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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="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> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#concrete'>Concrete</a></li> <li><a href='#the_atiyah_exact_sequence'>The Atiyah exact sequence</a></li> <li><a href='#GeneralAbstractDefinition'>General abstract</a></li> </ul> <li><a href='#application'>Application</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A jet can be thought of as the <a class="existingWikiWord" href="/nlab/show/infinitesimal">infinitesimal</a> <a class="existingWikiWord" href="/nlab/show/germ">germ</a> of a <a class="existingWikiWord" href="/nlab/show/section">section</a> of some <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> or of a map between spaces. Jets are a coordinate free version of Taylor-polynomials and Taylor series.</p> <h2 id="definition">Definition</h2> <h3 id="concrete">Concrete</h3> <p>For</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>≔</mo><mi>E</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> p \coloneqq E \to X </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/surjective+submersion">surjective submersion</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> and <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>, the bundle</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mi>k</mi></msup><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> J^k P \to X </annotation></semantics></math></div> <p>of <strong>order-<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> jets of sections of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math></strong> is the bundle whose <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> over a point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> is the space of equivalence classes of <a class="existingWikiWord" href="/nlab/show/germ">germs</a> of <a class="existingWikiWord" href="/nlab/show/section">sections</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, where two germs are considered equivalent if their first <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> partial <a class="existingWikiWord" href="/nlab/show/derivative">derivatives</a> at <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> coincide.</p> <p>In the case when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a trivial bundle <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>×</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">p:X\times Y \to X</annotation></semantics></math> its sections are canonically in bijection with maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and two sections have the same partial derivatives iff the partial derivatives of the corresponding maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> agree. So in this case the <a class="existingWikiWord" href="/nlab/show/jet+space">jet space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mi>k</mi></msup><mi>P</mi></mrow><annotation encoding="application/x-tex">J^k P</annotation></semantics></math> is called the space of jets of maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and commonly denoted with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mi>k</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J^k(X,Y)</annotation></semantics></math>.</p> <p>In order to pass to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>→</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">k \to \infty</annotation></semantics></math> to form the <em>infinite jet bundle</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>∞</mn></msup><mi>P</mi></mrow><annotation encoding="application/x-tex">J^\infty P</annotation></semantics></math> one forms the <a class="existingWikiWord" href="/nlab/show/projective+limit">projective limit</a> over the finite-order jet bundles,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>∞</mn></msup><mi>E</mi><mo>≔</mo><msub><munder><mi>lim</mi><mo>⟵</mo></munder> <mi>k</mi></msub><msup><mi>J</mi> <mi>k</mi></msup><mi>E</mi><mo>=</mo><munder><mi>lim</mi><mo>⟵</mo></munder><mrow><mo>(</mo><mi>⋯</mi><msup><mi>J</mi> <mn>3</mn></msup><mi>E</mi><mo>→</mo><msup><mi>J</mi> <mn>2</mn></msup><mi>E</mi><mo>→</mo><msup><mi>J</mi> <mn>1</mn></msup><mi>E</mi><mo>→</mo><mi>E</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> J^\infty E \coloneqq \underset{\longleftarrow}{\lim}_k J^k E = \underset{\longleftarrow}{\lim} \left( \cdots J^3 E \to J^2 E \to J^1 E \to E \right) </annotation></semantics></math></div> <p>but one has to decide in which category of <a class="existingWikiWord" href="/nlab/show/infinite-dimensional+manifolds">infinite-dimensional manifolds</a> to take this limit:</p> <ol> <li> <p>one may form the limit formally, i.e. in <a class="existingWikiWord" href="/nlab/show/pro-manifolds">pro-manifolds</a>. This is what is implicit for instance in <a href="#Anderson">Anderson, p.3-5</a>;</p> </li> <li> <p>one may form the limit in <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifolds">Fréchet manifolds</a>, this is farily explicit in (<a href="#Saunders89">Saunders 89, chapter 7</a>). See at <em><a href="Frechet+manifold#ProjectiveLimitsOfSmoothFiniteDimensionalManifolds">Fréchet manifold – Projective limits of finite-dimensional manifolds</a></em>. Beware that this is not equivalent to the pro-manifold structure (see the remark <a href="Frechet+manifold#DifferenceBetweenProManifoldAndFrecherManifoldStructure">here</a>). It makes sense to speak of <em><a class="existingWikiWord" href="/nlab/show/locally+pro-manifolds">locally pro-manifolds</a></em>.