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value="Search" 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="synthetic_differential_geometry">Synthetic 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="topos_theory">Topos Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/topos+theory">topos theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Toposes">Toposes</a></li> </ul> <h2 id="background">Background</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> </ul> </li> </ul> <h2 id="toposes">Toposes</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/Heyting+algebra">Heyting algebra</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos">topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable presheaf</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+sheaves">category of sheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/site">site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+pretopology">pretopology</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">topology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheafification">sheafification</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+topos">base topos</a>, <a class="existingWikiWord" href="/nlab/show/indexed+topos">indexed topos</a></p> </li> </ul> <h2 id="internal_logic">Internal Logic</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/subobject+classifier">subobject classifier</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+numbers+object">natural numbers object</a></p> </li> </ul> </li> </ul> <h2 id="topos_morphisms">Topos morphisms</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+morphism">logical morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/direct+image">direct image</a>/<a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/global+section">global sections</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+embedding">geometric embedding</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/surjective+geometric+morphism">surjective geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/essential+geometric+morphism">essential geometric morphism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+geometric+morphism">locally connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+geometric+morphism">connected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+geometric+morphism">totally connected geometric morphism</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+geometric+morphism">étale geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+geometric+morphism">open geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+geometric+morphism">proper geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/compact+topos">compact topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/separated+geometric+morphism">separated geometric morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hausdorff+topos">Hausdorff topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+geometric+morphism">local geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bounded+geometric+morphism">bounded geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/base+change">base change</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+geometric+morphism">localic geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperconnected+geometric+morphism">hyperconnected geometric morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atomic+geometric+morphism">atomic geometric morphism</a></p> </li> </ul> </li> </ul> <h2 id="extra_stuff_structure_properties">Extra stuff, structure, properties</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+locale">topological locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localic+topos">localic topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/petit+topos">petit topos/gros topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a>, <a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classifying+topos">classifying topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> </li> </ul> <h2 id="cohomology_and_homotopy">Cohomology and homotopy</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> </li> </ul> <h2 id="in_higher_category_theory">In higher category theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-topos">(0,1)-topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%280%2C1%29-site">(0,1)-site</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-site">2-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-sheaf">2-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf">(∞,1)-sheaf</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, <a class="existingWikiWord" href="/nlab/show/derived+stack">derived stack</a></p> </li> </ul> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Diaconescu%27s+theorem">Diaconescu's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Barr%27s+theorem">Barr's theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/topos+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="formal_geometry">Formal geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/formal+geometry">formal geometry</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/local-global+principle">local-global principle</a></p> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/infinitesimal+ring+extension">infinitesimal