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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="cohesive_toposes">Cohesive toposes</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></strong></p> <p><strong>Backround</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+cohesive+toposes">motivation for cohesive toposes</a></p> </li> </ul> <p><strong>Definition</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">strongly ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">totally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a> / <a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> / <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <p><strong>Presentation over a site</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+site">locally connected site</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+site">locally ∞-connected site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+site">connected site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+site">∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+site">strongly connected site</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+site">strongly ∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+site">totally connected site</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+site">totally ∞-connected site</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+site">local site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-local+site">∞-local site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-cohesive+site">∞-cohesive site</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, <a class="existingWikiWord" href="/nlab/show/discrete+groupoid">discrete groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-topological+%E2%88%9E-groupoid">D-topological ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/topological+groupoid">topological groupoid</a>, <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>, <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-groupoid">Lie 2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+super+%E2%88%9E-groupoid">smooth super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a>, <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>, <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a>, <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/synthetic+differential+super+%E2%88%9E-groupoid">synthetic differential super ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion+of+global-+over+G-equivariant+homotopy+theory">cohesion of global- over G-equivariant homotopy theory</a></strong></p> </li> </ul> </div></div> </div> </div> <h1 id="smooth_spaces">Smooth spaces</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#Properties'>Properties</a></li> <ul> <li><a href='#Cohesion'>Cohesion</a></li> <li><a href='#topos_points_and_stalks'>Topos points and stalks</a></li> <li><a href='#distribution_theory'>Distribution theory</a></li> </ul> <li><a href='#VariantsAndGeneralizations'>Variants and generalizations</a></li> <ul> <li><a href='#synthetic_differential_geometry'>Synthetic differential geometry</a></li> <li><a href='#higher_smooth_geometry'>Higher smooth geometry</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>The concept of a <em>smooth set</em> or <em>smooth space</em>, in the sense discussed here, is a generalization of that of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> beyond that of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>: A smooth set is a <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a> that may be <em>probed</em> by smooth <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>.</p> <p>For expository details see at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">geometry of physics – smooth sets</a></em>.</p> <p>Alternatively, the smooth test spaces may be taken to be more generally all <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>. But since <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> themselves are built from gluing together smooth <a class="existingWikiWord" href="/nlab/show/open+balls">open balls</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>D</mi> <mi>int</mi> <mi>n</mi></msubsup><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n_{int} \subset \mathbb{R}^n</annotation></semantics></math> or equivalently <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>, one may just as well consider Cartesian spaces test spaces. Finally, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> is diffeomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>, one can just as well take just the cartesian smooth spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> as test objects.</p> <h2 id="definition">Definition</h2> <p>The category of <strong>smooth spaces</strong> is the <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothSp</mi><mo>:</mo><mo>=</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Diff</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> SmoothSp := Sh(Diff) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a> on the <a class="existingWikiWord" href="/nlab/show/site">site</a> <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> of smooth manifolds equipped with its standard <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> (<a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>) given by open covers of manifolds.