</p> </li> </ol> <h3 id="the_atiyah_exact_sequence">The Atiyah exact sequence</h3> <p>When <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><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\mathcal{O}_X)</annotation></semantics></math> is a complex-analytic manifold with the structure sheaf of holomorphic functions, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> a locally free sheaf of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_X</annotation></semantics></math>-modules, we can be even more explicit. The first jet bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J^1(E)</annotation></semantics></math> fits into a short exact sequence, called the Atiyah exact sequence:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>E</mi><msub><mo>⊗</mo> <mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow></msub><msubsup><mi>Ω</mi> <mi>X</mi> <mn>1</mn></msubsup><mo>→</mo><msup><mi>J</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><mi>E</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0\to E\otimes_{\mathcal{O}_X}\Omega_X^1\to J^1(E)\to E\to 0 </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>J</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>E</mi><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>X</mi> <mn>1</mn></msubsup><mo stretchy="false">)</mo><mo>⊕</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">J^1(E) = (E\otimes\Omega_X^1)\oplus E</annotation></semantics></math> as a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>-module, but with an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_X</annotation></semantics></math>-action given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>s</mi><mo>⊗</mo><mi>ω</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>f</mi><mi>s</mi><mo>⊗</mo><mi>ω</mi><mo>+</mo><mi>t</mi><mo>⊗</mo><mi mathvariant="normal">d</mi><mi>f</mi><mo>,</mo><mi>f</mi><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> f(s\otimes\omega,t) = (f s\otimes\omega+t\otimes\mathrm{d}f, f t). </annotation></semantics></math></div> <p>The extension class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msup><mi>J</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>∈</mo><msubsup><mi mathvariant="normal">Ext</mi> <mrow><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow> <mn>1</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>E</mi><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>X</mi> <mn>1</mn></msubsup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[J^1(E)]\in\mathrm{Ext}_{\mathcal{O}_X}^1(E,E\otimes\Omega_X^1)</annotation></semantics></math> of this exact sequence is called the Atiyah class of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>, and is somewhat equivalent to the first Chern class of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>. Note that the Atiyah class is exactly the obstruction to the Atiyah exact sequence admitting a splitting, and a (holomorphic) splitting of the Atiyah exact sequence is exactly a Koszul connection.</p> <h3 id="GeneralAbstractDefinition">General abstract</h3> <p>We discuss a <a class="existingWikiWord" href="/nlab/show/general+abstract">general abstract</a> definition of jet bundles.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> equipped with <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> with <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><mi>ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math> (or rather a tower <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℑ</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\Im_k</annotation></semantics></math> of such, for each infinitesimal order <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><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">k \in \mathbb{N} \cup \{\infty\}</annotation></semantics></math> ).</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Im(X)</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a> object.</p> <p>Notice that we have the canonical morphism, the <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>-component of the <a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit</a> of the <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">\Im</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/monad">monad</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>ℑ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> i \colon X \to \Im(X) </annotation></semantics></math></div> <p>(“inclusion of constant paths into all infinitesimal paths”).