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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="differential_geometry">Differential geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> </li> </ul> <p><strong>Tangency</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a></p> </li> </ul> <p><strong>The magic algebraic facts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a></p> </li> </ul> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a></p> </li> <li> <p><a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a></strong></p> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> <p id="models_2"><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>)</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></p> </li> </ul> </div></div> <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="contents">Contents</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='#RelationToTopologicalSpaces'>Relation to topological and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math>-generated spaces</a></li> <li><a href='#OnTopologicalHomotopyTypeAndDiffeologicalShape'>Topological homotopy type and diffeological shape</a></li> <li><a href='#EmbeddingOfSmoothManifoldsIntoDiffeoloticalSpaces'>Embedding of smooth manifolds into diffeological spaces</a></li> <li><a href='#embedding_of_smooth_manifolds_with_boundary_into_diffeological_spaces'>Embedding of smooth manifolds with boundary into diffeological spaces</a></li> <li><a href='#EmbeddingOfBanachManifoldsIntoDiffeologicalSpaces'>Embedding of Banach manifolds into diffeological spaces</a></li> <li><a href='#RelationBetweenDeffeologicalAndFrechetStructure'>Embedding of Fréchet manifolds into diffeological spaces</a></li> <li><a href='#EmbeddingOfDiffeologicalSpacesIntoTheSheafTopos'>Embedding of diffeological spaces into smooth sets</a></li> <li><a href='#EmbeddingOfDiffeologicalSpacesIntoHigherDifferentialGeometry'>Embedding of diffeological spaces into higher differential geometry</a></li> <li><a href='#distribution_theory'>Distribution theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#ReferencesGeneral'>Foundations</a></li> <li><a href='#differential_geometry_of_diffeological_spaces'>Differential geometry of diffeological spaces</a></li> <li><a href='#full_subcategories'>Full subcategories</a></li> <li><a href='#ReferencesForOrbifolds'>For orbifolds</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>diffeological space</strong> is a type of <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a>. As with the other variants, it subsumes the notion of <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> but also naturally captures other spaces that one would like to think of as smooth spaces but aren’t manifolds; for example, the space of all smooth maps between two smooth manifolds can be made into a diffeological space. (These mapping spaces are rarely manifolds themselves, see <a class="existingWikiWord" href="/nlab/show/manifolds+of+mapping+spaces">manifolds of mapping spaces</a>.)</p> <p>In a little more detail, a <strong>diffeology</strong>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> on a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> on the <a class="existingWikiWord" href="/nlab/show/category+of+open+subsets">category of open subsets</a> of Euclidean spaces with smooth maps as morphisms. To each open set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊆</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">U \subseteq \mathbb{R}^n</annotation></semantics></math>, it assigns a subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Set</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Set(U,X)</annotation></semantics></math>. The functions in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Set</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Set(U,X)</annotation></semantics></math> are to be regarded as the “smooth functions” from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. A <strong>diffeological space</strong> is then a set together with a diffeology on it.</p> <p>Diffeological spaces were originally introduced in (<a href="#Souriau79">Souriau 79</a>). They have subsequently been developed in the textbook (<a href="#PIZ">Iglesias-Zemmour 13</a>)</p> <h2 id="definition">Definition</h2> <div class="num_defn" style="border:solid #cccccc;border-width:2px 1px;padding:0 1em;margin:0 1em;" id="DiffSp"> <h6 id="definition_2">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪𝓅</mi></mrow><annotation encoding="application/x-tex">\mathcal{Op}</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/site">site</a> whose objects are the open subsets of the Euclidean 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> and whose morphisms are <a class="existingWikiWord" href="/nlab/show/smooth+map">smooth map</a>s between these. The <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪𝓅</mi></mrow><annotation encoding="application/x-tex">\mathcal{Op}</annotation></semantics></math> is generated by the <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a> of open covers, i.e., a family of maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><msub><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>∈</mo><mi>I</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{U_i\to X\}_{i\in I}</annotation></semantics></math> is a covering family if every map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U_i\to X</annotation></semantics></math> is an open embedding and the union of the images of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">U_i</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>A <strong>diffeological space</strong> is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\mathcal{D})</annotation></semantics></math> where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a set</p> </li> <li> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><mo>∈</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>𝒪𝓅</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{D} \in Sh(\mathcal{Op})</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/diffeology">diffeology</a></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>:</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/subobject">subsheaf</a> of the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↦</mo><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U \mapsto Hom_{Set}(U,X)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}(*) = X</annotation></semantics></math></p> </li> <li> <p>equivalently: a <a class="existingWikiWord" href="/nlab/show/concrete+sheaf">concrete sheaf</a> on the <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪𝓅</mi></mrow><annotation encoding="application/x-tex">\mathcal{Op}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}(*) = X</annotation></semantics></math> - a <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete</a> <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a> (see there for more details).</p> </li> </ul> </li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> of diffeological spaces is a morphism of the corresponding <a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a>: we take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DiffeologicalSp</mi><mo>↪</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">DiffeologicalSp \hookrightarrow Sh(CartSp)</annotation></semantics></math> to be the full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> on the diffeological spaces in the sheaf topos.</p> </div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(X,\mathcal{D})</annotation></semantics></math> a diffeological space, and for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>𝒪𝓅</mi></mrow><annotation encoding="application/x-tex">U \in \mathcal{Op}</annotation></semantics></math>, the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{D}(U)</annotation></semantics></math> is also called the set of <strong><a class="existingWikiWord" href="/nlab/show/plots">plots</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> on <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>. This is to be thought of as the set of ways of mapping <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> smoothly into the would-be space <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>. This assignment <em>defined</em> what it means for a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \to X</annotation></semantics></math> of sets to be smooth.</p> <p>For some comments on the reasoning behind this kind of definition of generalized <a class="existingWikiWord" href="/nlab/show/space">space</a>s see <a class="existingWikiWord" href="/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks">motivation for sheaves, cohomology and higher stacks</a>.</p> <p>A sheaf on the site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪𝓅</mi></mrow><annotation encoding="application/x-tex">\mathcal{Op}</annotation></semantics></math> of open subsets of Euclidean spaces is completely specified by its restriction to <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>, the full subcategory of <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>: The <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CartSp</mi><mo>↪</mo><mi>𝒪𝓅</mi></mrow><annotation encoding="application/x-tex">CartSp \hookrightarrow \mathcal{Op}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/dense+subsite">dense subsite</a>-inclusion. Therefore in the sequel we shall often restrict our attention to <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>.