</p> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi></mrow><annotation encoding="application/x-tex">Diff</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the category of <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> <em>embedded</em> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^\infty</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi></mrow><annotation encoding="application/x-tex">Diff</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/essentially+small+category">essentially small category</a>, so there are no size issues involved in this definition.</p> <p>But since manifolds themselves are defined in terms of gluing conditons, the <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSp</mi></mrow><annotation encoding="application/x-tex">SmoothSp</annotation></semantics></math> depends on much less than all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi></mrow><annotation encoding="application/x-tex">Diff</annotation></semantics></math>.</p> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ball</mi><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msubsup><mi>D</mi> <mi>int</mi> <mi>n</mi></msubsup><mo>→</mo><msubsup><mi>D</mi> <mi>int</mi> <mi>m</mi></msubsup><mo stretchy="false">)</mo><mo>∈</mo><mi>Diff</mi><mo stretchy="false">|</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>∈</mo><mi>ℕ</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> Ball := \{ (D^n_{int} \to D^m_{int}) \in Diff | n,m \in \mathbb{N}\} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>m</mi></msup><mo stretchy="false">)</mo><mo>∈</mo><mi>Diff</mi><mo stretchy="false">|</mo><mi>n</mi><mo>,</mo><mi>m</mi><mo>∈</mo><mi>ℕ</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> CartSp := \{ (\mathbb{R}^n \to \mathbb{R}^m) \in Diff | n,m \in \mathbb{N}\} </annotation></semantics></math></div> <p>be the full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategories</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ball</mi></mrow><annotation encoding="application/x-tex">Ball</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi></mrow><annotation encoding="application/x-tex">Diff</annotation></semantics></math> on open balls and on cartesian spaces, respectively. Then the corresponding <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> <a class="existingWikiWord" href="/nlab/show/topos">toposes</a> are still those of smooth spaces:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>SmoothSp</mi></mtd> <mtd><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Ball</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} SmoothSp &\simeq Sh(Ball) \\ & \simeq Sh(CartSp) \end{aligned} \,. </annotation></semantics></math></div> <h2 id="examples">Examples</h2> <ul> <li> <p>The category of ordinary <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> is a full subcategory of smooth spaces:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Diff</mi><mo>↪</mo><mi>SmoothSp</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Diff \hookrightarrow SmoothSp \,. </annotation></semantics></math></div> <p>When one regards smooth spaces concretely as sheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi></mrow><annotation encoding="application/x-tex">Diff</annotation></semantics></math>, then this inclusion is of course just the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>.</p> </li> <li> <p>The full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DiffSp</mi><mo>⊂</mo><mi>SmoothSp</mi></mrow><annotation encoding="application/x-tex"> DiffSp \subset SmoothSp </annotation></semantics></math></div> <p>on <a class="existingWikiWord" href="/nlab/show/concrete+sheaf">concrete sheaves</a> is called the category of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>.</p> <ul> <li> <p>The standard class of examples of smooth spaces that motivate their use even in cases where one starts out being intersted just in <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> are <strong>mapping spaces</strong>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> two smooth spaces (possibly just ordinary smooth manifolds), by the <a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a> the <strong>mapping space</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math>, i.e. the space of smooth maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math> exists again naturally as a smooth. By the general formula it is given as a <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> by the assignment</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo><mo>:</mo><mi>U</mi><mo>↦</mo><mi>SmoothSp</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo>×</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\Sigma,X] : U \mapsto SmoothSp(\Sigma \times U, X) \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <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> are ordinary manifolds, then the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> on the right sits inside that of the underlying sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSp</mi><mo stretchy="false">(</mo><mi>Σ</mi><mo>×</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>Set</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>Σ</mi><mo stretchy="false">|</mo><mo>×</mo><mo stretchy="false">|</mo><mi>U</mi><mo stretchy="false">|</mo><mo>,</mo><mo stretchy="false">|</mo><mi>X</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SmoothSp(\Sigma \times U , X) \subset Set(|\Sigma| \times |U|, |X| )</annotation></semantics></math> so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X]</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>.