</p> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>i</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>i</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mover><munder><mo>⟶</mo><mrow><mi>Jet</mi><mo>:</mo><mo>=</mo><msub><mi>i</mi> <mo>*</mo></msub></mrow></munder><mover><mo>⟵</mo><mrow><msup><mi>i</mi> <mo>*</mo></msup></mrow></mover></mover><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>ℑ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> (i^\ast \dashv i_\ast) \;\colon\; \mathbf{H}_{/X} \stackrel{\overset{i^*}{\longleftarrow}}{\underset{Jet := i_*}{\longrightarrow}} \mathbf{H}_{/\Im(X)} </annotation></semantics></math></div> <div class="num_defn" id="GeneralAbstractDefinition"> <h6 id="definition_2">Definition</h6> <p>The <em><a class="existingWikiWord" href="/nlab/show/jet+comonad">jet comonad</a></em> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-comonad">(∞,1)-comonad</a></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> <mo>*</mo></msup><msub><mi>i</mi> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub><mo>⟶</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>X</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> i^\ast i_\ast \;\colon\; \mathbf{H}_{/X} \longrightarrow \mathbf{H}_{/X} </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> gives even an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>i</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>i</mi> <mo>*</mo></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(i_! \dashv i^\ast \dashv i_\ast)</annotation></semantics></math>, there is a <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><msub><mi>T</mi> <mi>inf</mi></msub><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T_{inf} X \times_X (-)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/jet+comonad">jet comonad</a> of def. <a class="maruku-ref" href="#GeneralAbstractDefinition"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>inf</mi></msub><mi>X</mi><msub><mo>×</mo> <mi>X</mi></msub><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>Jet</mi></mrow><annotation encoding="application/x-tex"> T_{inf}X \times_X (-) \;\dashv\; Jet </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>inf</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">T_{inf} X</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</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>, see at <em><a href="differential+cohesion#RelationOfInfinitesimalDiskBundleToJetBundle">differential cohesion – infinitesimal disk bundle – relation to jet bundles</a></em></p> </div> <div class="num_remark" id="LiteratureOnGeneralAbstractCharacterization"> <h6 id="remark_2">Remark</h6> <p>In the context of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> the fact that the jet bundle construction is a comonad was explicitly observed in (<a href="#Marvan86">Marvan 86</a>, see also <a href="#Marvan93">Marvan 93, section 1.1</a>, <a href="#Marvan89">Marvan 89</a>). It is almost implicit in (<a href="#KrasilshchikVerbovetsky98">Krasil’shchik-Verbovetsky 98, p. 13, p. 17</a>, <a href="#Krasilshchik99">Krasilshchik 99, p. 25</a>).</p> <p>In the context of <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> the fact that the jet bundle construction is <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> to the <a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a> construction is (<a href="#Kock80">Kock 80, prop. 2.2</a>).</p> <p id="InTheContextOf"> In the context of <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> and of <a class="existingWikiWord" href="/nlab/show/D-schemes">D-schemes</a> as in (<a href="#BeilinsonDrinfeld">BeilinsonDrinfeld, 2.3.2</a>, reviewed in <a href="#Paugam">Paugam, section 2.3</a>), the base change comonad formulation inf def. <a class="maruku-ref" href="#GeneralAbstractDefinition"></a> was noticed in (<a href="#Lurie">Lurie, prop. 0.9</a>).</p> </div> <p>In as in (<a href="#BeilinsonDrinfeld">BeilinsonDrinfeld, 2.3.2</a>, reviewed in <a href="#Paugam">Paugam, section 2.3</a>) jet bundles are expressed dually in terms of algebras in <a class="existingWikiWord" href="/nlab/show/D-modules">D-modules</a>. We now indicate how the translation works.</p> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>In terms of <a class="existingWikiWord" href="/nlab/show/differential+homotopy+type+theory">differential homotopy type theory</a> this means that forming “jet types” of <a class="existingWikiWord" href="/nlab/show/dependent+types">dependent types</a> over <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/dependent+product">dependent product</a> operation along the unit of the <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>jet</mi><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mi>X</mi><mo>→</mo><mi>ℑ</mi><mi>X</mi></mrow></munder><mi>E</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> jet(E) \coloneqq \underset{X \to \Im X}{\prod} E \,. </annotation></semantics></math></div></div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/quasicoherent+%28%E2%88%9E%2C1%29-sheaf">quasicoherent (∞,1)-sheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a morphism of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-sheaves">(∞,2)-sheaves</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>→</mo><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \to Mod \,. </annotation></semantics></math></div> <p>We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>QC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> QC(X) := Hom(X, Mod) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/quasicoherent+%28%E2%88%9E%2C1%29-sheaves">quasicoherent (∞,1)-sheaves</a>.