</p> <p>One may define a <em><a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a></em> to be <em>any</em> sheaf of <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>. A diffeological space is equivalently a <a class="existingWikiWord" href="/nlab/show/concrete+sheaf">concrete sheaf</a> on the <a class="existingWikiWord" href="/nlab/show/concrete+site">concrete site</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>. (For details see <a href="geometry+of+physics+--+smooth+sets#DiffeologicalSpacesAreTheConcreteSmoothSets">this Prop.</a> at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">geometry of physics – smooth sets</a></em>.)</p> <p>The <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DiffeologicalSpaces</mi><mo>↪</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> DiffeologicalSpaces \hookrightarrow Sh(CartSp) </annotation></semantics></math></div> <p>on all concrete sheaves is not a <a class="existingWikiWord" href="/nlab/show/topos">topos</a>, but is a <a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a>.</p> <p>This is Prop. <a class="maruku-ref" href="#DiffeologicalSpacesAreTheConcreteSmoothSets"></a> below.</p> <p>The concreteness condition on the sheaf is a reiteration of the fact that a diffeological space is a subsheaf of the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>↦</mo><msup><mi>X</mi> <mrow><mo stretchy="false">|</mo><mi>U</mi><mo stretchy="false">|</mo></mrow></msup></mrow><annotation encoding="application/x-tex">U \mapsto X^{|U|}</annotation></semantics></math>. In this way, one does not have to explicitly mention the underlying set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as it is determined by the sheaf on the one-point open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^0</annotation></semantics></math>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, i.e. every object of <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>, becomes a diffeological space by defining the <a class="existingWikiWord" href="/nlab/show/plots">plots</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">U \in CartSp</annotation></semantics></math> to be the ordinary <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>s from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, i.e. the morphisms in <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>:</mo><mi>U</mi><mo>↦</mo><msub><mi>Hom</mi> <mi>Diff</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X : U \mapsto Hom_{Diff}(U,X) \,. </annotation></semantics></math></div></li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> two diffeological spaces, their <a class="existingWikiWord" href="/nlab/show/product">product</a> as sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times Y</annotation></semantics></math> becomes a diffeological space whose <a class="existingWikiWord" href="/nlab/show/plots">plots</a> are pairs consisting of a <a class="existingWikiWord" href="/nlab/show/plot">plot</a> into <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 one into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>:</mo><mi>U</mi><mo>↦</mo><msub><mi>Hom</mi> <mi>DiffSp</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>×</mo><msub><mi>Hom</mi> <mi>DiffSp</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \times Y : U \mapsto Hom_{DiffSp}(U,X) \times Hom_{DiffSp}(U,Y) \,. </annotation></semantics></math></div></li> <li> <p>Given any two diffeological spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>, the set of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>DiffSp</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{DiffSp}(X,Y)</annotation></semantics></math> becomes a smooth space by taking the <a class="existingWikiWord" href="/nlab/show/plots">plots</a> on some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> to be the smooth morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>U</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \times U \to Y</annotation></semantics></math>, i.e. the smooth <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 families of smooth maps from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">]</mo><mo>:</mo><mi>U</mi><mo>↦</mo><msub><mi>Hom</mi> <mi>DiffSp</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>U</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [X,Y] : U \mapsto Hom_{DiffSp}(X \times U, Y) \,. </annotation></semantics></math></div> <p>In this formula we regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>CartSp</mi><mo>↪</mo><mi>Diff</mi></mrow><annotation encoding="application/x-tex">U \in CartSp \hookrightarrow Diff</annotation></semantics></math> as a diffeological space according to the above example. In fact, we apply secretly here the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> and use the general formula for the cartesian <a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a>.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="RelationToTopologicalSpaces">Relation to topological and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math>-generated spaces</h3> <div> <div class="num_prop" id="AdjunctionBetweenTopologicalSpacesAndDiffeologicalSpaces"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/adjunction+between+topological+spaces+and+diffeological+spaces">adjunction between topological spaces and diffeological spaces</a>)</strong></p> <p>There is a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a></p> <div class="maruku-equation" id="eq:AdjointFunctorsBetweenTopSpAndDifflgSp"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>TopSp</mi><munderover><mrow><mphantom><mi>AA</mi></mphantom><mo>⊥</mo><mphantom><mi>AA</mi></mphantom></mrow><munder><mo>⟶</mo><mi>Cdfflg</mi></munder><mover><mo>⟵</mo><mi>Dtplg</mi></mover></munderover><mi>DifflgSp</mi></mrow><annotation encoding="application/x-tex"> TopSp \underoverset{ \underset{ Cdfflg }{\longrightarrow} }{ \overset{ Dtplg }{\longleftarrow} }{\phantom{AA}\bot\phantom{AA}} DifflgSp </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/categories">categories</a> of <a class="existingWikiWord" href="/nlab/show/Top">TopologicalSpaces</a> and of <a class="existingWikiWord" href="/nlab/show/DiffeologicalSpaces">DiffeologicalSpaces</a>, where</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cdfflg</mi></mrow><annotation encoding="application/x-tex">Cdfflg</annotation></semantics></math> takes a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the <strong>continuous diffeology</strong>, namely the diffeological space on the same underlying set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">X_s</annotation></semantics></math> whose plots <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>s</mi></msub><mo>→</mo><msub><mi>X</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">U_s \to X_s</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> (from the underlying <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> of the <a class="existingWikiWord" href="/nlab/show/domain">domain</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>).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi></mrow><annotation encoding="application/x-tex">Dtplg</annotation></semantics></math> takes a <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a> to the <strong>diffeological topology</strong> (<a class="existingWikiWord" href="/nlab/show/D-topology">D-topology</a>), namely the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> with the same underlying set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">X_s</annotation></semantics></math> and with the <a class="existingWikiWord" href="/nlab/show/final+topology">final topology</a> that makes all its plots <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>s</mi></msub><mo>→</mo><msub><mi>X</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">U_{s} \to X_{s}</annotation></semantics></math> into <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a>: called the <em><a class="existingWikiWord" href="/nlab/show/D-topology">D-topology</a></em>.</p> <p>Hence a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>O</mi><mo>⊂</mo><mo>♭</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">O \subset \flat X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> in the <a class="existingWikiWord" href="/nlab/show/D-topology">D-topology</a> precisely if for each plot <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">f \colon U \to X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>O</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">f^{-1}(O) \subset U</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> in 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><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>.</p> </li> </ul> <p>Moreover:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/fixed+point+of+an+adjunction">fixed points of this adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Top">TopologicalSpaces</a> (those for which the <a class="existingWikiWord" href="/nlab/show/counit+of+an+adjunction">counit</a> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, hence here: a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>) are precisely the <a class="existingWikiWord" href="/nlab/show/Delta-generated+topological+spaces">Delta-generated topological spaces</a> (<a href="Delta-generated+topological+space#AsDTopologicalSpaces">i.e.