ring extension</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Artin+algebra">Artin algebra</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/formal+neighbourhood">formal neighbourhood</a>, <a class="existingWikiWord" href="/nlab/show/formal+spectrum">formal spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/completion+of+a+ring">completion of a ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adic+topology">adic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> <p><a class="existingWikiWord" href="/nlab/show/formal+deformation+quantization">formal deformation quantization</a></p> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#axiomatics'>Axiomatics</a></li> <li><a href='#models'>Models</a></li> <ul> <li><a href='#wellAdaptedModels'>Well adapted models</a></li> </ul> <li><a href='#variations'>Variations</a></li> <ul> <li><a href='#higher_categorical_versions'>Higher categorical versions</a></li> <li><a href='#supergeometric_versions'>Supergeometric versions</a></li> </ul> <li><a href='#Constructions'>Constructions in synthetic differential geometry</a></li> <ul> <li><a href='#tangent_bundle'>Tangent bundle</a></li> <li><a href='#differential_equation'>Differential equation</a></li> <li><a href='#differential_forms'>Differential forms</a></li> <li><a href='#flow_of_a_vector_field'>Flow of a vector field</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#books'>Books</a></li> <li><a href='#expositions'>Expositions</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <blockquote> <p>Der unendlich kleinste Theil des Raumes ist immer ein Raum, etwas, das Continuität hat, nicht aber ein blosser Punct, oder die Grenze zwischen bestimmten Stellen im Raume; (<a href="Grundriss+des+Eigenthümlichen+der+Wissenschaftslehre#4IVUnendlichKleinsterTeilDesRaumes">Fichte 1795, Grundriss, §4.IV</a>)</p> </blockquote> <p>In <em>synthetic differential geometry</em> one formulates <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> <a class="existingWikiWord" href="/nlab/show/axiom">axiomatically</a> in <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a> – called <a class="existingWikiWord" href="/nlab/show/smooth+toposes">smooth toposes</a> – of <a class="existingWikiWord" href="/nlab/show/generalized+smooth+spaces">generalized smooth spaces</a> by assuming the explicit existence of <a class="existingWikiWord" href="/nlab/show/infinitesimal+neighbourhoods">infinitesimal neighbourhoods</a> of points.</p> <p>The main point of the axioms is to ensure that a well defined notion of the <a class="existingWikiWord" href="/nlab/show/infinitesimal+spaces">infinitesimal spaces</a> exists in the topos, whose existence concretely and usefully formalizes the wide-spread but often vague intuition about the role of infinitesimals in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>.</p> <p>In particular, in such toposes <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> there exists an <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> that behaves like the <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal interval</a> in such a way that for any space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">X \in E</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent 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>, is, again as an object of the topos, just the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mspace width="thickmathspace"></mspace><mtext>:=</mtext><mspace width="thickmathspace"></mspace><msup><mi>X</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">T X \;\text{:=}\; X^D</annotation></semantics></math> (using the notation of <a class="existingWikiWord" href="/nlab/show/exponential+object"> exponential objects</a> in the <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a> <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>). So a tangent vector in this context is literally an <em>infinitesimal path</em> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>This way, in <a class="existingWikiWord" href="/nlab/show/smooth+topos"> smooth toposes</a> it is possible to give precise well-defined meaning to many of the familiar computations – wide-spread in particular in the <a class="existingWikiWord" href="/nlab/show/physics">physics</a> literature – that compute with supposedly “infinitesimal” quantities.</p> <div class="num_remark" id="SophusLieQuote"> <h6 id="remark">Remark</h6> <p>As quoted by <a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a> in his <a href="https://users-math.au.dk/kock/sdg99.pdf#page=9">first book (p. 9)</a>, <a class="existingWikiWord" href="/nlab/show/Sophus+Lie">Sophus Lie</a> – one of the founding fathers of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> and, of course <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a> – once said that he found his main theorems in <a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a> using “synthetic reasoning”, but had to write them up in non-synthetic style (see <em><a class="existingWikiWord" href="/nlab/show/analytic+versus+synthetic">analytic versus synthetic</a></em>) just due to lack of a formalized language:</p> <blockquote> <p>“The reason why I have postponed for so long these investigations, which are basic to my other work in this field, is essentially the following: I found these theories originally by synthetic considerations. But I soon realized that, as expedient ( <em>zweckmässig</em> ) the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations, which lead to deal with objects that till now have almost exclusively been considered analytically. After long vacillations, I have decided to use a half synthetic, half analytic form. I hope my work will serve to bring justification to the synthetic method besides the analytical one.” (<a class="existingWikiWord" href="/nlab/show/Sophus+Lie">Sophus Lie</a>, <em>Allgemeine Theorie der partiellen Differentialgleichungen erster Ordnung</em>, Math. Ann. 9 (1876).)