</p> <p>A <em><a class="existingWikiWord" href="/nlab/show/D-module">D-module</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> is a morphism of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-sheaves">(∞,2)-sheaves</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Im (X) \to Mod \,. </annotation></semantics></math></div> <p>We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DQC</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Hom</mi><mo stretchy="false">(</mo><mi>ℑ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> DQC(X) := Hom(\Im (X), Mod) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> of D-modules.</p> </div> <p>The <strong>Jet algebra</strong> functor is the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> from <a class="existingWikiWord" href="/nlab/show/associative+algebra">commutative algebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{D}(X)</annotation></semantics></math> to those over the <a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(X)</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Jet</mi><mo>⊣</mo><mi>F</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Alg</mi> <mrow><mi>𝒟</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mover><munder><mo>→</mo><mi>F</mi></munder><mover><mo>←</mo><mi>Jet</mi></mover></mover><msub><mi>Alg</mi> <mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (Jet \dashv F) : Alg_{\mathcal{D}(X)} \stackrel{\overset{Jet}{\leftarrow}}{\underset{F}{\to}} Alg_{\mathcal{O}(X)} \,. </annotation></semantics></math></div> <h2 id="application">Application</h2> <p>Typical <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangians</a> in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a> are defined on jet bundles. Their <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> is governed by <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equations</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+prolongation">jet prolongation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+space">jet space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+group">jet group</a>, <a class="existingWikiWord" href="/nlab/show/jet+groupoid">jet groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/evolutionary+vector+field">evolutionary vector field</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/source+form">source form</a>, <a class="existingWikiWord" href="/nlab/show/Lepage+form">Lepage form</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+jet">metric jet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/h-principle">h-principle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic jet space</a></p> </li> </ul> <div> <p><strong>Examples of sequences of local structures</strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></th><th>point</th><th>first order <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/formal+geometry">formal</a> = arbitrary order infinitesimal</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>local = <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a>wise</th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊂</mo></mrow><annotation encoding="application/x-tex">\subset</annotation></semantics></math></th><th>finite</th></tr></thead><tbody><tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>←</mo></mrow><annotation encoding="application/x-tex">\leftarrow</annotation></semantics></math> <strong><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/integration">integration</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/derivative">derivative</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Taylor+series">Taylor series</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a> (path)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/tangent+vector">tangent vector</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/jet">jet</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ">germ</a> of <a class="existingWikiWord" href="/nlab/show/curve">curve</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/curve">curve</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhood">infinitesimal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+neighbourhood">formal neighbourhood</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/germ+of+a+space">germ of a space</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/square-0+ring+extension">square-0 ring extension</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nilpotent+ring+extension">nilpotent ring extension</a>/<a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring+extension">ring extension</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/finite+field">finite