</a> <a class="existingWikiWord" href="/nlab/show/D-topological+spaces">D-topological spaces</a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mspace width="thickmathspace"></mspace><mspace width="thinmathspace"></mspace><mtext>is</mtext><mspace width="thickmathspace"></mspace><mi>Δ</mi><mtext>-generated</mtext><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mo>≃</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>ϵ</mi> <mi>X</mi></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munderover><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \;\,\text{is}\;\Delta\text{-generated} \;\;\;\;\; \Leftrightarrow \;\;\;\;\; Dtplg(Cdfflg(X)) \underoverset{\simeq}{\;\;\epsilon_X\;\;}{\longrightarrow} X </annotation></semantics></math></div></li> <li> <p>this is an <a class="existingWikiWord" href="/nlab/show/idempotent+adjunction">idempotent adjunction</a>, which exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math>-generated/<a class="existingWikiWord" href="/nlab/show/D-topological+spaces">D-topological spaces</a> as a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> inside <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> and a <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a> inside all <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>:</p> </li> </ol> <div class="maruku-equation" id="eq:DeltaGeneratedSpacesInIdempotentAdjunction"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>TopologicalSpaces</mi><munderover><mrow><mphantom><mi>AA</mi></mphantom><mo>⊥</mo><mphantom><mi>AA</mi></mphantom></mrow><munder><mo>⟶</mo><mi>Cdfflg</mi></munder><mover><mo>↩</mo><mrow></mrow></mover></munderover><mi>DTopologicalSpaces</mi><munderover><mrow><mphantom><mi>AA</mi></mphantom><mo>⊥</mo><mphantom><mi>AA</mi></mphantom></mrow><munder><mo>↪</mo><mrow></mrow></munder><mover><mo>⟵</mo><mi>Dtplg</mi></mover></munderover><mi>DiffeologicalSpaces</mi></mrow><annotation encoding="application/x-tex"> TopologicalSpaces \underoverset { \underset{ Cdfflg }{\longrightarrow} } { \overset{ }{\hookleftarrow} } {\phantom{AA}\bot\phantom{AA}} DTopologicalSpaces \underoverset { \underset{ }{\hookrightarrow} } { \overset{ Dtplg }{\longleftarrow} } {\phantom{AA}\bot\phantom{AA}} DiffeologicalSpaces </annotation></semantics></math></div> <p>Finally, these adjunctions are a sequence of <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a> with respect to the:</p> <table><thead><tr><th></th><th></th><th></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+on+D-topological+spaces">model structure on D-topological spaces</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">model structure on diffeological spaces</a></td></tr> </tbody></table> <blockquote> <p>Caution: There was a gap in the original proof that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DTopologicalSpaces</mi><msub><mo>≃</mo> <mi>Quillen</mi></msub><mi>DiffeologicalSpaces</mi></mrow><annotation encoding="application/x-tex">DTopologicalSpaces \simeq_{Quillen} DiffeologicalSpaces</annotation></semantics></math>. The gap is claimed to be filled now, see the commented references <a href="model+structure+on+diffeological+spaces#References">here</a>.</p> </blockquote> </div> <p>Essentially these adjunctions and their properties are observed in <a href="diffeological+space#SYH10">Shimakawa, Yoshida & Haraguchi 2010, Prop. 3.1, Prop. 3.2, Lem. 3.3</a>, see also <a href="D-topology#CSW13">Christensen, Sinnamon & Wu 2014, Sec. 3.2</a>. The model structures and Quillen equivalences are due to <a href="#model+structure+on+Delta-generated+topological+spaces#Haraguchi13">Haraguchi 13, Thm. 3.3</a> (on the left) and <a href="model+structure+on+diffeological+spaces#HaraguchiShimakawa13">Haraguchi-Shimakawa 13, Sec. 7</a> (on the right).</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>We spell out the existence of the <a class="existingWikiWord" href="/nlab/show/idempotent+adjunction">idempotent adjunction</a> <a class="maruku-eqref" href="#eq:DeltaGeneratedSpacesInIdempotentAdjunction">(2)</a>:</p> <p>First, to see we have an <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi><mo>⊣</mo><mi>Cdfflg</mi></mrow><annotation encoding="application/x-tex">Dtplg \dashv Cdfflg</annotation></semantics></math>, we check the hom-isomorphism (<a href="adjoint+functor#eq:HomIsomorphismForAdjointFunctors">here</a>).</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>DiffeologicalSpaces</mi></mrow><annotation encoding="application/x-tex">X \in DiffeologicalSpaces</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>TopologicalSpaces</mi></mrow><annotation encoding="application/x-tex">Y \in TopologicalSpaces</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">(-)_s</annotation></semantics></math> for the underlying sets. Then a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>, hence a <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous function</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; Dtplg(X) \longrightarrow Y \,, </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>s</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>X</mi> <mi>s</mi></msub><mo>→</mo><msub><mi>Y</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">f_s \colon X_s \to Y_s</annotation></semantics></math> of the underlying <a class="existingWikiWord" href="/nlab/show/sets">sets</a> such that for every <a class="existingWikiWord" href="/nlab/show/open+subset">open subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊂</mo><msub><mi>Y</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">A \subset Y_s</annotation></semantics></math> and every <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon \mathbb{R}^n \to X</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/preimage">preimage</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>s</mi></msub><mo>∘</mo><msub><mi>ϕ</mi> <mi>s</mi></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊂</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(f_s \circ \phi_s)^{-1}(A) \subset \mathbb{R}^n</annotation></semantics></math> is open. But this means equivalently that for every such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">f \circ \phi</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>. This, in turn, means equivalently that the same underlying function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">f_s</annotation></semantics></math> constitutes a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo>˜</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>⟶</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde f \;\colon\; X \longrightarrow Cdfflg(Y)</annotation></semantics></math>.</p> <p>In summary, we thus have a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a> of <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msub><mi>f</mi> <mi>s</mi></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><mover><mi>f</mi><mo>˜</mo></mover><msub><mo stretchy="false">)</mo> <mi>s</mi></msub><mo>=</mo><msub><mi>f</mi> <mi>s</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Hom( Dtplg(X), Y ) &\simeq& Hom(X, Cdfflg(Y)) \\ f_s &\mapsto& (\widetilde f)_s = f_s } </annotation></semantics></math></div> <p>given simply as the <a class="existingWikiWord" href="/nlab/show/identity+function">identity</a> on the underlying <a class="existingWikiWord" href="/nlab/show/functions">functions</a> of underlying sets. This makes it immediate that this hom-isomorphism is <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and this establishes the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a>.</p> <p>Next, to see that the <a class="existingWikiWord" href="/nlab/show/D-topological+spaces">D-topological spaces</a> are the <a class="existingWikiWord" href="/nlab/show/fixed+point+of+an+adjunction">fixed points</a> of this adjunction, we apply the above <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> on hom-sets to the case</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo></mtd> <mtd><mi>Hom</mi><mo stretchy="false">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mi>Z</mi></msub><msub><mo stretchy="false">)</mo> <mi>s</mi></msub></mtd> <mtd><mo>↦</mo></mtd> <mtd><mo stretchy="false">(</mo><mi mathvariant="normal">id</mi><msub><mo stretchy="false">)</mo> <mi>s</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Hom( Dtplg(Cdfflg(Z)), Y ) &\simeq& Hom(Cdfflg(Z), Cdfflg(Y)) \\ (\epsilon_Z)_s &\mapsto& (\mathrm{id})_s } </annotation></semantics></math></div> <p>to find that the <a class="existingWikiWord" href="/nlab/show/counit+of+the+adjunction">counit of the adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>ϵ</mi> <mi>X</mi></msub></mrow></mover><mi>X</mi></mrow><annotation encoding="application/x-tex"> Dtplg(Cdfflg(X)) \overset{\epsilon_X}{\longrightarrow} X </annotation></semantics></math></div> <p>is given by the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a> on the underlying sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϵ</mi> <mi>X</mi></msub><msub><mo stretchy="false">)</mo> <mi>s</mi></msub><mo>=</mo><msub><mi>id</mi> <mrow><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>s</mi></msub><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">(\epsilon_X)_s = id_{(X_s)}</annotation></semantics></math>.</p> <p>Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\eta_X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>, namely a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a>, precisely if the open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">X_s</annotation></semantics></math> with respect to the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are precisely those with respect to the topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi><mo stretchy="false">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Dtplg(Cdfflg(X))</annotation></semantics></math>, which means equivalently that the open subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> coincide with those whose pre-images under all continuous functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon \mathbb{R}^n \to X</annotation></semantics></math> are open. This means equivalently that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a D-topological space.