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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> <h4 id="variational_calculus">Variational calculus</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></strong></p> <h2 id="differential_geometric_version">Differential geometric version</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+Lagrangian">local Lagrangian</a>, <a class="existingWikiWord" href="/nlab/show/local+action+functional">local action functional</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+form">Euler-Lagrange form</a>, <a class="existingWikiWord" href="/nlab/show/source+form">source form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lepage+form">Lepage form</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principle+of+extremal+action">principle of extremal action</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</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/covariant+phase+space">covariant phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Noether%27s+theorem">Noether's theorem</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/conserved+current">conserved current</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetry">symmetry</a></p> </li> </ul> </li> </ul> <h2 id="derived_differential_geometric_version">Derived differential geometric version</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+complex">BV-BRST complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/variational+calculus+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#review'>Review</a></li> <li><a href='#ReferencesAsJetCoalgebras'>As jet coalgebras</a></li> <li><a href='#Conferences'>Conferences</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>The concept of <em>diffiety</em> (<a href="#Vinogradov81">Vinogradov 81</a>) reflects the concept of <em><a class="existingWikiWord" href="/nlab/show/partial+differential+equation">partial differential equation</a></em> (generally non-linear) in analogy to how the concept of <em><a class="existingWikiWord" href="/nlab/show/algebraic+variety">algebraic variety</a></em> reflects that of <em><a class="existingWikiWord" href="/nlab/show/polynomial">polynomial</a> <a class="existingWikiWord" href="/nlab/show/equation">equation</a></em>:</p> <p>A <em>diffiety</em> is the <a class="existingWikiWord" href="/nlab/show/solution">solution</a>-locus <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mo>↪</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{E}\hookrightarrow J^\infty_\Sigma(E)</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/partial+differential+equation">partial differential equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mi>Φ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">D \Phi = 0</annotation></semantics></math> regarded as an ordinary <a class="existingWikiWord" href="/nlab/show/equation">equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\tilde D = 0</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J^\infty_\Sigma(E)</annotation></semantics></math> of some <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>fb</mi></mover><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \overset{fb}{\to} \Sigma</annotation></semantics></math>.</p> <p>Here <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 space of <a class="existingWikiWord" href="/nlab/show/free+variables">free variables</a> of the <a class="existingWikiWord" href="/nlab/show/PDE">PDE</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>fb</mi></mover><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \overset{fb}{\to} \Sigma</annotation></semantics></math> is the bundle of <a class="existingWikiWord" href="/nlab/show/dependent+variables">dependent variables</a>, and a <a class="existingWikiWord" href="/nlab/show/differential+operator">differential operator</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> D \;\colon\; \Gamma_\Sigma(E) \longrightarrow \Gamma_\Sigma(F) </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/space+of+smooth+sections">space of smooth sections</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>fb</mi></mrow><annotation encoding="application/x-tex">fb</annotation></semantics></math> is represented by a <a class="existingWikiWord" href="/nlab/show/bundle+morphism">bundle morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo lspace="verythinmathspace">:</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi></mrow><annotation encoding="application/x-tex"> \tilde D \colon J^\infty_\Sigma(E) \to F </annotation></semantics></math></div> <p>out of the <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> via <a class="existingWikiWord" href="/nlab/show/jet+prolongation">jet prolongation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>j</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo lspace="verythinmathspace">:</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Γ</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j^\infty_\Sigma \colon \Gamma_\Sigma(E) \to \Gamma_\Sigma(J^\infty_\Sigma(E))</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mi>Φ</mi><mo>=</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">(</mo><msubsup><mi>j</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>Φ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D \Phi = \tilde D (j^\infty_\Sigma(\Phi))</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> <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><mtext>diffiety</mtext></mtd> <mtd></mtd> <mtd><mtext>jet bundle</mtext></mtd> <mtd><mrow><mtable><mtr><mtd><mtext>differential</mtext></mtd></mtr> <mtr><mtd><mtext>operator</mtext></mtd></mtr></mtable></mrow></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>ℰ</mi></mtd> <mtd><mover><mo>↪</mo><mrow><mover><mi>D</mi><mo