field</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mrow><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_{(p)}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">localization at (p)</a></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/integers">integers</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/local+Lie+group">local Lie group</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a></td><td style="text-align: left;"></td><td style="text-align: left;">local strict deformation quantization</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+deformation+quantization">strict deformation quantization</a></td></tr> </tbody></table> </div> <ul> <li>in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>/<a class="existingWikiWord" href="/nlab/show/Goodwillie+calculus">Goodwillie calculus</a>: <a class="existingWikiWord" href="/nlab/show/jet+%28%E2%88%9E%2C1%29-category">jet (∞,1)-category</a></li> </ul> <h2 id="references">References</h2> <p>Jets were introduced by <a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a> in 1951 in a series of five short articles in <a class="existingWikiWord" href="/nlab/show/Comptes+Rendus">Comptes Rendus</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>: <em>Les prolongements d’une variété différentiable. I. Calcul des jets, prolongement principal.</em>, C. R. Acad. Sci. Paris 233 (1951), 598–600.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>: <em>Les prolongements d’une variété différentiable. II. L’espace des jets d’ordre <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">r</annotation></semantics></math> de <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">V_n</annotation></semantics></math> dans <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>m</mi></msub></mrow><annotation encoding="application/x-tex">V_m</annotation></semantics></math></em>, C. R. Acad. Sci. Paris 233 (1951), 777–779.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>: <em>Les prolongements d’une variété différentiable. III. Transitivité des prolongements</em>, C. R. Acad. Sci. Paris 233 (1951), 1081–1083.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>: <em>Les prolongements d’une variété différentiable. IV. Éléments de contact et éléments d’enveloppe</em>, C. R. Acad. Sci. Paris 234 (1952), 1028–1030.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>: <em>Les prolongements d’une variété différentiable. V. Covariants différentiels et prolongements d’une structure infinitésimale</em>, C. R. Acad. Sci. Paris 234 (1952), 1424–1425.</p> </li> </ul> <p>Exposition of <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> in terms of jet bundles and <a class="existingWikiWord" href="/nlab/show/Lepage+forms">Lepage forms</a> and aimed at examples from <a class="existingWikiWord" href="/nlab/show/physics">physics</a> is in</p> <ul> <li id="MusilovaHronek16"><a class="existingWikiWord" href="/nlab/show/Jana+Musilov%C3%A1">Jana Musilová</a>, <a class="existingWikiWord" href="/nlab/show/Stanislav+Hronek">Stanislav Hronek</a>, <em>The calculus of variations on jet bundles as a universal approach for a variational formulation of fundamental physical theories</em>, Communications in Mathematics, Volume 24, Issue 2 (Dec 2016) (<a href="https://doi.org/10.1515/cm-2016-0012">doi.org/10.1515/cm-2016-0012</a>)</li> </ul> <p>Lecture notes and textbook accounts:</p> <ul> <li id="Michor80"> <p><a class="existingWikiWord" href="/nlab/show/Peter+Michor">Peter Michor</a>, <em>Manifolds of differentiable mappings</em>, Shiva Publishing (1980) <a href="http://www.mat.univie.ac.at/~michor/manifolds_of_differentiable_mappings.pdf">pdf</a></p> </li> <li id="Saunders89"> <p><a class="existingWikiWord" href="/nlab/show/David+Saunders">David Saunders</a>, <em>The geometry of jet bundles</em>, London Mathematical Society Lecture Note Series <strong>142</strong>, Cambridge Univ. Press 1989.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Demeter+Krupka">Demeter Krupka</a>, <a class="existingWikiWord" href="/nlab/show/Josef+Jany%C5%A1ka">Josef Janyška</a>, Part 1 of: <em>Lectures on differential invariants</em>, Univerzita J. E. Purkyně, Brno (1990) [ISBN:80-210-165-8, <a href="https://www.researchgate.net/publication/36792711_Lectures_on_Differential_Invariants">researchgate</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean-Fran%C3%A7ois+Pommaret">Jean-François Pommaret</a>, Chapter II in: <em>Partial Differential Equations and Group Theory</em>, Springer (1994) [<a href="https://doi.org/10.1007/978-94-017-2539-2">doi:10.1007/978-94-017-2539-2</a>]</p> </li> <li id="Krasilshchik99"> <p><a class="existingWikiWord" href="/nlab/show/Joseph+Krasil%27shchik">Joseph Krasil'shchik</a> in collaboration with Barbara Prinari, <em>Lectures on Linear Differential Operators over Commutative Algebras</em>, 1998 (<a href="https://diffiety.mccme.ru/preprint/99/01_99.pdf">pdf</a>)</p> </li> <li> <p>Shihoko Ishii, <em>Jet schemes, arc spaces and the Nash problem</em>, <a href="http://arXiv.org/abs/0704.3327">arXiv:math.AG/0704.3327</a></p> </li> <li> <p>G. Sardanashvily, <em>Fibre bundles, jet manifolds and Lagrangian theory</em>, Lectures for