</p> <p>The above formula says that a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-parameterized family of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Sigma \to X</annotation></semantics></math> is smooth as a map into the smooth space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>Σ</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\Sigma,X ]</annotation></semantics></math> precisely if the corresponding map of sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>×</mo><mi>Σ</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \times \Sigma \to X</annotation></semantics></math> is an ordinary morphism of smooth manifolds.</p> </li> </ul> </li> <li> <p>The canonical examples of smooth spaces that are not diffeological spaces are the sheaves of (closed) differential forms:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>K</mi> <mi>n</mi></msup><mo>:</mo><mi>U</mi><mo>↦</mo><msubsup><mi>Ω</mi> <mi>closed</mi> <mi>n</mi></msubsup><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K^n : U \mapsto \Omega^n_{closed}(U) \,. </annotation></semantics></math></div></li> <li> <p>The category</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SimpSmoothSp</mi><mo>:</mo><mo>=</mo><msup><mi>SmoothSp</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex"> SimpSmoothSp := SmoothSp^{\Delta^{op}} </annotation></semantics></math></div> <p>equivalently that of sheaves on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi></mrow><annotation encoding="application/x-tex">Diff</annotation></semantics></math> with values in <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Diff</mi><mo>,</mo><mi>SSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \cdots \simeq Sh(Diff, SSet) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/simplicial+objects">simplicial objects</a> in smooth spaces naturally carries the structure of a <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> (for instance the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">model structure on simplicial sheaves</a> or that of a Brown <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> (if one restricts to locally Kan simplicial sheaves)) and as such is a <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentation</a> for the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-stacks">smooth ∞-stacks</a>.</p> </li> </ul> <h2 id="Properties">Properties</h2> <h3 id="Cohesion">Cohesion</h3> <div class="num_prop" id="SmoothSetsFormACohesiveTopos"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> form a <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi></mrow><annotation encoding="application/x-tex">SmoothSet</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> is a <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></p> <div class="maruku-equation" id="eq:SheafToposAdjointQuadruple"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mrow><mtable><mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AAA</mi></mphantom><msub><mi>Π</mi> <mn>0</mn></msub><mphantom><mi>AAA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>Disc</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟶</mo><mrow><mphantom><mi>AAA</mi></mphantom><mi>Γ</mi><mphantom><mi>AAA</mi></mphantom></mrow></mover></mtd></mtr> <mtr><mtd><mover><mo>⟵</mo><mrow><mphantom><mi>AA</mi></mphantom><mi>coDisc</mi><mphantom><mi>AA</mi></mphantom></mrow></mover></mtd></mtr></mtable></mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex"> SmoothSet \array{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\longleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\longleftarrow} } Set </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>First of all (by <a href="geometry+of+physics+--+smooth+sets#CategoryOfSmoothSets">this Prop</a>) smooth sets indeed form a <a class="existingWikiWord" href="/nlab/show/sheaf+topos">sheaf topos</a>, over the <a class="existingWikiWord" href="/nlab/show/site">site</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> between them, and equipped with the <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> of differentiably-<a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a> (<a href="geometry+of+physics+--+smooth+sets#DifferentiallyGoodOpenCover">this def.</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> SmoothSet \simeq Sh(CartSp) \,. </annotation></semantics></math></div> <p>Hence, by Prop. <a class="maruku-ref" href="#CategoriesOfSheavesOnCohesiveSiteIsCohesive"></a>, it is now sufficient to see that <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> is a <a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a> (Def. <a class="maruku-ref" href="#OneCohesiveSite"></a>).