stretchy="false">˜</mo></mover><mo>=</mo><mn>0</mn></mrow></mover></mtd> <mtd><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mover><mi>D</mi><mo stretchy="false">˜</mo></mover></mover></mtd> <mtd><mi>F</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ & \text{diffiety} && \text{jet bundle} & \array{\text{differential} \\ \text{operator}} & \\ & \mathcal{E} &\overset{\tilde D = 0}{\hookrightarrow}& J^\infty_\Sigma(E) &\overset{\tilde D}{\longrightarrow}& F } </annotation></semantics></math></div> <p>For instance in <a class="existingWikiWord" href="/nlab/show/Lagrangian+field+theory">Lagrangian field theory</a> the bundle in question is a <a class="existingWikiWord" href="/nlab/show/field+bundle">field bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mover><mo>→</mo><mi>fb</mi></mover><mi>Σ</mi></mrow><annotation encoding="application/x-tex">E \overset{fb}{\to} \Sigma</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/partial+differential+equation">partial differential equation</a> is the <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>EL</mi></msub><mstyle mathvariant="bold"><mi>L</mi></mstyle><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\delta_{EL}\mathbf{L} = 0</annotation></semantics></math>, and its diffiety solution locus <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">\mathcal{E}</annotation></semantics></math> inside the <a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">J^\infty_\Sigma(E)</annotation></semantics></math> is called the <em><a class="existingWikiWord" href="/nlab/show/shell">shell</a></em> of the field theory.</p> <p>In <a href="#Marvan86">Marvan 86</a> it was observed that Vinogradov’s <a class="existingWikiWord" href="/nlab/show/formally+integrable+PDE">formally integrable diffieties</a> are equivalently the <a class="existingWikiWord" href="/nlab/show/coalgebra+over+a+comonad">coalgebras</a> over the <a class="existingWikiWord" href="/nlab/show/jet+comonad">jet comonad</a> acting on <a class="existingWikiWord" href="/nlab/show/locally+pro-manifold">locally pro-manifold</a>-<a class="existingWikiWord" href="/nlab/show/bundles">bundles</a> (over a base space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/free+variables">free variables</a>). This statement generalizes to the <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> of the <a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a> <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> (<a href="#KhavkineSchreiber17">Khavkine-Schreiber 17</a>), where the <a class="existingWikiWord" href="/nlab/show/jet+comonad">jet comonad</a> is realized as the comonad corresponding to <a class="existingWikiWord" href="/nlab/show/base+change">base change</a> along the <a class="existingWikiWord" href="/nlab/show/de+Rham+shape">de Rham shape</a> projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>Σ</mi></msub></mrow></mover><mi>ℑ</mi><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma \overset{\eta_\Sigma}{\longrightarrow} \Im \Sigma</annotation></semantics></math>. By <a class="existingWikiWord" href="/nlab/show/comonadic+descent">comonadic descent</a> this implies that over <a class="existingWikiWord" href="/nlab/show/formally+smooth">formally smooth</a> base spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/formally+integrable+PDE">formally integrable diffieties</a> are equivalently the <a class="existingWikiWord" href="/nlab/show/bundles">bundles</a> over the <a class="existingWikiWord" href="/nlab/show/de+Rham+shape">de Rham shape</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Im \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>PDEs</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><msub><mi>Diffeties</mi> <mi>Σ</mi></msub><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msubsup><mi>J</mi> <mi>Σ</mi> <mn>∞</mn></msubsup><mi>CoAlg</mi><mrow><mo>(</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>Σ</mi></mrow></msub><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>ℑ</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex"> PDEs_\Sigma(\mathbf{H}) \;=\; Diffeties_\Sigma(\mathbf{H}) \;\simeq\; J^\infty_\Sigma CoAlg\left(\mathbf{H}_{/\Sigma} \right) \;\simeq\; \mathbf{H}_{/\Im(\Sigma)} </annotation></semantics></math></div> <p>(<a href="#KhavkineSchreiber17">Khavkine-Schreiber 17, thorem 3.52, theorem 3.60</a>)</p> <p>This makes manifest how <a class="existingWikiWord" href="/nlab/show/diffieties">diffieties</a> are the analog in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> of concepts in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>: For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> a suitable <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> then a <a class="existingWikiWord" href="/nlab/show/quasicoherent+module">quasicoherent module</a> over its <a class="existingWikiWord" href="/nlab/show/de+Rham+shape">de Rham shape</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Im \Sigma</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/crystal">crystal</a>”) is called a <em><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></em> and represents an algebraic <a class="existingWikiWord" href="/nlab/show/linear+partial+differential+equation">linear partial differential equation</a>, while a <a class="existingWikiWord" href="/nlab/show/relative+scheme">relative scheme</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℑ</mi><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Im \Sigma</annotation></semantics></math> is called a <em><a class="existingWikiWord" href="/nlab/show/D-scheme">D-scheme</a></em> and represents a general algebraic <a class="existingWikiWord" href="/nlab/show/partial+differential+equation">partial differential equation</a>. See also at <em><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a></em> for more on this.