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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="manifolds_and_cobordisms">Manifolds and cobordisms</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a></strong> and <strong><a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</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> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <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/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> </ul> </div></div> <h4 id="geometry">Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong> / <strong><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#definition_via_pseudogroups'>Definition via pseudogroups</a></li> <ul> <li><a href='#definitions_2'>Definitions</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#isomorphisms_of_manifolds'>Isomorphisms of manifolds</a></li> </ul> <li><a href='#definition_via_cartologies'>Definition via cartologies</a></li> </ul> <li><a href='#generalizations'>Generalizations</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <div style="float:right;margin:0 10px 10px 0;"> <img src="https://ncatlab.org/nlab/files/ChartsOfAManifold.png" width="500" /> </div> <p>A <em>manifold</em> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> which looks locally like a <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>, commonly a finite-dimensional Cartesian space <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>, in which case one speaks of a <em>manifold of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <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></em> or <em><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>-fold</em>, but possibly an infinite-dimensional <a class="existingWikiWord" href="/nlab/show/topological+vector+space">topological vector space</a>, in which case one has an <a class="existingWikiWord" href="/nlab/show/infinite-dimensional+manifold">infinite-dimensional manifold</a>.</p> <p>What “locally looks like” means depends on what sort of structure we are considering a Cartesian space to embody. At one extreme, we can think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math> as merely a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>. Or, <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> may be considered as carrying more rigid types of structure, such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>k</mi></msup></mrow><annotation encoding="application/x-tex">C^k</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/differential+structure">differential structure</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/PL+structure">piecewise-linear (PL) structure</a>, real <a class="existingWikiWord" href="/nlab/show/analytic+function">analytic structure</a>, affine structure, hyperbolic structure, foliated structure, etc., etc. Accordingly we have notions of <em><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></em>, <em><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a></em>, <em><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></em>, <em><a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a></em> etc. By default these are modeled on <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> spaces, but most notions have generalizations to a corresponding notion of <a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a>.</p> <blockquote> <p>graphics grabbed from <a class="existingWikiWord" href="/nlab/show/The+Geometry+of+Physics+-+An+Introduction">Frankel</a></p> </blockquote> <p>In any case, the type of geometry embodied in a particular flavor of manifold is controlled by a particular <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> or, more generally, <a class="existingWikiWord" href="/nlab/show/category">category</a> of transformations which preserves whatever geometric features one is interested in; cf. Felix Klein’s <em><a class="existingWikiWord" href="/nlab/show/Erlangen+program">Erlanger Programm</a></em>.</p> <h2 id="definitions">Definitions</h2> <p>Here we will focus on the general notion of a manifold. More concrete examples can be found in individual pages such as <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a> and <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>.</p> <p>We will present two possible definitions. The first, via <a class="existingWikiWord" href="/nlab/show/pseudogroups">pseudogroups</a>, has a simpler definition, but has two (rather serious) drawbacks:</p> <ol> <li>There is in general no notion of morphisms between manifolds. At best, we can only talk about isomorphisms of manifolds.</li> <li>We can only define smooth <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>-manifolds for each <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> separately, but not smooth manifolds in general (and the same applies to complex manifolds etc.)</li> </ol> <p>The second definition via cartologies was proposed by <a class="existingWikiWord" href="/nlab/show/Todd+Trimble">Todd Trimble</a> to solve the above two problems.</p> <h3 id="definition_via_pseudogroups">Definition via pseudogroups</h3> <p>The setting is 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> together with a <a class="existingWikiWord" href="/nlab/show/pseudogroup">pseudogroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> 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>. For the sake of concreteness, the reader may as well focus on the case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">X = \mathbb{R}^n</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> of <a class="existingWikiWord" href="/nlab/show/diffeomorphisms">diffeomorphisms</a> between 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>.