</p> <p>It clearly has <a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a>: The <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> is the <a class="existingWikiWord" href="/nlab/show/point">point</a>, given by the 0-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>=</mo><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex"> \ast = \mathbb{R}^0 </annotation></semantics></math></div> <p>and the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of two <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> is the Cartesian space whose <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> is the <a class="existingWikiWord" href="/nlab/show/sum">sum</a> of the two separate dimensions:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub></mrow></msup><mo>×</mo><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msup><mi>ℝ</mi> <mrow><msub><mi>n</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>n</mi> <mn>2</mn></msub></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{n_1} \times \mathbb{R}^{n_2} \;\simeq\; \mathbb{R}^{ n_1 + n_2 } \,. </annotation></semantics></math></div> <p>This establishes the first clause in Def. <a class="maruku-ref" href="#OneCohesiveSite"></a>.</p> <p>For the second clause, consider a differentiably-<a class="existingWikiWord" href="/nlab/show/good+open+cover">good open cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mover><mo>→</mo><mrow></mrow></mover><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \overset{}{\to} \mathbb{R}^n\}</annotation></semantics></math> (<a href="geometry+of+physics+--+smooth+sets#DifferentiallyGoodOpenCover">this def.</a>). This being a <a class="existingWikiWord" href="/nlab/show/good+cover">good cover</a> implies that its <a class="existingWikiWord" href="/nlab/show/Cech+groupoid">Cech groupoid</a> is, as an <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> (via <a href="geometry+of+physics+--+categories+and+toposes#PresheavesOfGroupoidsAsInternalGroupoidsInPresheaves">this remark</a>), of the form</p> <div class="maruku-equation" id="eq:CechGroupoidForCartSp"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><munder><mo>∩</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C(\{U_i\}_i) \;\simeq\; \left( \array{ \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} y(U_i) } \right) \,. </annotation></semantics></math></div> <p>where we used the defining property of <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a> to identify <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><msub><mo>×</mo> <mi>X</mi></msub><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo>∩</mo> <mi>X</mi></msub><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(U_i) \times_X y(U_j) \simeq y( U_i \cap_X U_j )</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of <a class="maruku-eqref" href="#eq:CechGroupoidForCartSp">(2)</a>, regarded just as a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> of <a class="existingWikiWord" href="/nlab/show/reflexive+graph">reflexive</a> <a class="existingWikiWord" href="/nlab/show/directed+graph">directed</a> <a class="existingWikiWord" href="/nlab/show/graphs">graphs</a> (hence ignoring <a class="existingWikiWord" href="/nlab/show/composition">composition</a> for the moment), is readily seen to be the <a class="existingWikiWord" href="/nlab/show/graph">graph</a> of the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of the components (the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> follows immediately from that of the component colimits):</p> <div class="maruku-equation" id="eq:ColimitOfCechGroupoidOverCartSp"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><munder><mo>∩</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><munder><mo>∩</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mo>*</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><mo>*</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \array{ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} y(U_i) } \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} \ast \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} \ast } \right) \end{aligned} \,. </annotation></semantics></math></div> <p>Here we first used that <a class="existingWikiWord" href="/nlab/show/colimits+commute+with+colimits">colimits commute with colimits</a>, hence in particular with <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> (<a href="geometry+of+physics+--+categories+and+toposes#LimitsCommuteWithLimits">this prop.</a>) and then that the colimit of a <a class="existingWikiWord" href="/nlab/show/representable+presheaf">representable presheaf</a> is <a class="existingWikiWord" href="/nlab/show/generalized+the">the</a> <a class="existingWikiWord" href="/nlab/show/singleton">singleton</a> set (<a href="geometry+of+physics+--+categories+and+toposes#ColimitOfRepresentableIsSingleton">this Lemma</a>).</p> <p>This colimiting <a class="existingWikiWord" href="/nlab/show/graph">graph</a> carries a unique <a class="existingWikiWord" href="/nlab/show/composition">composition</a> structure making it a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, since there is at most one morphism between any two objects, and every object carries a morphism from itself to itself. This implies that this groupoid is actually the colimiting groupoid of the Cech groupoid: hence the groupoid obtained from replacing each representable summand in the Cech groupoid by a point.