ref" href="#TransgressionOfDifferentialFormsToMappingSpacesViaEvaluationMap"></a> we get</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>γ</mi> <mo>*</mo></msup><msub><mo>∫</mo> <mi>Σ</mi></msub><msup><mi mathvariant="normal">ev</mi> <mo>*</mo></msup><mi>A</mi><mo>=</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>γ</mi><mo>,</mo><msub><mi mathvariant="normal">id</mi> <mi>Σ</mi></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><msup><mi mathvariant="normal">ev</mi> <mo>*</mo></msup><mi>A</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \gamma^\ast \int_\Sigma \mathrm{ev}^\ast A = \int_\Sigma (\gamma,\mathrm{id}_\Sigma)^\ast \mathrm{ev}^\ast A \; \in \Omega^n(U) </annotation></semantics></math></div> <p>Here we recognize in the integrand the pullback along the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>×</mo><mi>Σ</mi><mo>⊣</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">( (-)\times \Sigma \dashv [\Sigma,-])</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>γ</mi><mo stretchy="false">˜</mo></mover><mo>:</mo><mi>U</mi><mo>×</mo><mi>Σ</mi><mo>→</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\tilde \gamma : U \times \Sigma \to \Sigma</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math>, which is given by applying the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <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><mo>×</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">(-)\times \Sigma</annotation></semantics></math> and then postcomposing with the adjunction counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">ev</mi></mrow><annotation encoding="application/x-tex">\mathrm{ev}</annotation></semantics></math>:</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>U</mi><mo>×</mo><mi>Σ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>γ</mi><mo>,</mo><msub><mi mathvariant="normal">id</mi> <mi>Σ</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>×</mo><mi>Σ</mi></mtd> <mtd><mover><mo>⟶</mo><mi mathvariant="normal">ev</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ U \times \Sigma & \overset{(\gamma, \mathrm{id}_\Sigma)}{\longrightarrow} & [\Sigma,X] \times \Sigma & \overset{\mathrm{ev}}{\longrightarrow} & X } \,. </annotation></semantics></math></div> <p>Hence the integral is now</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>=</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><msup><mover><mi>γ</mi><mo stretchy="false">˜</mo></mover> <mo>*</mo></msup><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots = \int_{\Sigma} \tilde \gamma^\ast A \,. </annotation></semantics></math></div> <p>This is the operation of the top horizontal composite in the following <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality square</a> for <a class="existingWikiWord" href="/nlab/show/adjuncts">adjuncts</a>, and so the claim follows by its <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commutativity</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><mover><mi>γ</mi><mo stretchy="false">˜</mo></mover><mo>∈</mo></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><mi>Σ</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>U</mi><mo>×</mo><mi>Σ</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mi>Ω</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><mi>γ</mi><mo>∈</mo></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msup><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>n</mi></msup><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \tilde \gamma \in & Hom(U \times\Sigma, X) & \overset{Hom(U \times \Sigma,A)}{\longrightarrow} & Hom(U \times \Sigma, \mathbf{\Omega}^{n+k}) & \overset{\int_\Sigma(U)}{\longrightarrow} & \Omega^n(U) \\ & {}^{\mathllap{\simeq}}\big\downarrow && {}^{\mathllap{\simeq}}\big\downarrow && {}^{\mathllap{\simeq}}\big\downarrow \\ \gamma \in & Hom(U,[\Sigma,X]) & \overset{Hom(U,[\Sigma,A])}{\longrightarrow} & Hom(U,[\Sigma,\mathbf{\Omega}^{n+k}]) & \overset{Hom(U,\int_\Sigma)}{\longrightarrow} & Hom(U,\mathbf{\Omega}^n) } </annotation></semantics></math></div> <p>(here we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>Hom</mi> <mi>SmoothSet</mi></msub><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">Hom(-,-) \coloneqq Hom_{SmoothSet}(-,-)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>).