</p> <p>Finally, to see that we have an <a class="existingWikiWord" href="/nlab/show/idempotent+adjunction">idempotent adjunction</a>, it is sufficient to check (by <a href="idempotent+adjunction#EquivalentConditionsForIdempotency">this Prop.</a>) that the <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi><mo>∘</mo><mi>Cdfflg</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>TopologicalSpaces</mi><mo>→</mo><mi>TopologicalSpaces</mi></mrow><annotation encoding="application/x-tex"> Dtplg \circ Cdfflg \;\colon\; TopologicalSpaces \to TopologicalSpaces </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/idempotent+comonad">idempotent comonad</a>, hence that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Dtplg</mi><mo>∘</mo><mi>Cdfflg</mi><mover><mo>⟶</mo><mrow><mi>Dtplg</mi><mo>⋅</mo><mi>η</mi><mo>⋅</mo><mi>Cdfflg</mi></mrow></mover><mi>Dtplg</mi><mo>∘</mo><mi>Cdfflg</mi><mo>∘</mo><mi>Dtplg</mi><mo>∘</mo><mi>Cdfflg</mi></mrow><annotation encoding="application/x-tex"> Dtplg \circ Cdfflg \overset{ Dtplg \cdot \eta \cdot Cdfflg }{\longrightarrow} Dtplg \circ Cdfflg \circ Dtplg \circ Cdfflg </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>. But, as before for the adjunction counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>, we have that also the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/identity+function">identity function</a> on the underlying sets. Therefore, this being a natural isomorphism is equivalent to the operation of passing to the D-topological refinement of the topology of a topological space being an idempotent operation, which is clearly the case.</p> </div> </div> <p>Further discussion of the <a class="existingWikiWord" href="/nlab/show/D-topology">D-topology</a> is in <a href="#CSW13">CSW 13</a>.</p> <h3 id="OnTopologicalHomotopyTypeAndDiffeologicalShape">Topological homotopy type and diffeological shape</h3> <div> <div class="num_defn" id="DiffeologicalSingularSimplicialSet"> <h6 id="definition">Definition</h6> <p><strong>(diffeological singular simplicial set)</strong></p> <p>Consider the <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial</a> <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Δ</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>Δ</mi> <mi>diff</mi> <mo>•</mo></msubsup></mrow></mover></mtd> <mtd><mi>DiffeologicalSpaces</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><msubsup><mi>Δ</mi> <mi>diff</mi> <mi>n</mi></msubsup><mpadded width="0"><mrow><mo>≔</mo><mo maxsize="1.2em" minsize="1.2em">{</mo><mover><mi>x</mi><mo stretchy="false">→</mo></mover><mo>∈</mo><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>i</mi></munder><msup><mi>x</mi> <mi>i</mi></msup><mo>=</mo><mn>1</mn><mo maxsize="1.2em" minsize="1.2em">}</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Delta & \overset{ \Delta^\bullet_{diff} }{ \longrightarrow } & DiffeologicalSpaces \\ [n] &\mapsto& \Delta^n_{diff} \mathrlap{ \coloneqq \big\{ \vec x \in \mathbb{R}^{n+1} \;\vert\; \underset{i}{\sum} x^i = 1 \big\} } } </annotation></semantics></math></div> <p>which in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> is the standard extended <a class="existingWikiWord" href="/nlab/show/n-simplex">n-simplex</a> inside <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> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^{n+1}</annotation></semantics></math>, equipped with its sub-<a class="existingWikiWord" href="/nlab/show/diffeology">diffeology</a>.</p> <p>This induces a <a class="existingWikiWord" href="/nlab/show/nerve+and+realization">nerve and realization</a> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> between <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> and <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>:</p> <div class="maruku-equation" id="eq:NerveAndRealizationForDiffeologicalSpaces"><span class="maruku-eq-number">(3)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DiffeologicalSpaces</mi><munderover><mrow><mphantom><mi>AA</mi></mphantom><mo>⊥</mo><mphantom><mi>AA</mi></mphantom></mrow><munder><mo>⟶</mo><mrow><msub><mi>Sing</mi> <mpadded width="0"><mi>diff</mi></mpadded></msub></mrow></munder><mover><mo>⟵</mo><mrow><msub><mrow><mo>|</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>|</mo></mrow> <mpadded width="0"><mi>diff</mi></mpadded></msub></mrow></mover></munderover><mi>SimplicialSets</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> DiffeologicalSpaces \underoverset { \underset{Sing_{\mathrlap{diff}}}{\longrightarrow} } { \overset{ \left\vert - \right\vert_{\mathrlap{diff}} }{\longleftarrow} } { \phantom{AA}\bot\phantom{AA} } SimplicialSets \,, </annotation></semantics></math></div> <p>where the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> is the <em>diffeological <a class="existingWikiWord" href="/nlab/show/singular+simplicial+set">singular simplicial set</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex">Sing_{diff}</annotation></semantics></math>.</p> </div> <p>(e.g. <a href="diffeological+space#ChristensenWu13">Christensen-Wu 13, Def. 4.3</a>)</p> <div class="num_remark" id="DiffeologicalSingularSetIsPathGroupoid"> <h6 id="remark">Remark</h6> <p><strong>(diffeological singular simplicial set as <a class="existingWikiWord" href="/nlab/show/path+%E2%88%9E-groupoid">path ∞-groupoid</a>)</strong></p> <p>Regarding <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> as <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">presenting</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoids">∞-groupoids</a>, we may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing_{diff}(X)</annotation></semantics></math> (Def. <a class="maruku-ref" href="#DiffeologicalSingularSimplicialSet"></a>) as the <a class="existingWikiWord" href="/nlab/show/path+%E2%88%9E-groupoid">path ∞-groupoid</a> of the <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>In fact, by the discussion at <a class="existingWikiWord" href="/nlab/show/shape+via+cohesive+path+%E2%88%9E-groupoid">shape via cohesive path ∞-groupoid</a> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex">Sing_{diff}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28%E2%88%9E%2C1%29-category">equvialent</a> to the <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a> of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> regarded as <a class="existingWikiWord" href="/nlab/show/objects">objects</a> of the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>diff</mi></msub><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>Shp</mi><mo>∘</mo><mi>i</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>DiffeologicalSpaces</mi><mover><mo>↪</mo><mi>i</mi></mover><msub><mi>SmoothGroupoids</mi> <mn>∞</mn></msub><mover><mo>⟶</mo><mi>Shape</mi></mover><msub><mi>Groupoids</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex"> Sing_{diff} \;\simeq\; Shp \circ i \;\;\colon\;\; DiffeologicalSpaces \overset{i}{\hookrightarrow} SmoothGroupoids_{\infty} \overset{Shape}{\longrightarrow} Groupoids_\infty </annotation></semantics></math></div></div> </div><div> <p> <div class="num_prop" id="TopologicalHomotopyTypeIsCohesiveShapeOfCdfflg"> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/topological+space">topological</a> <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> is <a class="existingWikiWord" href="/nlab/show/cohesive+shape">cohesive shape</a> of <a class="existingWikiWord" href="/nlab/show/continuous+diffeology">continuous diffeology</a>)</strong> <br /> For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/TopologicalSpaces">TopologicalSpaces</a>, the <a class="existingWikiWord" href="/nlab/show/cohesive+shape">cohesive shape</a>/<a class="existingWikiWord" href="/nlab/show/path+%E2%88%9E-groupoid">path ∞-groupoid</a> presented by its <em>diffeological singular simplicial set</em> (Def. <a class="maruku-ref" href="#DiffeologicalSingularSimplicialSet"></a>, Remark <a class="maruku-ref" href="#DiffeologicalSingularSetIsPathGroupoid"></a>) of its <a class="existingWikiWord" href="/nlab/show/continuous+diffeology">continuous diffeology</a> is <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturally</a><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><a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> presented by the ordinary <a class="existingWikiWord" href="/nlab/show/singular+simplicial+set">singular simplicial set</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>diff</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi mathvariant="normal">W</mi> <mi>wh</mi></msub></mrow><mrow></mrow></munderover><mi>Sing</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Sing_{diff} \big( Cdfflg(X) \big) \underoverset { \in \mathrm{W}_{wh} } {} {\longrightarrow} Sing(X) \,. </annotation></semantics></math></div> <p></p> </div> </p> <p>(<a href="diffeological+space#ChristensenWu13">Christensen & Wu 2013, Prop. 4.14</a>)</p> </div> <p> <div class='num_prop' id='InternalHomOnDTopSpacesHasCorrectWeakType'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological</a> <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> on <a class="existingWikiWord" href="/nlab/show/D-topological+spaces">D-topological spaces</a> has correct <a class="existingWikiWord" href="/nlab/show/shape+via+cohesive+path+%E2%88%9E-groupoid">cohesive shape</a>)</strong> <br /> For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>DTopSp</mi><mo>↪</mo><mi>TopSp</mi></mrow><annotation encoding="application/x-tex">X, A \,\in\, DTopSp \hookrightarrow TopSp</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> of <a class="existingWikiWord" href="/nlab/show/D-topological+spaces">D-topological spaces</a>, their <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> formed in diffeological spaces has diffeological <a class="existingWikiWord" href="/nlab/show/singular+simplicial+complex">singular simplicial complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>diff</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing_{diff}(-)</annotation></semantics></math> <a class="maruku-eqref" href="#eq:NerveAndRealizationForDiffeologicalSpaces">(3)</a> <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weakly homotopy equivalent</a> to the ordinary <a class="existingWikiWord" href="/nlab/show/singular+simplicial+set">singular simplicial set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Sing</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sing(-)</annotation></semantics></math> of the ordinary <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Maps</mi> <mi>Top</mi></msub></mrow><annotation encoding="application/x-tex">Maps_{Top}</annotation></semantics></math> with its <a class="existingWikiWord" href="/nlab/show/compact-open+topology">compact-open topology</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>k</mi><mi>TopSp</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Sing</mi> <mi>diff</mi></msub><mrow><mo>(</mo><msub><mi>Maps</mi> <mi>dfflg</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>)</mo></mrow><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi mathvariant="normal">W</mi> <mi>wh</mi></msub></mrow><mrow></mrow></munderover><mi>Sing</mi><mrow><mo>(</mo><msub><mi>Maps</mi> <mi>Top</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> X,\, A \;\in\; k TopSp \;\;\;\;\;\; \vdash \;\;\;\;\;\; Sing_{diff} \left( Maps_{dfflg} \big( X ,\, A \big) \right) \underoverset {\in \mathrm{W}_{wh}} {} {\longrightarrow} Sing \left( Maps_{Top} \big( X ,\, A \big) \right) </annotation></semantics></math></div> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p>By <a href="#SYH10">SYH 10</a> we have the following morphism:</p> <div class="maruku-equation" id="eq:AComparisonMoprphismForShapeOfDiffeologicalMappingSpace"><span class="maruku-eq-number">(4)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Cdfflg</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>Maps</mi> <mi>Top</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>Cdfflg</mi><mo stretchy="false">(</mo><msub><mi mathvariant="normal">W</mi> <mi>wh</mi></msub><mo stretchy="false">)</mo></mrow><mrow><mi>Cdfflg</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow></munderover><mi>Cdfflg</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mstyle mathvariant="bold"><mi>smap</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msub><mi>Maps</mi> <mi>dfflg</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Cdfflg \big( Maps_{Top}(X,\,A) \big) \underoverset { \in Cdfflg(\mathrm{W}_{wh}) } { Cdfflg(\phi) } {\longrightarrow} Cdfflg \big( \mathbf{smap}(X ,\, A) \big) \;\; \simeq \;\; Maps_{dfflg} (X,\, A) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>smap</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{smap}</annotation></semantics></math> is some topologization of the set of maps (defined on their <a href="https://arxiv.org/pdf/1010.3336.pdf#page=6">p. 6</a> ) of which all we need to know is that:</p> <ol> <li> <p>(shown on the right of <a class="maruku-eqref" href="#eq:AComparisonMoprphismForShapeOfDiffeologicalMappingSpace">(4)</a>) its image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cdfflg</mi></mrow><annotation encoding="application/x-tex">Cdfflg</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to the internal hom <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Maps</mi> <mi>dfflg</mi></msub></mrow><annotation encoding="application/x-tex">Maps_{dfflg}</annotation></semantics></math> in diffeological spaces, according to their Prop. 4.7 (<a href="https://arxiv.org/pdf/1010.3336.pdf#page=7">p. 7</a>),</p> </li> <li> <p>(shown on the left of <a class="maruku-eqref" href="#eq:AComparisonMoprphismForShapeOfDiffeologicalMappingSpace">(4)</a>) it is <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalent</a>, via some map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> according to their Prop. 5.4 (<a href="https://arxiv.org/pdf/1010.3336.pdf#page=9">p. 9</a>) to the compact-open topology.</p> </li> </ol> <p>Hence the claim follows by using <a class="existingWikiWord" href="/nlab/show/2-out-of-3">2-out-of-3</a> in the <a class="existingWikiWord" href="/nlab/show/naturality+square">naturality square</a> of the natural weak homotopy equivalence from Prop. <a class="maruku-ref" href="#TopologicalHomotopyTypeIsCohesiveShapeOfCdfflg"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Sing</mi> <mi>diff</mi></msub><mo>∘</mo><mi>Cdfflg</mi><mspace width="thickmathspace"></mspace><munderover><mo>⟶</mo><mrow><mo>∈</mo><msub><mi mathvariant="normal">W</mi> <mi>wh</mi></msub></mrow><mrow></mrow></munderover><mspace width="thickmathspace"></mspace><mi>Sing</mi></mrow><annotation encoding="application/x-tex"> Sing_{diff} \circ Cdfflg \; \underoverset {\in \mathrm{W}_{wh}} {} {\longrightarrow} \; Sing </annotation></semantics></math></div> <p>applied to <a class="maruku-eqref" href="#eq:AComparisonMoprphismForShapeOfDiffeologicalMappingSpace">(4)</a>.</p> </div> </p> <h3 id="EmbeddingOfSmoothManifoldsIntoDiffeoloticalSpaces">Embedding of smooth manifolds into diffeological spaces</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>The obvious functor from the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/SmoothManifolds">SmoothManifolds</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> to the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/DiffeologicalSpaces">DiffeologicalSpaces</a> of diffeological spaces is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmoothManifolds</mi><mo>↪</mo><mi>DiffeologicalSpacess</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> SmoothManifolds \hookrightarrow DiffeologicalSpacess \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof">Proof</h6> <p>This is a direct consequence of the fact that <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><mo></mo><mi>smooth</mi></msub></mrow><annotation encoding="application/x-tex">_{smooth}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/dense+sub-site">dense sub-site</a> of <a class="existingWikiWord" href="/nlab/show/SmoothManifolds">SmoothManifolds</a> and the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>.</p> <p>It may nevertheless be useful to spell out the elementary proof directly:</p> <p>To see that the functor is faithful, notice that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f,g : X \to Y</annotation></semantics></math> are two <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>s that differ at some point, then they must differ in some <a class="existingWikiWord" href="/nlab/show/open+neighbourhood">open neighbourhood</a> of that point. This <a class="existingWikiWord" href="/nlab/show/open+ball">open ball</a> is a <a class="existingWikiWord" href="/nlab/show/plot">plot</a>, hence the corresponding diffeological spaces differ on that <a class="existingWikiWord" href="/nlab/show/plot">plot</a>.</p> <p>To see that the functor is full, we need to show that a map of sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> that sends <a class="existingWikiWord" href="/nlab/show/plots">plots</a> to <a class="existingWikiWord" href="/nlab/show/plots">plots</a> is necessarily a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a>, hence that all its <a class="existingWikiWord" href="/nlab/show/derivative">derivative</a>s exist. This can be tested already on all smooth curves <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>:</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\gamma : (0,1) \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. By <a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a>, a function that takes all smooth curves to smooth curves is necessarily a smooth function. But curves are in particular <a class="existingWikiWord" href="/nlab/show/plots">plots</a>, so a function that takes all <a class="existingWikiWord" href="/nlab/show/plots">plots</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/plots">plots</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> must be smooth.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>The proof shows that we could restrict attention to the full sub-site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CartSp</mi> <mrow><mi>dim</mi><mo>≤</mo><mn>1</mn></mrow></msub><mo>⊂</mo><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">CartSp_{dim \leq 1} \subset CartSp</annotation></semantics></math> on the objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>0</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^1</annotation></semantics></math> and still have a full and faithful embedding</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>Sh</mi><mo stretchy="false">(</mo><msub><mi>CartSp</mi> <mrow><mi>dim</mi><mo>≤</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Diff \hookrightarrow Sh(CartSp_{dim \leq 1}) \,. </annotation></semantics></math></div> <p>This fact plays a role in the definition of <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a>s, which are <a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a>s defined by <a class="existingWikiWord" href="/nlab/show/plots">plots</a> by curves into and out of them.