</p> <p>Precisely this operation on <a class="existingWikiWord" href="/nlab/show/Cech+groupoids">Cech groupoids</a> of <a class="existingWikiWord" href="/nlab/show/good+open+covers">good open covers</a> of <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> is what <em><a class="existingWikiWord" href="/nlab/show/Borsuk%27s+nerve+theorem">Borsuk's nerve theorem</a></em> is about, a classical result in <a class="existingWikiWord" href="/nlab/show/topology">topology</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>. This theorem implies directly that the set of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> of the groupoid <a class="maruku-eqref" href="#eq:ColimitOfCechGroupoidOverCartSp">(4)</a> is in <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> with the set of <a class="existingWikiWord" href="/nlab/show/connected+components">connected components</a> of the <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>, regarded as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. But this is evidently a <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected topological space</a>, which finally shows that, indeed</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><munder><mi>lim</mi><munder><mo>⟶</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0 \underset{\underset{CartSp^{op}}{\longrightarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; \ast \,. </annotation></semantics></math></div> <p>The second item of the second clause in Def. <a class="maruku-ref" href="#OneCohesiveSite"></a> follows similarly, but more easily: The <a class="existingWikiWord" href="/nlab/show/limit">limit</a> of the <a class="existingWikiWord" href="/nlab/show/Cech+groupoid">Cech groupoid</a> is readily seen to be, as before, the unique groupoid structure on the limiting underlying graph of presheaves. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>=</mo><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\ast = \mathbb{R}^0</annotation></semantics></math>, which is hence an <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CartSp^{op}</annotation></semantics></math>, limits over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CartSp^{op}</annotation></semantics></math> yield simply the evaluation on that object:</p> <div class="maruku-equation" id="eq:ColimitOfCechGroupoidOverCartSp"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><munder><mo>∩</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><mi>y</mi><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom><mo>)</mo></mrow></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>Hom</mi> <mi>CartSp</mi></msub><mrow><mo>(</mo><mo>*</mo><mo>,</mo><msub><mi>U</mi> <mi>i</mi></msub><munder><mo>∩</mo><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow></munder><msub><mi>U</mi> <mi>j</mi></msub><mo>)</mo></mrow></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mo maxsize="1.2em" minsize="1.2em">↑</mo><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><msub><mi>Hom</mi> <mi>CartSp</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) & \simeq \left( \array{ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i,j}{\coprod} y(U_i \underset{\mathbb{R}^n}{\cap} U_j) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} \underset{i}{\coprod} y(U_i) } \phantom{A} \right) \\ & \simeq \left( \array{ \underset{i,j}{\coprod} Hom_{CartSp}\left( \ast, U_i \underset{\mathbb{R}^n}{\cap} U_j \right) \\ \big\downarrow \big\uparrow \big\downarrow \\ \underset{i}{\coprod} Hom_{CartSp}( \ast, U_i ) } \right) \end{aligned} \,. </annotation></semantics></math></div> <p>Here we used that <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> (here <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a>) of <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> are computed objectwise, and then the definition of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math>.</p> <p>But the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> induced by this graph on its set of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>i</mi></munder><msub><mi>Hom</mi> <mi>CartSp</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{i}{\coprod} Hom_{CartSp}( \ast, U_i )</annotation></semantics></math> precisely identifies pairs of points, one in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> the other in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">U_j</annotation></semantics></math>, that are actually the same point of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> being covered. Hence the set of <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> is the set of points of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>, which is just what remained to be shown:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><munder><mi>lim</mi><munder><mo>⟵</mo><mrow><msup><mi>CartSp</mi> <mi>op</mi></msup></mrow></munder></munder><mi>C</mi><mo stretchy="false">(</mo><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo stretchy="false">}</mo> <mi>i</mi></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>CartSp</mi></msub><mo stretchy="false">(</mo><mo>*</mo><mo>,</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0 \underset{\underset{CartSp^{op}}{\longleftarrow}}{\lim} C(\{U_i\}_i) \;\simeq\; Hom_{CartSp}(\ast, \mathbb{R}^n) \,. </annotation></semantics></math></div></div> <h3 id="topos_points_and_stalks">Topos points and