theoreticians, <a href="http://xxx.lanl.gov/abs/0908.1886">arXiv:0908.1886</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Peter+Olver">Peter Olver</a>, <em>Lectures on Lie groups and differential equation</em>, chapter 3, <em>Jets and differential invariants</em>, 2012 (<a href="http://www.math.umn.edu/~olver/sm_/j.pdf">pdf</a>)</p> </li> </ul> <p>Early accounts include</p> <ul> <li>Hubert Goldschmidt, <em>Integrability criteria for systems of nonlinear partial differential equations</em>, J. Differential Geom. Volume 1, Number 3-4 (1967), 269-307 (<a href="http://projecteuclid.org/euclid.jdg/1214428094">Euclid</a>)</li> </ul> <p>The algebra of smooth functions of just <em>locally</em> finite order on the jet bundle (“<a class="existingWikiWord" href="/nlab/show/locally+pro-manifold">locally pro-manifold</a>”) was maybe first considered in</p> <ul> <li id="Takens79"><a class="existingWikiWord" href="/nlab/show/Floris+Takens">Floris Takens</a>, <em>A global version of the inverse problem of the calculus of variations</em>, J. Differential Geom. Volume 14, Number 4 (1979), 543-562. (<a href="https://projecteuclid.org/euclid.jdg/1214435235">Euclid</a>)</li> </ul> <p id="DiscussionOfFrechetManifoldStructure"> Discussion of the <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifold">Fréchet manifold</a> structure on infinite jet bundles includes</p> <ul> <li id="Saunders89"> <p><a class="existingWikiWord" href="/nlab/show/David+Saunders">David Saunders</a>, chapter 7 <em>Infinite jet bundles</em> of <em>The geometry of jet bundles</em>, London Mathematical Society Lecture Note Series <strong>142</strong>, Cambridge Univ. Press 1989.</p> </li> <li> <p>M. Bauderon, <em>Differential geometry and Lagrangian formalism in the calculus of variations</em>, in <em>Differential Geometry, Calculus of Variations, and their Applications</em>, Lecture Notes in Pure and Applied Mathematics, 100, Marcel Dekker, Inc., N.Y., 1985, pp. 67-82.</p> </li> <li id="DodsonGalanisVassiliou15"> <p><a class="existingWikiWord" href="/nlab/show/C.+T.+J.+Dodson">C. T. J. Dodson</a>, George Galanis, Efstathios Vassiliou,, p. 109 and section 6.3 of <em>Geometry in a Fréchet Context: A Projective Limit Approach</em>, Cambridge University Press (2015)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Andrew+Lewis">Andrew Lewis</a>, <em>The bundle of infinite jets</em> (2006) (<a href="http://www.mast.queensu.ca/~andrew/notes/pdf/2006a.pdf">pdf</a>)</p> </li> </ul> <p>Discussion of finite-order jet bundles in tems of <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> is in</p> <ul> <li id="Kock80"> <p><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>Formal manifolds and synthetic theory of jet bundles</em>, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1980) Volume: 21, Issue: 3 (<a href="http://www.numdam.org/item?id=CTGDC_1980__21_3_227_0">Numdam</a>)</p> </li> <li id="Kock10"> <p><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, section 2.7 of <em>Synthetic geometry of manifolds</em>, Cambridge Tracts in Mathematics 180 (2010). (<a href="http://home.imf.au.dk/kock/SGM-final.pdf">pdf</a>)</p> </li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/jet+comonad">jet comonad</a> structure on the jet operation in the context of differential geometry is made explicit in</p> <ul> <li id="Marvan86"> <p><a class="existingWikiWord" href="/nlab/show/Michal+Marvan">Michal Marvan</a>, <em>A note on the category of partial differential equations</em>, in <em>Differential geometry and its applications</em>, Proceedings of the Conference August 24-30, 1986, Brno (<a class="existingWikiWord" href="/nlab/files/MarvanJetComonad.pdf" title="pdf">pdf</a>)</p> <p>(notice that prop. 1.3 there is wrong, the correct version is in the thesis of the author)</p> </li> </ul> <p>with further developments in</p> <ul> <li id="Marvan89"> <p><a class="existingWikiWord" href="/nlab/show/Michal+Marvan">Michal Marvan</a><em>On the horizontal cohomology with general coefficients</em>, 1989 (<a href="http://old.math.slu.cz/People/MichalMarvan/Annotations/horizontal.php">web announcement</a>, <a href="http://dml.cz/dmlcz/701469">web archive</a>)</p> <p><strong>Abstract:</strong> In the present paper the horizontal cohomology theory is interpreted as a special case of the Van Osdol bicohomology theory applied to what we call a “jet comonad”. It follows that differential equations have well-defined cohomology groups with coefficients in linear differential equations.