</p> <p>While the site <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CartSp</mi> <mrow><mi>dim</mi><mo>≤</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">CartSp_{dim \leq 1}</annotation></semantics></math> is more convenient for some purposes, it is not so useful for other purposes, mostly when diffeological spaces are regarded from the point of view of the full sheaf topos: the sheaf topos <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> <mrow><mi>dim</mi><mo>≤</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(CartSp_{dim \leq 1})</annotation></semantics></math> lacks some non-<a class="existingWikiWord" href="/nlab/show/concrete+sheaf">concrete</a> sheaves of interest, such as the sheaves of differential forms of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\geq 2</annotation></semantics></math>.</p> </div> <h3 id="embedding_of_smooth_manifolds_with_boundary_into_diffeological_spaces">Embedding of smooth manifolds with boundary into diffeological spaces</h3> <div class="num_prop" id="SmoothManifoldsWithBoundaryEmbedIntoDiffeologicalSpaces"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/manifolds+with+boundaries+and+corners">manifolds with boundaries and corners</a> form <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>)</strong></p> <p>The evident <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>SmthMfdWBdrCrn</mi><mover><mo>↪</mo><mphantom><mi>AAAA</mi></mphantom></mover><mi>DiffeologicalSpaces</mi></mrow><annotation encoding="application/x-tex"> SmthMfdWBdrCrn \overset{\phantom{AAAA}}{\hookrightarrow} DiffeologicalSpaces </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth</a> <a class="existingWikiWord" href="/nlab/show/manifolds+with+boundaries+and+corners">manifolds with boundaries and corners</a> to that of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> is <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a>, hence is a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a>-embedding.</p> </div> <p>(<a href="#PIZ">Iglesias-Zemmour 13, 4.16</a>, <a href="#GurerIZ19">Gürer & Iglesias-Zemmour 19</a>)</p> <h3 id="EmbeddingOfBanachManifoldsIntoDiffeologicalSpaces">Embedding of Banach manifolds into diffeological spaces</h3> <p>Also <a class="existingWikiWord" href="/nlab/show/Banach+manifolds">Banach manifolds</a> embed <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">fully faithfully</a> into the category of diffeological spaces. In (<a href="#Hain">Hain</a>) this is discussed in terms of Chen smooth spaces.</p> <h3 id="RelationBetweenDeffeologicalAndFrechetStructure">Embedding of Fréchet manifolds into diffeological spaces</h3> <p>We discuss a natural embedding of <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifolds">Fréchet manifolds</a> into the category of diffeological spaces.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>Define a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo lspace="verythinmathspace">:</mo><mi>FrechetManifolds</mi><mo>→</mo><mi>DiffeologicalSpaces</mi></mrow><annotation encoding="application/x-tex"> \iota \colon FrechetManifolds \to DiffeologicalSpaces </annotation></semantics></math></div> <p>in the evident way by taking for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifold">Fréchet manifold</a> for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> the set of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/plots">plots</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\iota(X)</annotation></semantics></math> to be the set of smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \to X</annotation></semantics></math>.</p> </div> <div class="num_prop" id="FrechetEmbedding"> <h6 id="proposition_3">Proposition</h6> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo lspace="verythinmathspace">:</mo><mi>FrechetManifolds</mi><mo>↪</mo><mi>DiffeologicalSpaces</mi></mrow><annotation encoding="application/x-tex">\iota \colon FrechetManifolds \hookrightarrow DiffeologicalSpaces</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a>.</p> </div> <p>This appears as (<a href="#Losik94">Losik 94, theorem 3.1.1</a>, following <a href="#Losik92">Losik 92</a>), as variant of the analogous statement for <a class="existingWikiWord" href="/nlab/show/Banach+manifolds">Banach manifolds</a> in (<a href="#Hain">Hain</a>). The fact that maps between Fréchet spaces are smooth if and only if they send smooth curves to smooth curves was proved earlier in (<a href="#Frolicher">Frölicher 81, théorème 1</a>)</p> <p>The statement is also implied by (<a href="#KrieglMichor">Kriegl-Michor 97, cor. 3.14</a>) which states that functions between <a class="existingWikiWord" href="/nlab/show/locally+convex+vector+spaces">locally convex vector spaces</a> are diffeologically smooth precisely if they send smooth <a class="existingWikiWord" href="/nlab/show/curves">curves</a> to smooth curves. This is not true if one uses <a class="existingWikiWord" href="/nlab/show/Michal-Bastiani+smooth+map">Michal-Bastiani smoothness</a> (<a href="#Glockner06">Glöckner 06</a>), in which case one merely has a <a class="existingWikiWord" href="/nlab/show/faithful+functor">faithful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>lctvs</mi><mo>→</mo><mi>DiffeologicalSpaces</mi></mrow><annotation encoding="application/x-tex">lctvs \to DiffeologicalSpaces</annotation></semantics></math>. Notice that the choice of topology in (<a href="#KrieglMichor">Kriegl-Michor 97</a>) is such that this equivalence of notions reduces to the above just for Fréchet manifolds.</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi><mo>∈</mo><mi>SmoothManifold</mi></mrow><annotation encoding="application/x-tex">X, Y \in SmoothManifold</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/compact+manifold">compact manifold</a>.</p> <p>Then under this embedding, the diffeological mapping space structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub></mrow><annotation encoding="application/x-tex">C^\infty(X,Y)_{diff}</annotation></semantics></math> on the mapping space coincides with the Fréchet manifold structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>Fr</mi></msub></mrow><annotation encoding="application/x-tex">C^\infty(X,Y)_{Fr}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ι</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>Fr</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><msub><mo stretchy="false">)</mo> <mi>diff</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \iota(C^\infty(X,Y)_{Fr}) \simeq C^\infty(X,Y)_{diff} \,. </annotation></semantics></math></div></div> <p>This appears as (<a href="#Waldorf09">Waldorf 09, lemma A.1.7</a>).</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <h3 id="EmbeddingOfDiffeologicalSpacesIntoTheSheafTopos">Embedding of diffeological spaces into smooth sets</h3> <p>We discuss how diffeological spaces are equivalently the <a class="existingWikiWord" href="/nlab/show/concrete+objects">concrete objects</a> in the <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> (see <a href="smooth+set#Cohesion">there</a>).</p> <div class="num_prop" id="DiffeologicalSpacesAreTheConcreteSmoothSets"> <h6 id="proposition_5">Proposition</h6> <p><strong>(diffeological spaces are the concrete smooth sets)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> on the <a class="existingWikiWord" href="/nlab/show/concrete+objects">concrete objects</a> in the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothSet</mi><mo>≔</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>Cart</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SmoothSet \coloneqq Sh(Cart)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a> is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the category of diffeological spaces</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/concrete+sheaves">concrete sheaves</a> for the <a class="existingWikiWord" href="/nlab/show/local+topos">local 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><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(CartSp)</annotation></semantics></math> are by definition those objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> for which the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Γ</mi><mo>⊣</mo><mi>CoDisc</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Gamma \dashv CoDisc)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>CoDisc</mi><mi>Γ</mi><mi>X</mi></mrow><annotation encoding="application/x-tex"> X \to CoDisc \Gamma X </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>. Monomorphisms of sheaves are tested objectwise, so that means equivalently that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>CartSp</mi></mrow><annotation encoding="application/x-tex">U \in CartSp</annotation></semantics></math> we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>Sh</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mi>Sh</mi></msub><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>Codisc</mi><mi>Γ</mi><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><mi>Γ</mi><mi>U</mi><mo>,</mo><mi>Γ</mi><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> X(U) \simeq Hom_{Sh}(U,X) \to Hom_{Sh}(U, Codisc \Gamma X) \simeq Hom_{Set}(\Gamma U, \Gamma X) </annotation></semantics></math></div> <p>is a monomorphism. This is precisely the condition on a sheaf to be a diffeological space.</p> </div> <p>For a fully detailed proof see <a href="geometry+of+physics+--+smooth+sets#DiffeologicalSpacesAreTheConcreteSmoothSets">this Prop.