</p> </blockquote> </div> <p>Synthetic differential geometry provides this formalized language.</p> <div class="num_remark" id="Peirce"> <h6 id="remark_2">Remark</h6> <p>Another advocate of the use of infinitesimals in the late 19th century was the American philosopher <a class="existingWikiWord" href="/nlab/show/Charles+Sanders+Peirce">Charles Sanders Peirce</a>:</p> <blockquote> <p>The illumination of the subject by a strict notation for the logic of relatives had shown me clearly and evidently that the idea of an infinitesimal involves no contradiction…As a mathematician, I prefer the method of infinitesimals to that of limits, as far easier and less infested with snares. Charles Sanders Peirce, <em>The Law of Mind</em>, The Monist <strong>2</strong> (1892)</p> </blockquote> <p>According to Bell (<a href="#Bell98">1998, p.5</a>):</p> <blockquote> <p>Peirce was aware, even before Brouwer, that a faithful account of the truly continuous will involve jettisoning the law of excluded(.)</p> </blockquote> </div> <h2 id="axiomatics">Axiomatics</h2> <p>The axioms of synthetic differential geometry demand that the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <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> of smooth spaces is</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a> (see there for details)</li> </ul> <p>in which in particular</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space"> infinitesimal spaces</a> exist and</p> </li> <li> <p>satisfy the <a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a>.</p> </li> </ul> <p>Depending on applications one imposes further axioms, such as the</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a>.</li> </ul> <p>With that little bit of axiomatics alone, a large amount of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> may be formulated. This has been carried through quite comprehensively by <a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, see the <a href="#References">reference</a> below.</p> <p>In his work he particularly makes use of the fact that as sophisticated as a <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a> may be when explicitly constructed (see the section on <a href="#models">models</a>), being a <a class="existingWikiWord" href="/nlab/show/topos">topos</a> means that one can reason inside it almost literally as in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. Using this Kock’s work gives descriptions of synthetic differential geometry which are entirely intuitive and have no esoteric topos-theoretic flavor. All he needs is the assumption that the <a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a> is satisfied for “numbers”. Here “numbers” is really to be interpreted in the topos, but if one just accepts that they satisfy the KL axiom, one may work with infinitesimals in this context in essentially precisely the naive way, with the topos theory in the background just ensuring that everything makes good sense.</p> <h2 id="models">Models</h2> <p>Being axiomatic, reasoning in synthetic differential geometry applies in every <strong>model</strong> for the axioms, i.e. in every concrete choice of <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>.</p> <p>Models of <a class="existingWikiWord" href="/nlab/show/smooth+toposes">smooth toposes</a> tend to be inspired by, but more general than, constructions familiar from <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>. In particular the old insight promoted by <a class="existingWikiWord" href="/nlab/show/Grothendieck">Grothendieck</a> in his work, that <a class="existingWikiWord" href="/nlab/show/nilpotent+ideals">nilpotent ideals</a> in <a class="existingWikiWord" href="/nlab/show/rings">rings</a> are formal duals of spaces with infinitesimal extension is typically used to model <a class="existingWikiWord" href="/nlab/show/infinitesimal+spaces">infinitesimal spaces</a> in synthetic differential geometry.</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry+applied+to+algebraic+geometry">synthetic differential geometry applied to algebraic geometry</a></em> for more on this.</p> <p>The main difference between models for <a class="existingWikiWord" href="/nlab/show/smooth+toposes">smooth toposes</a> and <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a> from this perspective is that models for <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a> tend to employ test spaces that are <em>richer</em> than plain formal duals to commutative <a class="existingWikiWord" href="/nlab/show/ring"> rings</a> or algebras, as in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>: typical models for synthetic differential geometry use test spaces given by formal duals of <a class="existingWikiWord" href="/nlab/show/generalized+smooth+algebras">generalized smooth algebras</a> that remember “smooth structure” in the usual sense of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> (and different from, though not entirely unrelated to, the notion of <a class="existingWikiWord" href="/nlab/show/smooth+scheme">smooth scheme</a> in algebraic geometry). This is in particular true for the <a href="#wellAdaptedModels">well adapted models</a>.