stalks</h3> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">n \in N</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">topos point</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mo>:</mo><mi>Set</mi><mover><mover><mo>→</mo><mrow><msubsup><mi>D</mi> <mo>*</mo> <mi>n</mi></msubsup></mrow></mover><mover><mo>←</mo><mrow><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></mover></mover><mi>SmoothSp</mi></mrow><annotation encoding="application/x-tex"> D^n : Set \stackrel{\stackrel{(D^n)^*}{\leftarrow}} {\stackrel{D^n_*}{\to}} SmoothSp </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/inverse+image">inverse image</a> morphism – the <a class="existingWikiWord" href="/nlab/show/stalk">stalk</a> – is given on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>SmoothSp</mi></mrow><annotation encoding="application/x-tex">A \in SmoothSp</annotation></semantics></math> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><msup><mo stretchy="false">)</mo> <mo>*</mo></msup><mi>A</mi><mo>:</mo><mo>=</mo><msub><mo lspace="0em" rspace="thinmathspace">colim</mo> <mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>⊃</mo><mi>U</mi><mo>∋</mo><mn>0</mn></mrow></msub><mi>A</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (D^n)^* A := \colim_{\mathbb{R}^n \supset U \ni 0} A(U) \,, </annotation></semantics></math></div> <p>where the colimit is over all open neighbourhoods of the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>.</p> </div> <div class="num_lemma"> <h6 id="lemma_2">Lemma</h6> <p>SmoothSp has <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a>: they are given by the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">D^n</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>.</p> </div> <h3 id="distribution_theory">Distribution theory</h3> <p>Since a space of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> is canonically a smooth set, it is natural to consider the <em>smooth</em> <a class="existingWikiWord" href="/nlab/show/linear+functionals">linear functionals</a> on such <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a>. These turn out to be equivalent to the <a class="existingWikiWord" href="/nlab/show/continuous+linear+functionals">continuous linear functionals</a>, hence to <a class="existingWikiWord" href="/nlab/show/distributional+densities">distributional densities</a>. See at <em><a class="existingWikiWord" href="/nlab/show/distributions+are+the+smooth+linear+functionals">distributions are the smooth linear functionals</a></em> for details.</p> <h2 id="VariantsAndGeneralizations">Variants and generalizations</h2> <h3 id="synthetic_differential_geometry">Synthetic differential geometry</h3> <p>The <a class="existingWikiWord" href="/nlab/show/site">site</a> <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><msub><mrow></mrow> <mi>smooth</mi></msub></mrow><annotation encoding="application/x-tex">{}_{smooth}</annotation></semantics></math> may be replaced by the site <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><msub><mrow></mrow> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">{}_{th}</annotation></semantics></math> (see there) whose objects are products of smooth Cartesian spaces with <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+points">infinitesimally thickened points</a>. The corresponding <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><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>CartSp</mi> <mi>th</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(CartSp_{th})</annotation></semantics></math> is called the <em><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></em>. It contains smooth spaces with possibly infinitesimal extension and is a model for <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a> (a “<a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>”), which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(CartSp)</annotation></semantics></math> is not.</p> <p>The two toposes are related by an <a class="existingWikiWord" href="/nlab/show/adjoint+quadruple">adjoint quadruple</a> of functors that witness the fact that the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><msub><mi>CartSp</mi> <mi>th</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(CartSp_{th})</annotation></semantics></math> are possiby infinitesimal extensions of objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(CartSp)</annotation></semantics></math>. For more discussion of this see <a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a>.</p> <h3 id="higher_smooth_geometry">Higher smooth geometry</h3> <p>The topos of smooth spaces has an evident generalization from <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> to <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>, hence from <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> to <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>: to an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> of <em><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a></em>. See there for more details.</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> <h2 id="related_concepts">Related concepts</h2> <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> <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> <h2 id="references">References</h2> <p>The category of sheaves of the site of smooth manifolds appears as a model for <a class="existingWikiWord" href="/nlab/show/homotopy+types">homotopy types</a> in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, Ch. 6 in: <em>Faisceaux localement asphériques</em> (2003) [<a href="https://cisinski.app.uni-regensburg.de/mtest2.