</p> <h4 id="definitions_2">Definitions</h4> <div class="num_defn" id="Chart"> <h6 id="definition">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/chart">chart</a></strong> on 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>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is an <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>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> together with an open embedding</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>U</mi><mo>→</mo><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi: U \to X\,. </annotation></semantics></math></div> <p>Two charts <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><mi>U</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi: U \to X</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\psi: V \to X</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<strong>compatible</strong> if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>∘</mo><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>:</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>U</mi><mo>∩</mo><mi>V</mi><mo stretchy="false">)</mo><mo>→</mo><mi>ψ</mi><mo stretchy="false">(</mo><mi>U</mi><mo>∩</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\psi \circ \phi^{-1}: \phi(U \cap V) \to \psi(U \cap V)</annotation></semantics></math></div> <p>belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> </div> <div class="num_defn" id="Atlas"> <h6 id="definition_2">Definition</h6> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<strong>atlas</strong> on 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>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a family of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-compatible charts <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>α</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>α</mi></msub><mo>→</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">(\phi_\alpha: U_\alpha \to X)_\alpha</annotation></semantics></math>, def. <a class="maruku-ref" href="#Chart"></a>, such that <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>α</mi></msub><msub><mo stretchy="false">)</mo> <mi>α</mi></msub></mrow><annotation encoding="application/x-tex">(U_\alpha)_\alpha</annotation></semantics></math> covers <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. The (restricted) maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mrow><mi>α</mi><mi>β</mi></mrow></msub><mo>=</mo><msub><mi>ϕ</mi> <mi>β</mi></msub><mo>∘</mo><msubsup><mi>ϕ</mi> <mi>α</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\phi_{\alpha \beta} = \phi_\beta \circ \phi_{\alpha}^{-1}</annotation></semantics></math> are called <strong>transition functions</strong> between the charts of the atlas.</p> </div> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<strong>manifold</strong> is a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> equipped with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-atlas (definition <a class="maruku-ref" href="#Atlas"></a>).</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This means that we can think of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-manifold as a space which is <em>locally modeled</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> according to the geometry <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>It is almost invariably the case in classical manifold theory that one requires some technical <a class="existingWikiWord" href="/nlab/show/nice+topological+space">niceness</a> properties on the <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> underlying a manifold.</p> <p>Usually, in the definition of manifold it is understood that the underlying topological space</p> <ol> <li> <p>is a <a class="existingWikiWord" href="/nlab/show/Hausdorff+topological+space">Hausdorff topological space</a> (if not one usually speaks explicitly of a <em><a class="existingWikiWord" href="/nlab/show/non-Hausdorff+manifold">non-Hausdorff manifold</a></em>)</p> </li> <li> <p>is a <a class="existingWikiWord" href="/nlab/show/paracompact+topological+space">paracompact topological space</a>;</p> </li> </ol> <p>Often it is also assumed that the topology has a countable <a class="existingWikiWord" href="/nlab/show/basis+of+a+topology">basis</a> as well.</p> <p>In many of the typical cases, this will mean that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/metric+space">metrizable</a>. In many studies, for example in <a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, one goes even further and assumes the manifolds are <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>An atlas is <em>not</em> considered an essential part of the structure of a manifold: two different atlases may yield the same manifold structure. This is encoded by the following definition <a class="maruku-ref" href="#IsomorphismOfManifolds"></a> of <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> between manifolds.</p> </div> <h4 id="examples">Examples</h4> <p>If the term “manifold” appears without further qualification, what is usually meant is a <strong>smooth <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>-manifold</strong> of some <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> <strong>dimension</strong> <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>: a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-manifold where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is the pseudogroup of invertible <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> maps between open sets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>. Replacing <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> here by a half-space <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><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>:</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>≥</mo><mn>0</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{x \in \mathbb{R}^n: x_1 \geq 0\}</annotation></semantics></math>, one obtains the notion of smooth <strong>manifold with boundary</strong>. Or, replacing <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> here by the <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>-cube <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>I</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">I^n</annotation></semantics></math>, one obtains the notion of (smooth) <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>-<strong>manifold with (cubical) corners</strong>. Morphisms of manifolds are here called <strong>smooth maps</strong>, and isomorphisms are called <strong>diffeomorphisms</strong>. (In manifold theory, one usually reserves the term <strong>smooth function</strong> for smooth maps to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>.)</p> <p>A <strong>topological <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>-manifold</strong> is a manifold with respect to the pseudogroup of homeomorphisms between open sets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^n</annotation></semantics></math>. Any continuous function between topological manifolds is a morphisms, and any homeomorphism is an isomorphism. A <strong><a class="existingWikiWord" href="/nlab/show/piecewise-linear+manifold">piecewise-linear (PL) <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>-manifold</a></strong> is where the pseudogroup consists of piecewise-linear homeomorphisms between such open sets; morphisms are called <strong>piecewise-linear (PL) maps</strong>.