</p> <p>However, with a a sufficiently general perspective on <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> one finds that algebraic geometry and synthetic differential geometry are both special cases of a more general notion of theories of generalized spaces. For more on this see <a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a>.</p> <h3 id="wellAdaptedModels">Well adapted models</h3> <p>A <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <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> modelling the axioms of synthetic differential geometry is called <strong>(well) adapted</strong> if the ordinary <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> of <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> embeds into it, in particular if there is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a> <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\to E</annotation></semantics></math> from the category of ordinary <a class="existingWikiWord" href="/nlab/show/smooth+manifold"> smooth manifolds</a> into <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>.</p> <p>A standard model for well adapted synthetic toposes is obtained in terms of sheaves on duals of “germ determined” <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>-rings. This is described in great detail in the textbook <em><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></em>.</p> <p>The conception and discussion of these well adapted toposes goes back to <a class="existingWikiWord" href="/nlab/show/Eduardo+Dubuc">Eduardo Dubuc</a>, who studied them in a long series of articles. He <a href="http://north.ecc.edu/alsani/ct99-00%288-12%29/msg00218.html">asks</a> people to refer to this topos as the <strong><a class="existingWikiWord" href="/nlab/show/Dubuc+topos">Dubuc topos</a></strong>.</p> <p>This theory of well-adapted models was later summarized and extended in the standard textbook</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <a class="existingWikiWord" href="/nlab/show/Gonzalo+Reyes">Gonzalo Reyes</a>: <em><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></em>, Springer (1991) [<a href="https://doi.org/10.1007/978-1-4757-4143-8">doi:10.1007/978-1-4757-4143-8</a>]</li> </ul> <h2 id="variations">Variations</h2> <h3 id="higher_categorical_versions">Higher categorical versions</h3> <ul> <li>Synthetic differential geometry may be thought of as embedded in the general theory of <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifolds">derived smooth manifolds</a> and, generally, that of <a class="existingWikiWord" href="/nlab/show/generalized+schemes">generalized schemes</a>.</li> </ul> <h3 id="supergeometric_versions">Supergeometric versions</h3> <p>The notion of synthetic differential geometry extends to the context of <a class="existingWikiWord" href="/nlab/show/supergeometry+-+contents">supergeometry</a>. See</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic differential supergeometry</a>.</li> </ul> <h2 id="Constructions">Constructions in synthetic differential geometry</h2> <h3 id="tangent_bundle">Tangent bundle</h3> <p>The <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a> of an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a> is just the <a class="existingWikiWord" href="/nlab/show/exponential+object">exponential object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>:</mo><mo>=</mo><msup><mi>X</mi> <mi>D</mi></msup></mrow><annotation encoding="application/x-tex">T X := X^D</annotation></semantics></math>. The unique inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">{*} \to D</annotation></semantics></math> induces a canonical projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">T X \to X</annotation></semantics></math>. A <a class="existingWikiWord" href="/nlab/show/section">section</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>T</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">X \to T X</annotation></semantics></math> of that projection is a <a class="existingWikiWord" href="/nlab/show/tangent+vector+field">tangent vector field</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>. Its <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>→</mo><msup><mi>X</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">D \to X^X</annotation></semantics></math> that sends the unique point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> to the identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Id</mi> <mi>X</mi></msub><mo>∈</mo><msup><mi>X</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">Id_X \in X^X</annotation></semantics></math>.</p> <h3 id="differential_equation">Differential equation</h3> <p>A <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a> is an extension problem in a <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a> along a morphism that includes an <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal object</a> into another object.