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></li> </ul> <h2 id="references">References</h2> <h3 id="general">General</h3> <ul> <li id="Vinogradov81"> <p><a class="existingWikiWord" href="/nlab/show/Alexandre+Vinogradov">Alexandre Vinogradov</a>, <em>Geometry of nonlinear differential equations</em>, Journal of Soviet Mathematics 17 (1981) 1624–1649 (<a href="https://doi.org/10.1007/BF01084594">doi:10.1007/BF01084594</a>)</p> </li> <li id="Vinogradov84"> <p><a class="existingWikiWord" href="/nlab/show/Alexandre+Vinogradov">Alexandre Vinogradov</a>, <em>Local symmetries and conservation laws</em>, Acta Appl. Math., Vol. 2, 1984, p. 21 (<a href="http://www.ams.org/mathscinet-getitem?mr=736872">MR85m:58192</a>, <a href="http://dx.doi.org/10.1007/BF01405491">doi</a>)</p> </li> <li id="Vinogradov89"> <p><a class="existingWikiWord" href="/nlab/show/Alexandre+Vinogradov">Alexandre Vinogradov</a>, <em>Symmetries and conservation laws of partial differential equations: basic notions and results</em>, Acta Appl. Math., Vol. 15, 1989, p. 3. <a href="http://www.ams.org/mathscinet-getitem?mr=1007340">MR91b:58282</a>, <a href="http://dx.doi.org/10.1007/BF00131928">doi</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexandre+Vinogradov">Alexandre Vinogradov</a>, <em>Scalar differential invariants, diffieties and characteristic classes</em>, in: Mechanics, Analysis and Geometry: 200 Years after, 379–414, <a href="http://www.ams.org/mathscinet-getitem?mr=1098525">MR92e:58244</a></p> </li> <li> <p>G. Cicogna, G. Gaeta, <em>Lie-point symmetries in bifurcation problems</em>, Annales de l’institut Henri Poincaré (A) Physique théorique, 56 no. 4 (1992), p. 375-414 <a href="http://www.numdam.org/item?id=AIHPA_1992__56_4_375_0">numdam</a></p> </li> <li> <p>L. Vitagliano, <em>Hamilton-Jacobi diffieties</em>, <a href="http://arxiv.org/abs/1104.0162">arxiv/1104.0162</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexandre+M.+Vinogradov">Alexandre M. Vinogradov</a>‘s <a href="http://diffiety.ac.ru/curvita/amv.htm">webpage</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Joseph+Krasil%27shchik">Joseph Krasil'shchik</a>‘s <a href="http://diffiety.ac.ru/curvita/isk.htm">webpage</a> (with links to some papers) and <a href="http://gdeq.org/index.php?title=Joseph_Krasil%27shchik">wiki list of publications</a></p> </li> </ul> <h3 id="review">Review</h3> <ul> <li> <p><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://xxx.lanl.gov/abs/math/9808130v2">arxiv:math.DG/9808130</a>, 150 p.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Joseph+Krasil%27shchik">Joseph Krasil'shchik</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Verbovetsky">Alexander Verbovetsky</a>, Geometry of jet spaces and integrable systems, J. Geom. Phys. (2010), <a href="doi:10.1016/j.geomphys.2010.10.012">doi</a>, <a href="http://arxiv.org/abs/1002.0077">arXiv:1002.0077</a></p> </li> </ul> <p>See also</p> <ul> <li>wikipedia: <a href="http://en.wikipedia.org/wiki/Diffiety">diffiety</a>, <a href="http://en.wikipedia.org/wiki/Secondary_calculus_and_cohomological_physics">secondary calculus and cohomological physics</a></li> </ul> <h3 id="ReferencesAsJetCoalgebras">As jet coalgebras</h3> <p>Diffieties as <a class="existingWikiWord" href="/nlab/show/coalgebra+over+a+comonad">coalgebras</a> over the <a class="existingWikiWord" href="/nlab/show/jet+comonad">jet comonad</a> are discussed 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/MarvanJetComonadPDE.pdf" title="pdf">pdf</a>)</p> </li> <li id="MarvanThesis"> <p><a class="existingWikiWord" href="/nlab/show/Michal+Marvan">Michal Marvan</a>, thesis, 1989 (<a class="existingWikiWord" href="/nlab/files/MarvanThesis.pdf" title="pdf">pdf</a>, <a href="http://www.slu.cz/math/cz/lide/marvan-michal/docs/mar89-ocr.pdf">web</a>)</p> </li> <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> </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 id="KhavkineSchreiber17"> <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> <h3 id="Conferences">Conferences</h3> <ul> <li> <p><a href="http://school.diffiety.org">diffiety school</a>, <a href="http://diffiety.ac.ru">diffiety institute</a>, <a href="http://gdeq.org/index.php?title=Welcome_to_GDEq.org!">gdeq.org wiki</a></p> </li> <li><div style="float:right;margin:0 10px 10px 0;"> <a href="http://www.dipmat2.unisa.it/people/vitagliano/www/DS2018_poster.png"><img src="http://www.dipmat2.unisa.it/people/vitagliano/www/DS2018_poster.png" width="150" /></a></div> <p><a href="https://sites.google.com/site/levicivitainstitute/Activities/DiffietySchools/xxi-summer-diffiety-school">XXI Summer Diffiety School School on Geometry of PDEs</a>, July 19 - 31, 2018</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on December 11, 2017 at 11:26:30. See the <a href="/nlab/history/diffiety" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/diffiety" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/8191/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/diffiety/14" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/diffiety" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/diffiety" accesskey="S" class="navlink" id="history" rel="nofollow">History (14 revisions)</a> <a href="/nlab/show/diffiety/cite" style="color: black">Cite</a> <a href="/nlab/print/diffiety" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/diffiety" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>