</p> <p>One can go on to define, in a straightforward way, real <a class="existingWikiWord" href="/nlab/show/analytic+manifolds">analytic manifolds</a>, complex analytic manifolds, elliptic manifolds, hyperbolic manifolds, and so on, using the general notion of pseudogroup.</p> <p>Any 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> can always be turned into a manifold modelled on itself, using any pseudogroup <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>. Simply take the inclusions of open sets as charts.</p> <h4 id="isomorphisms_of_manifolds">Isomorphisms of manifolds</h4> <div class="num_defn" id="IsomorphismOfManifolds"> <h6 id="definition_4">Definition</h6> <p>An <strong><a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-manifolds <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">f: M \to N</annotation></semantics></math> (defined by chosen atlas structures, def. <a class="maruku-ref" href="#Atlas"></a>) is a <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo stretchy="false">(</mo><mi>U</mi><mo>∩</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msup><mi>ϕ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mover><mi>U</mi><mo>∩</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>f</mi></mover><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>V</mi><mover><mo>→</mo><mi>ψ</mi></mover><mi>ψ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>∩</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \phi(U \cap f^{-1}(V)) \overset{\phi^{-1}}{\to} U \cap f^{-1}(V) \overset{f}{\to} f(U) \cap V \overset{\psi}{\to} \psi(f(U) \cap V) </annotation></semantics></math></div> <p>is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U, \phi)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, def. <a class="maruku-ref" href="#Chart"></a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">x \in M</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(V, \psi)</annotation></semantics></math> is a coordinate chart of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">f(x) \in N</annotation></semantics></math>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">M_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_2</annotation></semantics></math> are two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-manifold structures on the same topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">M_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_2</annotation></semantics></math> are considered <strong>equal</strong> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-manifolds if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">id: M \to M</annotation></semantics></math> is an isomorphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">M_1</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_2</annotation></semantics></math> (and hence also from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">M_2</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>M</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">M_1</annotation></semantics></math>).</p> </div> <p>Alternatively, atlases are ordered by inclusion, and two atlases define the same manifold structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> if they have a common upper bound. Equivalently, two atlases define the same manifold structure if each chart of one is compatible with each chart of the other. Or, one could extend any atlas to the (unique) maximal atlas containing it, which consists of all charts compatible with each of the charts in the original atlas, and simply identify a manifold structure with a maximal atlas.</p> <div class="query"> <p><em>Rafael</em>: Can one define a manifold object in a category C as a G-manifold with G related to C? What would the relation between G and C be to obtain G-manifolds in C as manifold objects?</p> <p><em>Toby</em>: Yes, I think that this would make perfect sense; I think that we'd want <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> to be an <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Note that defining things like ‘smooth manifold’ in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> might still be difficult, but we've reduced it to internalising <a class="existingWikiWord" href="/nlab/show/Cart+Sp">Cart Sp</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. (There's also the matter that the above definition takes a notion of <a class="existingWikiWord" href="/nlab/show/space">space</a> for granted, so you'd have to internalise that into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> too, but I'm not sure how important that really is, when I think about how the topology on a smooth manifold can be recovered from the smooth structure.)</p> <p><em>Rafael</em>: Can someone that knows more than me about this add the result of this question to this article so nobody have to ask again.</p> <p><em>Toby</em>: I'd rather not, since it's all ‘I think’ and ‘might be difficult’; it's better as a query box, moved to the bottom if necessary. But if Todd agrees with me, then maybe he'll add it.</p> </div> <h3 id="definition_via_cartologies">Definition via cartologies</h3> <div class="query"> <p>Note: the following is tentative “original research”. It is prompted by the desire to extend the pseudogroup approach for defining general notions of manifold, so as to cover also an appropriate general notion of “map”. Comments, improvements, and corrections are encouraged – <em>Todd</em>.</p> <p>I've read through it once, and it makes sense. I'll read through it again more carefully later. —Toby</p> </div> <p>We begin by defining the <a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> (i.e., locally <a class="existingWikiWord" href="/nlab/show/preorder">preorder</a>ed <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a>) of <strong>regions</strong>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Reg</mi></mrow><annotation encoding="application/x-tex">Reg</annotation></semantics></math>. The objects are topological spaces (or locales if you prefer); the morphisms are <a class="existingWikiWord" href="/nlab/show/partial+functions">partial functions</a> with open domain, that is spans</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mi>i</mi></mover><mi>U</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \overset{i}{\leftarrow} U \overset{f}{\to} Y</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is continuous and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is an open embedding. The spans are locally (that is, for fixed <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>) ordered by inclusion.