</p> <p>For instance the ordinary first order homogeneous differential equation that asks the derivative of a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">f : X \to A</annotation></semantics></math> along a <a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>:</mo><mi>D</mi><mo>→</mo><msup><mi>X</mi> <mi>X</mi></msup></mrow><annotation encoding="application/x-tex">v : D \to X^X</annotation></semantics></math> to be given by a specified map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>T</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">\alpha: X \to T A</annotation></semantics></math> is given by a diagram of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>D</mi></mtd> <mtd><mover><mo>→</mo><mi>v</mi></mover></mtd> <mtd><msup><mi>X</mi> <mi>X</mi></msup></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>α</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>f</mi></mpadded></msub></mtd></mtr> <mtr><mtd><msup><mi>A</mi> <mi>X</mi></msup></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ D &\stackrel{v}{\to}& X^X \\ {}^{\mathllap{\alpha}}\downarrow & \swarrow_{\mathrlap{f}} \\ A^X } \,, </annotation></semantics></math></div> <p>where we have freely identified morphisms with their <a class="existingWikiWord" href="/nlab/show/adjunct"> adjuncts</a>. See <a class="existingWikiWord" href="/nlab/show/differential+equation">differential equation</a> for details.</p> <h3 id="differential_forms">Differential forms</h3> <p>A <a class="existingWikiWord" href="/nlab/show/differential+form">differential 1-form</a> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>:</mo><mi>T</mi><mi>X</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\omega : T X \to R</annotation></semantics></math> that is “fiberwise linear”. One elegant way to say this is obtained by considering all higher differential forms at once:</p> <p>for a sufficiently well behaved object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, there is the <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> which is the <a class="existingWikiWord" href="/nlab/show/infinitesimal+singular+simplicial+complex">infinitesimal singular simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mi>inf</mi> <mo>•</mo></msubsup><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">X^{(\Delta^\bullet_{inf})}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Taking functions on this produces the <a class="existingWikiWord" href="/nlab/show/cosimplicial+algebra">cosimplicial algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><msubsup><mi>Δ</mi> <mi>inf</mi> <mo>•</mo></msubsup></mrow></msup><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(X^{\Delta^\bullet_{inf}}, R)</annotation></semantics></math>. Its <a class="existingWikiWord" href="/nlab/show/Moore+complex">normalized Moore cochain complex</a> is isomorphic to the <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham dg-algebra</a> of differential forms on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>N</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>Hom</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><mo stretchy="false">(</mo><msubsup><mi>Δ</mi> <mo>•</mo> <mi>inf</mi></msubsup><mo stretchy="false">)</mo></mrow></msup><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>Ω</mi> <mi>dR</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N^\bullet(Hom(X^{(\Delta^{inf}_\bullet)},R)) = \Omega^\bullet_{dR}(X) \,. </annotation></semantics></math></div> <p>This is discussed at</p> <ul> <li> <p><a href="http://ncatlab.org/nlab/show/infinitesimal+object#SpacOfInfSimpl">Spaces of infinitesimal k-simplices</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms in synthetic differential geometry</a>.</p> </li> </ul> <h3 id="flow_of_a_vector_field">Flow of a vector field</h3> <p>See at <a href="flow+of+a+vector+field#SyntheticDefinition">flow of a vector field</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+mathematics">synthetic mathematics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+geometry">synthetic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+topology">synthetic topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+algebraic+geometry">synthetic algebraic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+topology">synthetic differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+homotopy+theory">synthetic homotopy theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+%28infinity%2C1%29-category+theory">synthetic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>∞</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(\infty,1)</annotation> </semantics> </math>-category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+probability+theory">synthetic probability theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+differential+geometry">derived differential geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/synthetic+differential+infinity-groupoid">synthetic differential infinity-groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topos+of+laws+of+motion">topos of laws of motion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonstandard+analysis">nonstandard analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+topology">synthetic differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/logical+topology">logical topology</a></p> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometries of physics</a></strong></p> <table><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>(<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher</a>) <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/site">site</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-topos">∞-sheaf ∞-topos</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/discrete+geometry">discrete geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/terminal+category">Point</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Set">Set</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Discrete%E2%88%9EGroupoid">Discrete∞Grpd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/SmoothSet">SmoothSet</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGroupoid">Smooth∞Grpd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">formal geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FormalCartSp">FormalCartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FormalSmoothSet">FormalSmoothSet</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FormalSmooth%E2%88%9EGroupoid">FormalSmooth∞Grpd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/SuperFormalCartSp">SuperFormalCartSp</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/SuperFormalSmoothSet">SuperFormalSmoothSet</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/SuperFormalSmooth%E2%88%9EGroupoid">SuperFormalSmooth∞Grpd</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div> <h2 id="References">References</h2> <p>The idea of axiomatizing <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> using ideas inspired by <a class="existingWikiWord" href="/nlab/show/topos">topos</a> theory originates in</p> <ul> <li id="Lawvere67"><a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Bill Lawvere</a>, <em><a class="existingWikiWord" href="/nlab/show/Categorical+dynamics">Categorical dynamics</a></em>, lecture (Chicago 1967). (<a href="http://www.mat.uc.pt/~ct2011/abstracts/lawvere_w.pdf">abstract pdf</a>)</li> </ul> <p>They were published as pp.1-28 in</p> <ul> <li id="Kocke79"><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a> (ed.), <em>Topos Theoretic Methods in Geometry</em> , Aarhus Univ. Var. Pub. Ser. 30 (1979).</li> </ul> <p>The first model for the axioms presented there served to demonstrate that the theory is non-empty, but was hard to work with. Much of the later work was concerned with refining the model-building. For instance</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Eduardo+Dubuc">Eduardo Dubuc</a>, <em>Sur les modèles de la géométrie différentielle synthétique</em>, Cahier Top et Géom. Diff. <strong>XX-3</strong> (1979) pp.231-279. (<a href="http://archive.numdam.org/article/CTGDC_1979__20_3_231_0.pdf">pdf</a>)</li> </ul> <p>These models are constructed in terms of <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> <a class="existingWikiWord" href="/nlab/show/topos"> toposes</a> on the category of <a class="existingWikiWord" href="/nlab/show/smooth+loci">smooth loci</a>, formal duals to <a class="existingWikiWord" href="/nlab/show/generalized+smooth+algebra">C^∞-rings</a>. See there for a detailed list of references.</p> <p>Transcripts or notes of further talks by Bill Lawvere on the subject are</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Bill Lawvere</a>, <em><a class="existingWikiWord" href="/nlab/show/Toposes+of+laws+of+motion">Toposes of laws of motion</a></em> , transcript of a talk in Montreal, Sept. 1997 (<a class="existingWikiWord" href="/nlab/files/LawvereToposesOfLawsOfMotions.pdf" title="pdf">pdf</a>)</p> <p>(on the description of <a class="existingWikiWord" href="/nlab/show/differential+equation"> differential equations</a> in terms of synthetic differential geometry)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">Bill Lawvere</a>, <em>Outline of synthetic differential geometry</em> , lectures in Buffalo (1998). (<a class="existingWikiWord" href="/nlab/files/LawvereSDGOutline.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>Two articles that exhibit the link to <a class="existingWikiWord" href="/nlab/show/continuum+mechanics">continuum mechanics</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">F. W. Lawvere</a>, <em>Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body</em> , Cah. Top. Géom. Diff. Cat. <strong>21</strong> no.4 (1980) pp.377-392. (<a href="http://archive.numdam.org/article/CTGDC_1980__21_4_377_0.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bill+Lawvere">F. W. Lawvere</a>, <em>Categorical algebra for continuum microphysics</em> , JPAA <strong>175</strong> (2002) pp.267-287.</p> </li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Marta+Bunge">Marta Bunge</a>, <a class="existingWikiWord" href="/nlab/show/Eduardo+Dubuc">Eduardo Dubuc</a>, <em>Archimedian local <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>-rings and models of synthetic differential geometry</em> Cahiers de Topologie et Géométrie Différentielle Catégoriques, <strong>XXVII-3</strong> (1986) pp.3-22. (<a href="http://www.numdam.org/item?id=CTGDC_1986__27_3_3_0">numdam</a>).