</p> </div> <div class="num_prop"> <h6 id="proposition_23">Proposition</h6> <p><strong>(relative transgression over <a class="existingWikiWord" href="/nlab/show/manifolds+with+boundary">manifolds with boundary</a>)</strong></p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/manifold+with+boundary">boundary</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\partial \Sigma</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>k</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \geq k \in \mathbb{N}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><msubsup><mi>Ω</mi> <mi>X</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\omega \in \Omega^n_{X}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/closed+differential+form">closed differential form</a>.</p> </li> </ol> <p>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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mo>∂</mo><mi>Σ</mi><mo>↪</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⟶</mo><mo stretchy="false">[</mo><mo>∂</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> (-)\vert_{\partial \Sigma} \;\coloneqq\; [\partial \Sigma \hookrightarrow \Sigma, X] \;\colon\; [\Sigma, X] \longrightarrow [\partial \Sigma, X] </annotation></semantics></math></div> <p>for the smooth function that restricts smooth functions 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> to smooth functions on the <a class="existingWikiWord" href="/nlab/show/boundary">boundary</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\partial \Sigma</annotation></semantics></math>.</p> <p>Then the operations of transgression of differential forms (def. <a class="maruku-ref" href="#TransgressionOfDifferentialFormsToMappingSpaces"></a>) to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> and to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\partial \Sigma</annotation></semantics></math>, respectively, are related by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mrow><mo>(</mo><msub><mi>τ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><msub><mi>τ</mi> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mphantom><mi>AAAAAAAA</mi></mphantom><mrow><mtable><mtr><mtd><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>τ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo stretchy="false">|</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>d</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mo>∂</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>τ</mi> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d \left( \tau_{\Sigma}(\omega) \right) = (-1)^{k+1} ((-)\vert_{\partial \Sigma})^\ast \tau_{\partial \Sigma}(\omega) \phantom{AAAAAAAA} \array{ [\Sigma, X] &\overset{ \tau_{\Sigma}(\omega) }{\longrightarrow}& \mathbf{\Omega}^{n-k} \\ {}^{\mathllap{(-)\vert_{\partial \Sigma} }}\downarrow && \downarrow^{\mathrlap{ (-1)^{k+1} d}} \\ [\partial \Sigma, X] &\underset{ \tau_{\partial\Sigma}(\omega) }{\longrightarrow}& \mathbf{\Omega}^{n-k+1} } \,. </annotation></semantics></math></div> <p>In particular this means that if the compact manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> happens to have no boundary (is a <a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a>) then transgression over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> takes closed differential forms to closed differential forms.</p> </div> <div class="proof"> <h6 id="proof_14">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>×</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi_{(-)}(-) \colon U \times \Sigma \to X</annotation></semantics></math> be a plot of the mapping space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma, X]</annotation></semantics></math>. Notice that the <a class="existingWikiWord" href="/nlab/show/de+Rham+differential">de Rham differential</a> on 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><mi>U</mi><mo>×</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">U \times \Sigma</annotation></semantics></math> decomposes as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><msub><mi>d</mi> <mi>U</mi></msub><mo>+</mo><msub><mi>d</mi> <mi>Σ</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d = d_U + d_\Sigma \,. </annotation></semantics></math></div> <p>Now we compute 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 displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>d</mi><msub><mi>τ</mi> <mi>Σ</mi></msub><mi>ω</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>ϕ</mi> <mo