</p> <p>These local posets are not cocomplete, but they admit certain obvious <a class="existingWikiWord" href="/nlab/show/joins">joins</a>: given a family of regional maps</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><msub><mi>U</mi> <mi>α</mi></msub><mo>,</mo><msub><mi>f</mi> <mi>α</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">(U_\alpha, f_\alpha): X \to Y</annotation></semantics></math></div> <p>the join <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">⋁</mo> <mi>α</mi></msub><mo stretchy="false">(</mo><msub><mi>U</mi> <mi>α</mi></msub><mo>,</mo><msub><mi>f</mi> <mi>α</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\bigvee_\alpha (U_\alpha, f_\alpha)</annotation></semantics></math> exists iff we have local compatibility:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>α</mi></msub><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>α</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>β</mi></msub></mrow></msub><mo>=</mo><msub><mi>f</mi> <mi>β</mi></msub><msub><mo stretchy="false">|</mo> <mrow><msub><mi>U</mi> <mi>α</mi></msub><mo>∩</mo><msub><mi>U</mi> <mi>β</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">f_{\alpha}|_{U_\alpha \cap U_\beta} = f_{\beta}|_{U_\alpha \cap U_\beta}</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>,</mo><mi>β</mi></mrow><annotation encoding="application/x-tex">\alpha, \beta</annotation></semantics></math>. Notice that composition on either side with a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-cell preserves any local joins which exist.</p> <p>Every coreflexive morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≤</mo><msub><mn>1</mn> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">r \leq 1_X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Reg</mi></mrow><annotation encoding="application/x-tex">Reg</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/retract">splits</a>: there is a map in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Reg</mi></mrow><annotation encoding="application/x-tex">Reg</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>Ext</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mover><mo>←</mo><mi>id</mi></mover><mi>Ext</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>i</mi></mover><mi>X</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">Ext(r) \overset{id}{\leftarrow} Ext(r) \overset{i}{\to} X,</annotation></semantics></math></div> <p>whose opposite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mi>op</mi></msup><mo>:</mo><mi>X</mi><mo>→</mo><mi>Ext</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i^op: X \to Ext(r)</annotation></semantics></math> also belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Reg</mi></mrow><annotation encoding="application/x-tex">Reg</annotation></semantics></math> (that is, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is an open embedding), and the equations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mi>op</mi></msup><mo>∘</mo><mi>i</mi><mo>=</mo><msub><mn>1</mn> <mrow><mi>Ext</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="2em"></mspace><mi>i</mi><mo>∘</mo><msup><mi>i</mi> <mi>op</mi></msup><mo>=</mo><mi>r</mi></mrow><annotation encoding="application/x-tex">i^{op} \circ i = 1_{Ext(r)} \qquad i \circ i^{op} = r</annotation></semantics></math></div> <p>hold. The object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ext</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ext(r)</annotation></semantics></math> may be called the <em>extension</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math>. This splitting is a kind of <span class="newWikiWord">comprehension principle<a href="/nlab/new/comprehension+principle">?</a></span> familiar from the theory of <a class="existingWikiWord" href="/nlab/show/allegory">allegories</a>, among other things.</p> <p>A <strong>cartology</strong> is a (locally <a class="existingWikiWord" href="/nlab/show/full+subcategory">full</a>) subbicategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>C</mi><mo>↪</mo><mi>Reg</mi></mrow><annotation encoding="application/x-tex">i: C \hookrightarrow Reg</annotation></semantics></math> such that</p> <ul> <li>(Closure under open subspaces) If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Ob</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in Ob(C)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>≤</mo><msub><mn>1</mn> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">r \leq 1_X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Reg</mi></mrow><annotation encoding="application/x-tex">Reg</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>Ext</mi><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">i: Ext(r) \to X</annotation></semantics></math> and its opposite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>i</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">i^{op}</annotation></semantics></math> are morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>.</li> <li>(“Sheaf condition”) The inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>Reg</mi></mrow><annotation encoding="application/x-tex">i: C \to Reg</annotation></semantics></math> reflects and preserves local joins.</li> </ul> <p>Intended examples include the case where the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> are 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 morphisms are spans</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>U</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>ℝ</mi> <mi>m</mi></msup><mo>→</mo><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">(U, f): \mathbb{R}^m \to \mathbb{R}^n</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is smooth.</p> <p>Given a cartology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>=</mo><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">r = (U, f): X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <strong>pseudo-invertible</strong> if there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>=</mo><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Y</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">s = (V, g): Y \to X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∘</mo><mi>r</mi><mo>=</mo><msub><mn>1</mn> <mi>U</mi></msub></mrow><annotation encoding="application/x-tex">s \circ r = 1_U</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∘</mo><mi>s</mi><mo>=</mo><msub><mn>1</mn> <mi>V</mi></msub></mrow><annotation encoding="application/x-tex">r \circ s = 1_V</annotation></semantics></math>.