</a> at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+sets">geometry of physics – smooth sets</a></em>.</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>The category of diffeological spaces is a <a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a>.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>This follows from the discussion at <a href="#Locality">Locality</a>.</p> </div> <p>This has some immediate general abstract consequences</p> <div class="num_cor"> <h6 id="corollary_2">Corollary</h6> <p>The category of diffeological spaces is</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a>.</p> </li> </ul> </div> <h3 id="EmbeddingOfDiffeologicalSpacesIntoHigherDifferentialGeometry">Embedding of diffeological spaces into higher differential geometry</h3> <p>In the last section we saw the embedding of diffeological spaces as precisely the <a class="existingWikiWord" href="/nlab/show/concrete+objects">concrete objects</a> is the <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><mi>CartSp</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Sh</mi><mo stretchy="false">(</mo><mi>SmthMfd</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh(CartSp) \simeq Sh(SmthMfd)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>. This is a general context for <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>. From there one can pass further to <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>: the topos of smooth sets in turn embeds</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Sh</mi><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>Smooth</mi><mn>∞</mn><mi>Grpd</mi><mo>≔</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>CartSp</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Sh(CartSp) \hookrightarrow Smooth \infty Grpd \coloneqq Sh_\infty(CartSp) </annotation></semantics></math></div> <p>into the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> of “higher <a class="existingWikiWord" href="/nlab/show/smooth+sets">smooth sets</a>” –<a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a> – as precisely the <a class="existingWikiWord" href="/nlab/show/truncated+object+in+an+%28%E2%88%9E%2C1%29-category">0-truncated objects</a>.</p> <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 diffeological space, 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="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">model structure on diffeological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+set">smooth set</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Diff">Diff</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+groupoid">diffeological groupoid</a>, <a class="existingWikiWord" href="/nlab/show/diffeological+%E2%88%9E-groupoid">diffeological ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connectology">connectology</a></p> </li> </ul> <h2 id="References">References</h2> <h3 id="ReferencesGeneral">Foundations</h3> <p>The basic idea of understanding <a class="existingWikiWord" href="/nlab/show/generalized+smooth+spaces">generalized smooth spaces</a> as <a class="existingWikiWord" href="/nlab/show/concrete+sheaves">concrete sheaves</a> on a <a class="existingWikiWord" href="/nlab/show/site">site</a> of smooth test spaces originates in work of <a class="existingWikiWord" href="/nlab/show/Kuo+Tsai+Chen">Kuo Tsai Chen</a> (see also at <em><a class="existingWikiWord" href="/nlab/show/Chen+space">Chen space</a></em>):</p> <ul> <li id="Chen73"><a class="existingWikiWord" href="/nlab/show/Kuo+Tsai+Chen">Kuo Tsai Chen</a>, <em>Iterated integrals of differential forms and loop space homology</em>, Ann. Math. 97 (1973), 217–246 (<a href="https://www.jstor.org/stable/1970846">jstor:1970846</a>)</li> </ul> <p>Chen considered (apart from <a class="existingWikiWord" href="/nlab/show/iterated+integrals">iterated integrals</a>) effectively <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> on a site of <a class="existingWikiWord" href="/nlab/show/convex+subsets">convex subsets</a> of <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a>. In</p> <ul> <li id="Chen75"><a class="existingWikiWord" href="/nlab/show/Kuo+Tsai+Chen">Kuo Tsai Chen</a>, <em>Iterated integrals, fundamental groups and covering spaces</em>, Trans. Amer. Math. Soc. 206 (1975), 83–98 (<a href="https://www.jstor.org/stable/1997148">jstor:1997148</a>)</li> </ul> <p>roughly the <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> condition was added (without using any of this sheaf-theoretic terminology). The definition of <em><a class="existingWikiWord" href="/nlab/show/Chen+spaces">Chen spaces</a></em> stabilized in</p> <ul> <li id="Chen77"><a class="existingWikiWord" href="/nlab/show/Kuo+Tsai+Chen">Kuo Tsai Chen</a>, <em>Iterated path integrals</em>, Bull. Amer. Math. Soc. 83, (1977), 831–879 (<a href="https://projecteuclid.org/euclid.bams/1183539443">euclid:1183539443</a>)</li> </ul> <p>and served as the basis of a celebrated theorem on the <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> of <a class="existingWikiWord" href="/nlab/show/loop+spaces">loop spaces</a>.</p> <p>A brief review is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kuo-Tsai+Chen">Kuo-Tsai Chen</a>, <em>On differentiable spaces</em>, in: <a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Schanuel">Stephen Schanuel</a> (eds.), <em><a class="existingWikiWord" href="/nlab/show/Categories+in+Continuum+Physics">Categories in Continuum Physics</a></em>, Lectures given at a Workshop held at SUNY, Buffalo 1982, Lecture Notes in Mathematics 1174, 1986 (<a href="https://link.springer.com/book/10.1007/BFb0076928">doi:10.1007/BFb0076928</a>)</li> </ul> <p>(which, curiously, still does not make the connection to the theory of sheaves).</p> <p id="However"> However, Chen does not require the domains of his plots to be <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a>, which makes Chen spaces be closely related to but different from <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> (see <a href="#Stacey11">Stacey 11, p. 32</a>)</p> <p>The proper concept of diffeological spaces was introduced, under the name <em>difféologie</em> and apparently independently from Chen, in:</p> <ul> <li id="Souriau79"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>, <em>Groupes différentiels</em>, in <em>Differential Geometrical Methods in Mathematical Physics</em> (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Math. 836, Springer, Berlin, (1980), pp. 91–128. (<a href="https://doi.org/10.1007/BFb0089728">doi:10.1007/BFb0089728</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=607688">mr:607688</a>)</p> </li> <li id="Souriau84"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Marie+Souriau">Jean-Marie Souriau</a>, <em>Groupes différentiels et physique mathématique</em>, In: Denardo G., Ghirardi G., Weber T. (eds.) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 201. Springer 1984 (<a href="https://doi.org/10.1007/BFb0016198">doi:10.1007/BFb0016198</a>)</p> </li> </ul> <p>motivated there by <a class="existingWikiWord" href="/nlab/show/diffeological+groups">diffeological groups</a> arising in <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>.</p> <p>The article</p> <ul> <li id="BaezHoffnung11"><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Hoffnung">Alexander Hoffnung</a>, <em>Convenient Categories of Smooth Spaces</em>, Transactions of the American Mathematical Society <strong>363</strong> 11 (2011) [<a href="http://arxiv.org/abs/0807.1704">arXiv:0807.1704</a>, <a href="https://www.jstor.org/stable/41307457">jstor:41307457</a>]</li> </ul> <p>proved that diffeological spaces are <a class="existingWikiWord" href="/nlab/show/concrete+sheaves">concrete sheaves</a> forming a <a class="existingWikiWord" href="/nlab/show/quasi-topos">quasi-topos</a>.</p> <p>A discussion of the relations of variants of the definition is in</p> <ul> <li id="Stacey11"><a class="existingWikiWord" href="/nlab/show/Andrew+Stacey">Andrew Stacey</a>, <em>Comparative Smootheology</em>, Theory and Applications of Categories, Vol. 25, 2011, No. 4, pp 64-117. (<a href="http://www.tac.mta.ca/tac/volumes/25/4/25-04abs.html">tac:25-04</a>)</li> </ul> <h3 id="differential_geometry_of_diffeological_spaces">Differential geometry of diffeological spaces</h3> <p>Following Souriau, a comprehensive textbook account of <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> formulated in terms of <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> (and coining that term) is</p> <ul> <li id="PIZ"><a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, <em>Diffeology</em>, Mathematical Surveys and Monographs, AMS (2013) (<a href="http://math.huji.ac.il/~piz/Site/The%20Book.html">web</a>, <a href="https://bookstore.ams.org/surv-185">ISBN:978-0-8218-9131-5</a>)</li> </ul> <p>following the thesis</p> <ul> <li id="IglesiasZemmour85"><a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, <em>Fibrations difféologiques et Homotopie</em>, Dissertation (1985) (<a href="http://math.huji.ac.il/~piz/Site/The%20Articles/D9DD15EE-6993-4CA3-8B9B-4FC1DEF4A418.html">web</a>, <a href="http://math.huji.ac.il/~piz/documents/TheseEtatPI.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/IglesiasZemmourFibrationsDiffeologiques1985.pdf" title="pdf">pdf</a>)</li> </ul> <p>which contains some useful material that may not yet have made it into the book.