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Cisinski-FaisceauxLocAsph.pdf" title="pdf">pdf</a>]</li> </ul> <p>and in the context of <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial homotopy theory</a> in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, section 3.4, from page 29 on in: <em>Sheaves and Homotopy Theory</em> [<a href="http://www.uoregon.edu/~ddugger/cech.html">web</a>, <a href="http://ncatlab.org/nlab/files/cech.pdf">pdf</a>]</p> <blockquote> <p>(the <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">topos points</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>Diff</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(Diff)</annotation></semantics></math> are discussed there in example 4.1.2 on p. 36, mentioned before on p. 31)</p> </blockquote> </li> </ul> <p>The equivalent incarnation over the smaller site <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> and discussion as a <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> appears in</p> <ul> <li id="Schreiber13"><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, Def. 1.2.16, Def. 1.3.58 of: <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em> [<a href="https://arxiv.org/abs/1310.7930">arXiv:1310.7930</a>]</li> </ul> <p id="TerminologyOrigin"> The terminology “smooth set” is due to</p> <ul> <li> <p>nLab: <em><a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a></em> — from <a href="https://ncatlab.org/nlab/revision/diff/smooth+set/10">revision 10</a> on (Feb 2013)</p> </li> <li> <p><a href="#dcct">Schreiber 2013</a>, <a href="https://arxiv.org/pdf/1310.7930v1#page=48">§1.2.2, p. 48</a></p> <p>(which otherwise speaks of “smooth <a class="existingWikiWord" href="/nlab/show/0-types">0-types</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+--+smooth+sets">geometry of physics – smooth sets</a></em> — from <a href="https://ncatlab.org/nlab/revision/diff/geometry+of+physics+--+smooth+sets/3">revision 3</a> on (Nov 2014)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Igor+Khavkine">Igor Khavkine</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, Def. 2.1 in: <em><a class="existingWikiWord" href="/schreiber/show/Synthetic+variational+calculus">Synthetic geometry of differential equations: I. Jets and comonad structure</a></em> [<a href="https://arxiv.org/abs/1701.06238">arXiv:1701.06238</a>]</p> </li> </ul> <p>Discussion of smooth sets as a <a class="existingWikiWord" href="/nlab/show/convenient+category+of+spaces">convenient category</a> for <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a> of <a class="existingWikiWord" href="/nlab/show/Lagrangian+quantum+field+theory">Lagrangian</a> <a class="existingWikiWord" href="/nlab/show/classical+field+theory">classical field theory</a>:</p> <ul> <li id="GiotopoulosSati23"><a class="existingWikiWord" href="/nlab/show/Grigorios+Giotopoulos">Grigorios Giotopoulos</a>, <a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <em>Field Theory via Higher Geometry I: <a class="existingWikiWord" href="/schreiber/show/Smooth+Sets+of+Fields">Smooth Sets of Fields</a></em>, Journal of Geometry and Physics (2025) 105462 [<a href="https://arxiv.org/abs/2312.16301">arXiv:2312.16301</a>, <a href="https://doi.org/10.1016/j.geomphys.2025.105462">doi:10.1016/j.geomphys.2025.105462</a>]</li> </ul> <p>Exposition:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Grigorios+Giotopoulos">Grigorios Giotopoulos</a>: <em>Classical field theory in the topos of smooth sets</em>, <a href="Center+for+Quantum+and+Topological+Systems#GiotopoulosOct2023">talk at</a> <a class="existingWikiWord" href="/nlab/show/CQTS">CQTS</a> (Oct 2023) [<a class="existingWikiWord" href="/nlab/files/Giotopoulos-FieldTheoryInSmoothSets.pdf" title="pdf">pdf</a>, video:<a href="https://youtu.be/7Bw9CJct8QY">YT</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grigorios+Giotopoulos">Grigorios Giotopoulos</a>: <em>Towards Non-Perturbative Lagrangian Field Theory via the Topos of Smooth Sets</em>, <a href="CQTS##Giotopoulos2024">talk at</a> <em><a href="M-Theory+and+Mathematics#2024">M-Theory and Mathematics 2024</a></em> (Jan 2024) [video: <a href="https://cdnapisec.kaltura.com/html5/html5lib/v2.73.2/mwEmbedFrame.php/p/1674401/uiconf_id/23435151/entry_id/1_z8xmdmu5?wid=_1674401&iframeembed=true&playerId=kaltura_player&entry_id=1_z8xmdmu5">kt</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grigorios+Giotopoulos">Grigorios Giotopoulos</a>: <em>Sheaf Topos Theory as a setting for Physics</em>, talk at <em><a href="http://www.physics.ntua.gr/corfu2024/nc.html">Workshop on Noncommutative and Generalized Geometry in String Theory</a></em>, Corfu Summer Institute (2024) [<a class="existingWikiWord" href="/nlab/files/Giotopoulos-ToposTheoryForPhysics.pdf" title="pdf">pdf</a>]</p> <blockquote> <p>(also on <a class="existingWikiWord" href="/nlab/show/super+smooth+sets">super smooth sets</a>)</p> </blockquote> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 8, 2025 at 12:17:58. 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