</p> <div class="num_lemma"> <h6 id="lemma">Lemma</h6> <p>In a cartology, the pseudo-invertible morphisms from an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to itself form a pseudogroup (as defined earlier).</p> </div> <p>The notion of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-manifold modeled on an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is defined just as before, using the pseudogroup 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> implied by the previous lemma. In particular, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-<strong>charts</strong> of an atlas structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, which are morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Reg</mi></mrow><annotation encoding="application/x-tex">Reg</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><mover><mo>←</mo><mi>i</mi></mover><mi>U</mi><mover><mo>→</mo><mi>ϕ</mi></mover><mi>M</mi></mrow><annotation encoding="application/x-tex">X \overset{i}{\leftarrow} U \overset{\phi}{\to} M</annotation></semantics></math></div> <p>satisfying the expected properties. We can thus speak of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-<strong>manifolds</strong> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C, X)</annotation></semantics></math>-manifolds if we want to make explicit the modeling 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>).</p> <p>Now, given a cartology <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, we define the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-manifolds. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> be a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C, X)</annotation></semantics></math>-manifold and <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> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C, Y)</annotation></semantics></math>-manifold. Then, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>-<strong>morphism</strong> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> to <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 a continuous map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">f: M \to N</annotation></semantics></math> such that the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Reg</mi></mrow><annotation encoding="application/x-tex">Reg</annotation></semantics></math>-composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>U</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>M</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>V</mi></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>i</mi><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo><mi>ϕ</mi></mtd> <mtd></mtd> <mtd><mn>1</mn><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo><mi>f</mi></mtd> <mtd></mtd> <mtd><mi>ψ</mi><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo><mi>j</mi></mtd> <mtd></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>M</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>N</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ && U &&&& M &&&& V && \\ & i \swarrow && \searrow \phi && 1 \swarrow && \searrow f && \psi \swarrow && \searrow j & \\ X &&&& M &&&& N &&&& Y } </annotation></semantics></math></div> <p>belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, for every pair of charts <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>X</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">(U, \phi): X \to M</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>ψ</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Y</mi><mo>→</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">(V, \psi): Y \to N</annotation></semantics></math>.</p> <p>These definitions need to be carefully checked against known examples (e.g., the categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PL</mi></mrow><annotation encoding="application/x-tex">PL</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Smooth</mi></mrow><annotation encoding="application/x-tex">Smooth</annotation></semantics></math>, among others).</p> <h2 id="generalizations">Generalizations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a> (notice that the definition of that is very much along the lines of the Klein-program-style definition above).</li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semialgebraic+manifold">semialgebraic manifold</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/dimension+of+a+manifold">dimension of a manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/low-dimensional+topology">low-dimensional topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/projective+manifold">projective manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+with+boundary">manifold with boundary</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+manifold">closed manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/product+manifold">product manifold</a></p> </li> </ul> <h2 id="references">References</h2> <blockquote> <p>For more see the references at <em><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a></em> and <em><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></em>.</p> </blockquote> <p>The vision of the modern notion of manifolds is attributed to:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Bernhard+Riemann">Bernhard Riemann</a>, <em><a class="existingWikiWord" href="/nlab/show/%C3%9Cber+die+Hypothesen%2C+welche+der+Geometrie+zu+Grunde+liegen">Über die Hypothesen, welche der Geometrie zu Grunde liegen</a></em>, Göttingen (1845) [<a href="https://doi.org/10.1007/978-3-642-35121-1">doi:10.1007/978-3-642-35121-1</a>]</p> <p>Engl. transl: <a class="existingWikiWord" href="/nlab/show/William+Clifford">William Clifford</a>: <em><a class="existingWikiWord" href="/nlab/show/On+the+hypotheses+which+underlie+geometry">On the hypotheses which underlie geometry</a></em>, Nature <strong>VIII</strong> (1873) 183-184 [<a href="https://doi.org/10.1007/978-3-319-26042-6">doi:10.1007/978-3-319-26042-6</a>]</p> </li> </ul> <p>See also:</p> <ul> <li>John Loftin, <em>The real definition of a smooth manifold</em> (<a href="http://andromeda.rutgers.edu/~loftin/difffal03/manifold.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 19, 2024 at 09:16:32. 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