</p> <p>Further exposition and lecture notes are in</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, <em>Diffeologies</em>, talk at <em><a class="existingWikiWord" href="/nlab/show/New+Spaces+for+Mathematics+and+Physics">New Spaces for Mathematics and Physics</a></em>, IHP Paris 2015 (<a href="https://www.youtube.com/watch?v=4sZDmiVOhaA">video recording</a>)</p> </li> <li id="IglesiasZemmour18"> <p><a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, <em>An introduction to diffeology</em>, lecture at <em><a href="http://www.turkmath.org/beta/konferans.php?id_konferans=365">Modern Mathematics Methods in Physics: Diffeology, Categories and Toposes and Non-commutative Geometry Summer School</a></em>, 2018, to appear in <em><a class="existingWikiWord" href="/nlab/show/New+Spaces+for+Mathematics+and+Physics">New Spaces for Mathematics and Physics</a></em> (<a href="http://math.huji.ac.il/~piz/documents/AITD.pdf">pdf</a>)</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Cartan+calculus">Cartan calculus</a> for <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a> is developed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Christian+Blohmann">Christian Blohmann</a>, <em>Elastic diffeological spaces</em>, <a href="https://arxiv.org/abs/2301.02583">arXiv:2301.02583</a>.</li> </ul> <p>Application to <a class="existingWikiWord" href="/nlab/show/classifying+spaces">classifying spaces</a> and <a class="existingWikiWord" href="/nlab/show/universal+connections">universal connections</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mark+Mostow">Mark Mostow</a>, <em>The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations</em>, J. Differential Geom. Volume 14, Number 2 (1979), 255-293 (<a href="https://projecteuclid.org/euclid.jdg/1214434974">euclid:jdg/1214434974</a>)</li> </ul> <p>More pointers and monthly seminars at:</p> <ul> <li><a href="http://diffeology.net/">diffeology.net</a></li> </ul> <h3 id="full_subcategories">Full subcategories</h3> <p>The <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a>-inclusion of <a class="existingWikiWord" href="/nlab/show/Banach+manifolds">Banach manifolds</a> into the category of diffeological spaces is due to</p> <ul> <li id="Hain"><a class="existingWikiWord" href="/nlab/show/Richard+Hain">Richard Hain</a>, <em>A characterization of smooth functions defined on a Banach space</em>, Proc. Amer. Math. Soc. 77 (1979), 63-67 (<a href="http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/home.html">web</a>, <a href="http://www.ams.org/journals/proc/1979-077-01/S0002-9939-1979-0539632-8/S0002-9939-1979-0539632-8.pdf">pdf</a>)</li> </ul> <p>The (non-full) embedding of <a class="existingWikiWord" href="/nlab/show/locally+convex+vector+spaces">locally convex vector spaces</a> and <a class="existingWikiWord" href="/nlab/show/Michal-Bastiani+smooth+maps">Michal-Bastiani smooth maps</a> into diffeological spaces is discussed around corollary 3.14 in</p> <ul> <li id="KrieglMichor"><a class="existingWikiWord" href="/nlab/show/Andreas+Kriegl">Andreas Kriegl</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Michor">Peter Michor</a>: <em><a class="existingWikiWord" href="/nlab/show/The+convenient+setting+of+global+analysis">The convenient setting of global analysis</a></em>, AMS (1997)</li> </ul> <p>That there are diffeologically-smooth maps between locally convex vector spaces that are not continuous, and a fortiori not smooth in the sense of Michal-Bastiani is given, for instance, in</p> <ul> <li id="Glockner06">Helge Glöckner, <em>Discontinuous non-linear mappings on locally convex direct limits</em>, Publ. Math. Debrecen 68 (2006) 1-13, <a href="http://arxiv.org/abs/math/0503387">arXiv:math/0503387</a>.</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a>-inclusion of <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9chet+manifolds">Fréchet manifolds</a> into diffeological spaces is discussed in</p> <ul> <li id="Losik92"><a class="existingWikiWord" href="/nlab/show/Mark+Losik">Mark Losik</a>, <em>Fréchet manifolds as diffeologic spaces</em>, Russian Mathematics 36:5 (1992), 36–42. English translation: <a href="https://dmitripavlov.org/scans/losik-frechet-manifolds-as-diffeologic-spaces.pdf">PDF</a>. Russian original: (<a href="http://mi.mathnet.ru/eng/ivm4812">mathnet:ivm4812</a>)</li> </ul> <p>and reviewed in</p> <ul> <li id="Losik94"><a class="existingWikiWord" href="/nlab/show/Mark+Losik">Mark Losik</a>, Section 3 of: <em>Categorical Differential Geometry</em>. Cah. Topol. Géom. Différ. Catég., 35(4):274–290, 1994 (<a href="http://www.numdam.org/item/CTGDC_1994__35_4_274_0">numdam:CTGDC_1994__35_4_274_0</a>)</li> </ul> <p>The proof can in fact be deduced from théorème 1 of</p> <ul> <li id="Frolicher"><a class="existingWikiWord" href="/nlab/show/Alfred+Fr%C3%B6licher">Alfred Frölicher</a>, <em>Applications lisses entre espaces et variétés de Fréchet</em>, C. R. Acad. Sci. Paris Sér. I Math. <strong>293</strong> (1981), no. 2, 125–127. <a href="http://gallica.bnf.fr/ark:/12148/bpt6k5533894s/f31.image">BnF</a></li> </ul> <p>The preservation of <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> under this embedding is due to</p> <ul> <li id="Waldorf09"><a class="existingWikiWord" href="/nlab/show/Konrad+Waldorf">Konrad Waldorf</a>, <em>Transgression to Loop Spaces and its Inverse I</em>, Cah. Topol. Geom. Differ. Categ., 2012, Vol. LIII, 162-210 (<a href="http://arxiv.org/abs/0911.3212">arXiv:0911.3212</a>)</li> </ul> <p>The largest <a class="existingWikiWord" href="/nlab/show/topological+space">topology</a> on the set which underlies a diffeological space with respect to which all <a class="existingWikiWord" href="/nlab/show/plots">plots</a> are <a class="existingWikiWord" href="/nlab/show/continuous+functions">continuous functions</a> (the “<a class="existingWikiWord" href="/nlab/show/D-topology">D-topology</a>”) is studied in</p> <ul> <li id="SYH10"> <p><a class="existingWikiWord" href="/nlab/show/Kazuhisa+Shimakawa">Kazuhisa Shimakawa</a>, K. Yoshida, <a class="existingWikiWord" href="/nlab/show/Tadayuki+Haraguchi">Tadayuki Haraguchi</a>, <em>Homology and cohomology via enriched bifunctors</em>, Kyushu Journal of Mathematics, 2018 Volume 72 Issue 2 Pages 239-252 (<a href="https://arxiv.org/abs/1010.3336">arXiv:1010.3336</a>, <a href="https://doi.org/10.2206/kyushujm.72.239">doi:10.2206/kyushujm.72.239</a>)</p> </li> <li id="CSW13"> <p><a class="existingWikiWord" href="/nlab/show/J.+Daniel+Christensen">J. Daniel Christensen</a>, Gord Sinnamon, <a class="existingWikiWord" href="/nlab/show/Enxin+Wu">Enxin Wu</a>, <em>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>-topology for diffeological spaces</em>, Pacific Journal of Mathematics 272 (1), 87-110, 2014 (<a href="http://arxiv.org/abs/1302.2935">arXiv:1302.2935</a>, <a href="https://msp.org/pjm/2014/272-1/p04.xhtml">doi:10.2140/pjm.2014.272.87</a>)</p> </li> </ul> <p>Some <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> modeled on diffeological spaces (instead of on <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>) via their <a class="existingWikiWord" href="/nlab/show/cohesion">cohesive</a> <a class="existingWikiWord" href="/nlab/show/shape+modality">shape</a> is discussed in</p> <ul> <li id="ChristensenWu13"><a class="existingWikiWord" href="/nlab/show/J.+Daniel+Christensen">J. Daniel Christensen</a>, <a class="existingWikiWord" href="/nlab/show/Enxin+Wu">Enxin Wu</a>, <em>The homotopy theory of diffeological spaces, I. Fibrant and cofibrant objects</em>, New York J. Math. 20 (2014), 1269-1303 (<a href="http://arxiv.org/abs/1311.6394">arXiv:1311.6394</a>)</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a>-inclusion of <a class="existingWikiWord" href="/nlab/show/manifolds+with+boundaries+and+corners">manifolds with boundaries and corners</a> is discussed in</p> <ul> <li id="GurerIZ19"><a class="existingWikiWord" href="/nlab/show/Serap+G%C3%BCrer">Serap Gürer</a>, <a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, <em>Differential forms on manifolds with boundary and corners</em>, Indagationes Mathematicae, Volume 30, Issue 5, September 2019, Pages 920-929 (<a href="https://doi.org/10.1016/j.indag.2019.07.004">doi:10.1016/j.indag.2019.07.004</a>)</li> </ul> <h3 id="ReferencesForOrbifolds">For orbifolds</h3> <p>On <a class="existingWikiWord" href="/nlab/show/orbifolds">orbifolds</a> regarded as naive local <a class="existingWikiWord" href="/nlab/show/quotient+spaces">quotient spaces</a> (instead of <a class="existingWikiWord" href="/nlab/show/homotopy+quotients">homotopy quotients</a>/<a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a>/<a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a>) but as such formed in <a class="existingWikiWord" href="/nlab/show/diffeological+spaces">diffeological spaces</a>:</p> <ul> <li id="IKZ10"> <p><a class="existingWikiWord" href="/nlab/show/Patrick+Iglesias-Zemmour">Patrick Iglesias-Zemmour</a>, <a class="existingWikiWord" href="/nlab/show/Yael+Karshon">Yael Karshon</a>, Moshe Zadka, <em>Orbifolds as diffeologies</em>, Transactions of the American Mathematical Society 362 (2010), 2811-2831 (<a href="https://arxiv.org/abs/math/0501093">arXiv:math/0501093</a>)</p> </li> <li id="Watts15"> <p><a class="existingWikiWord" href="/nlab/show/Jordan+Watts">Jordan Watts</a>, <em>The Differential Structure of an Orbifold</em>, Rocky Mountain Journal of Mathematics, Vol. 47, No. 1 (2017), pp. 289-327 (<a href="https://arxiv.org/abs/1503.01740">arXiv:1503.01740</a>)</p> </li> </ul> <p>On this approach seen in the broader context of <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/Proper+Orbifold+Cohomology">Proper Orbifold Cohomology</a></em> (<a href="https://arxiv.org/abs/2008.01101">arXiv:2008.01101</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 21, 2023 at 10:03:04. 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