stretchy="false">(</mo></msub><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mtd> <mtd><mo>=</mo><msub><mi>d</mi> <mi>U</mi></msub><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mi>d</mi> <mi>U</mi></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>d</mi><mo>−</mo><mi>d</mi><mi>Σ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mo>∫</mo> <mi>Σ</mi></msub><mi>d</mi><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mi>d</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mo>∫</mo> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><munder><munder><mrow><mi>d</mi><mi>ω</mi></mrow><mo>⏟</mo></munder><mrow><mo>=</mo><mn>0</mn></mrow></munder><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mi>d</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mo>∫</mo> <mi>Σ</mi></msub><msub><mi>d</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mo>∫</mo> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>ω</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>k</mi></msup><msub><mi>τ</mi> <mrow><mo>∂</mo><mi>Σ</mi></mrow></msub><mi>ω</mi><msub><mo stretchy="false">|</mo> <mrow><msub><mi>ϕ</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></msub></mrow></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} d \tau_{\Sigma}\omega\vert_{\phi_(-)} & = d_U \int_\Sigma (\phi_{(-)}(-))^\ast \omega \\ & = (-1)^k \int_\Sigma d_U (\phi_{(-)}(-))^\ast \omega \\ & = (-1)^k \int_\Sigma (d - d \Sigma) (\phi_{(-)}(-))^\ast \omega \\ & = (-1)^k \int_\Sigma d (\phi_{(-)}(-))^\ast \omega - (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega \\ & = (-1)^k \int_\Sigma (\phi_{(-)}(-))^\ast \underset{= 0}{\underbrace{d \omega}} - (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega \\ & = - (-1)^k \int_\Sigma d_\Sigma (\phi_{(-)}(-))^\ast \omega \\ & = -(-1)^k \int_{\partial \Sigma} (\phi_{(-)}(-))^\ast \omega \\ & = -(-1)^k \tau_{\partial \Sigma} \omega \vert_{\phi_{(-)}} \end{aligned} </annotation></semantics></math></div> <p>where in the second but last step we used <a class="existingWikiWord" href="/nlab/show/Stokes%27+theorem">Stokes' theorem</a>.</p> </div> <p>(…)</p> <p>(…)</p> <div class="footnotes"><hr /><ol><li id="fn:1"> <p>In view of the <em><a class="existingWikiWord" href="/nlab/show/smooth+homotopy+types">smooth homotopy types</a></em> to be discussed in <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">geometry of physics – smooth homotopy types</a></em>, the structures discussed now are properly called <em>smooth <a class="existingWikiWord" href="/nlab/show/0-types">0-types</a></em> or maybe <em>smooth <a class="existingWikiWord" href="/nlab/show/h-sets">h-sets</a></em> or just <em>smooth sets</em>. While this subsumes <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> which are indeed sets equipped with (particularly nice) <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, it is common in practice to speak of manifolds as “spaces” (indeed as <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> equipped with smooth structure). Historically the <em><a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a></em> and <em><a class="existingWikiWord" href="/nlab/show/Euclidean+space">Euclidean space</a></em> of <a class="existingWikiWord" href="/nlab/show/Newtonian+physics">Newtonian physics</a> are the archetypical examples of smooth manifolds and modern <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> developed very much via motivation by the study of the <em>spaces</em> in <a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a>, namely <em><a class="existingWikiWord" href="/nlab/show/spacetimes">spacetimes</a></em>. Unfortunately, in a parallel development the word “space” has evolved in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> to mean (just) the <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> <em>represented</em> by an actual <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> (their <a class="existingWikiWord" href="/nlab/show/fundamental+infinity-groupoids">fundamental infinity-groupoids</a>). Ironically, with this meaning of the word “space” the original <a class="existingWikiWord" href="/nlab/show/Euclidean+spaces">Euclidean spaces</a> become equivalent to the point, signifying that the modern meaning of “space” in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> is quite orthogonal to the original meaning, and that in homotopy theory therefore one should better stick to “<a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a>”. Since historically grown terminology will never be fully logically consistent, and since often the less well motivated terminology is more widely understood, we will follow tradition here and take the liberty to use “smooth sets” and “smooth spaces” synonymously, the former when we feel more formalistic, the latter when we feel more relaxed. <a href="#fnref:1" rev="footnote">↩</a></p> </li></ol></div></body></html> </div> <div class="revisedby"> <p> Last revised on November 4, 2024 at 01:05:27. 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