</p> </li> <li id="Marvan93"> <p><a class="existingWikiWord" href="/nlab/show/Michal+Marvan">Michal Marvan</a>, section 1.1 of <em>On Zero-Curvature Representations of Partial Differential Equations</em>, (1993) (<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.5631">web</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Khavkine">Igor Khavkine</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Synthetic+variational+calculus">Synthetic geometry of differential equations: I. Jets and comonad structure</a></em> (<a href="https://arxiv.org/abs/1701.06238">arXiv:1701.06238</a>)</p> </li> </ul> <p>In the context of algebraic geometry, the abstract characterization of jet bundles as the direct images of base change along the de Rham space projection is noticed in</p> <ul> <li id="Lurie"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, p. 6 of: <em>Notes on crystals and algebraic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>-modules</em> (2010) [<a class="existingWikiWord" href="/nlab/files/Lurie-NotesOnCrystals.pdf" title="pdf">pdf</a>]</li> </ul> <p>The explicit description in terms of formal duals of <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> in <a class="existingWikiWord" href="/nlab/show/D-module">D-modules</a> is in</p> <ul> <li id="BeilinsonDrinfeld"><a class="existingWikiWord" href="/nlab/show/Alexander+Beilinson">Alexander Beilinson</a>, <a class="existingWikiWord" href="/nlab/show/Vladimir+Drinfeld">Vladimir Drinfeld</a>, <em><a class="existingWikiWord" href="/nlab/show/Chiral+Algebras">Chiral Algebras</a></em></li> </ul> <p>Exposition:</p> <ul> <li id="Paugam"><a class="existingWikiWord" href="/nlab/show/Fr%C3%A9d%C3%A9ric+Paugam">Frédéric Paugam</a>, Section 2.3 of: <em>Homotopical Poisson Reduction of gauge theories</em> (<a href="http://people.math.jussieu.fr/~fpaugam/documents/homotopical-poisson-reduction-of-gauge-theories.pdf">pdf</a>)</li> </ul> <p>A discussion of jet bundles with an eye towards discussion of the <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a> on them :</p> <ul> <li id="Anderson"><a class="existingWikiWord" href="/nlab/show/Ian+Anderson">Ian Anderson</a>, chapter 1, section A of: <em>The variational bicomplex</em> [<a class="existingWikiWord" href="/nlab/files/AndersonVariationalBicomplex.pdf" title="pdf">pdf</a>]</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a> and <a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a> of jet bundles is discussed in</p> <ul> <li>G. Giachetta, L. Mangiarotti, <a class="existingWikiWord" href="/nlab/show/Gennadi+Sardanashvily">Gennadi Sardanashvily</a>, <em>Cohomology of the variational bicomplex on the infinite order jet space</em>, Journal of Mathematical Physics 42, 4272-4282 (2001) (<a href="http://arxiv.org/abs/math/0006074">arXiv:math/0006074</a>)</li> </ul> <p>where both versions (smooth functions being globally or locally of finite order) are discussed and compared.</p> <p>Discussion of all this in the convenient context of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>:</p> <ul> <li id="GiotopoulosSati23"><a class="existingWikiWord" href="/nlab/show/Grigorios+Giotopoulos">Grigorios Giotopoulos</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, §4 in: <em>Field Theory via Higher Geometry I: <a class="existingWikiWord" href="/schreiber/show/Smooth+Sets+of+Fields">Smooth Sets of Fields</a></em> [<a href="https://arxiv.org/abs/2312.16301">arXiv:2312.16301</a>]</li> </ul> <p>Discussion of jet-restriction of the <a class="existingWikiWord" href="/nlab/show/Haefliger+groupoid">Haefliger groupoid</a> is in</p> <ul> <li>Arne Lorenz, <em>Jet Groupoids, Natural Bundles and the Vessiot Equivalence Method</em>, Thesis (<a href="http://wwwb.math.rwth-aachen.de/~arne/Dissertation_Lorenz_Arne.pdf">pdf</a>)</li> </ul> <p>Discussion of jet bundles in <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> includes</p> <ul> <li> <p>Arthemy V. Kiselev, Andrey O. Krutov, appendix of <em>On the (non)removability of spectral parameters in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-graded zero-curvature representations and its applications</em> (<a href="http://arxiv.org/abs/1301.7143">arXiv:1301.7143</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gennadi+Sardanashvily">Gennadi Sardanashvily</a>, <em>Graded infinite order jet manifolds</em>, <em>Int. J. Geom. Methods Mod. Phys. v.4 (2007) 1335-1362</em> (<a href="https://arxiv.org/abs/0708.2434">arXiv:0708.2434</a>)</p> </li> </ul> <p>See also</p> <ul> <li id="KrasilshchikVerbovetsky98"><a class="existingWikiWord" href="/nlab/show/Joseph+Krasil%27shchik">Joseph Krasil'shchik</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Verbovetsky">Alexander Verbovetsky</a>, <em>Homological Methods in Equations of Mathematical Physics</em> (<a href="http://arxiv.org/abs/math/9808130">arXiv:math/9808130</a>)</li> </ul> <p>On jet bundles 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">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+manifold">graded manifolds</a> and their <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundles</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jan+Vysoky">Jan Vysoky</a>. <em>Graded Jet Geometry</em>. (2023). (<a href="https://arxiv.org/abs/2311.15754">arXiv:2311.15754</a>)</li> </ul> <p>On jet bundles in <a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Shahn+Majid">Shahn Majid</a>, Francisco Simão, <em>Quantum jet bundles</em>, Lett. Math. Phys. <strong>113</strong> 120 (2023) [<a href="https://arxiv.org/abs/2202.03067">arXiv:2202.03067</a>, <a href="https://doi.org/10.1007/s11005-023-01738-z">doi:10.1007/s11005-023-01738-z</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 19, 2025 at 03:47:36. 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