</li> </ul> <p>For the early French connection see:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Weil">André Weil</a>, <em>Théorie des points proches sur les variétés différentiables</em> , Colloq. Top. et Géom. Diff., Strasbourg (1953) pp.111-117.</p> </li> <li> <p>Jean Penon, <em>De l’infinitésimal au local (Thèse de Doctorat d’État)</em> Diagrammes <strong>S13</strong> (1985), pp.1-191. (<a href="http://archive.numdam.org/article/DIA_1985__S13__1_0.pdf">pdf</a>)</p> </li> </ul> <p>Discussion in <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> is in</p> <ul> <li><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>)</li> </ul> <p>Discussion in differentially cohesive <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> is in</p> <ul> <li id="Wellen17"> <p><a class="existingWikiWord" href="/nlab/show/Felix+Wellen">Felix Wellen</a>, <em><a class="existingWikiWord" href="/schreiber/show/thesis+Wellen">Formalizing Cartan Geometry in Modal Homotopy Type Theory</a></em>, 2017</p> </li> <li id="JazMyers22"> <p><a class="existingWikiWord" href="/nlab/show/David+Jaz+Myers">David Jaz Myers</a>, <em>Orbifolds as microlinear types in synthetic differential cohesive homotopy type theory</em> [<a href="https://arxiv.org/abs/2205.15887">arXiv:2205.15887</a>]</p> </li> </ul> <h3 id="books">Books</h3> <p>A nice elementary introduction which emphasizes calculations and the application as <em>engineering mathematics</em> can be found in</p> <ul> <li id="Bell98"><a class="existingWikiWord" href="/nlab/show/John+Lane+Bell">John Lane Bell</a>, <em>A Primer of Infinitesimal Analysis</em>, Cambridge UP 1998 (<a href="https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/primer-infinitesimal-analysis-2nd-edition?format=HB&isbn=9780521887182">ISBN:9780521887182</a>)</li> </ul> <p>The textbooks</p> <ul> <li id="Kock81"> <p><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>Synthetic Differential Geometry</em>, Cambridge University Press (1981, 2006) [<a href="https://users-math.au.dk/kock/sdg99.pdf">pdf</a>, <a href="https://doi.org/10.1017/CBO9780511550812">doi:10.1017/CBO9780511550812</a>]</p> </li> <li id="Kock10"> <p><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>Synthetic geometry of manifolds</em>, Cambridge Tracts in Mathematics <strong>180</strong> (2010) [<a href="https://doi.org/10.1017/CBO9780511691690">doi:10.1017/CBO9780511691690</a>, <a href="https://tildeweb.au.dk/au76680/SGM-final.pdf">pdf</a>]</p> </li> </ul> <p>develop the theory of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> using the axioms of synthetic differential geometry. The main goal in these books is to demonstrate how these axioms lead to a very elegant, very intuitive and very comprehensive conception of differential geometry. Accordingly, concrete models (whose explicit description is typically much more evolved than the nice axiomatics that holds once they have been constructed) play a minor role in these books.</p> <p>Somewhat complementary to that, the book</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <a class="existingWikiWord" href="/nlab/show/Gonzalo+Reyes">Gonzalo Reyes</a>, <em><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></em> , Springer Heidelberg 1991.</li> </ul> <p>focuses on concrete constructions of well-adapted models using the technology of <a class="existingWikiWord" href="/nlab/show/generalized+smooth+algebra"> generalized smooth algebras</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>-rings).</p> <p>Another textbook is</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ren%C3%A9+Lavendhomme">René Lavendhomme</a>, <em>Basic concepts of synthetic differential geometry</em>, Kluwer Texts in the Mathematical Sciences <strong>13</strong>, Springer (1996) [<a href="https://doi.org/10.1007/978-1-4757-4588-7">doi:10.1007/978-1-4757-4588-7</a>]</li> </ul> <h3 id="expositions">Expositions</h3> <p>Introductory survey includes</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>Synthetic differential geometry - new methods for old spaces</em>, talk at <em><a class="existingWikiWord" href="/nlab/show/New+Spaces+for+Mathematics+and+Physics">New Spaces for Mathematics and Physics</a></em>, IHP Paris (2015) [<a href="https://www.youtube.com/watch?v=AXz7xu3WrPE">video recording</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a>, <em>New methods for old spaces: synthetic differential geometry</em> [<a href="https://arxiv.org/abs/1610.00286">arxiv/1610.00286</a>]</p> </li> </ul> <p>Introductory expositions of basic ideas of synthetic differential geometry are</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Mike+Shulman">Mike Shulman</a>, <em>Chicago Pizza-Seminar: Synthetic Differential Geometry</em> (<a href="http://home.sandiego.edu/~shulman/papers/sdg-pizza-seminar.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Lane+Bell">John Lane Bell</a>, <em>An invitation to smooth infinitesimal analysis</em> (<a href="http://publish.uwo.ca/~jbell/invitation%20to%20SIA.pdf">pdf</a>)</p> </li> <li> <p>R. P. Kostecki, <em>Differential Geometry in Toposes</em>, ms. University of Warsaw (2009) <a href="http://www.fuw.edu.pl/~kostecki/sdg.pdf">pdf</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></em></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 25, 2024 at 20:05:10. 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