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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="higher_geometry">Higher 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> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In <a class="existingWikiWord" href="/nlab/show/general+topology">basic topology</a> and <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, by an <em>atlas</em> of/for a <a class="existingWikiWord" href="/nlab/show/topological+manifold">topological</a>-, <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable</a>- or <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> one means a collection of <a class="existingWikiWord" href="/nlab/show/coordinate+charts">coordinate charts</a> <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 \subset X</annotation></semantics></math> which form an <a class="existingWikiWord" href="/nlab/show/open+cover">open cover</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>.</p> <p>If one considers here the <a class="existingWikiWord" href="/nlab/show/disjoint+union">disjoint union</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mo>≔</mo><munder><mo>⊔</mo><mi>i</mi></munder><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{U} \coloneqq \underset{i}{\sqcup} U_i</annotation></semantics></math> of all the <a class="existingWikiWord" href="/nlab/show/coordinate+charts">coordinate charts</a>, then the separate chart embeddings <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 \subset X</annotation></semantics></math> give rise to a single <a class="existingWikiWord" href="/nlab/show/map">map</a> (<a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a>/<a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mo>⟶</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> \mathcal{U} \longrightarrow X </annotation></semantics></math></div> <p>and now the condition for an atlas is that this is a <a class="existingWikiWord" href="/nlab/show/surjective+function">surjective</a> <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+map">étale map</a>/<a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>.</p> <p>If, next, one regards this morphism, under the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>, inside the <a class="existingWikiWord" href="/nlab/show/topos">topos</a> of <a class="existingWikiWord" href="/nlab/show/formal+smooth+sets">formal smooth sets</a>, then these conditions on an atlas say that this morphism is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a>.</p> </li> </ol> <p>In this abstract form the concept of an atlas generalizes to any <a class="existingWikiWord" href="/nlab/show/cohesion">cohesive</a> <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> (<a href="#KhavkineSchreiber17">KS 17, Def. 3.3</a>, <a href="#Wellen18">Wellen 18, Def 4.13</a>, <a href="#SatiSchreiber20">Sati & Schreiber 2020, p. 27</a>).</p> <p>Next, for a <a class="existingWikiWord" href="/nlab/show/geometric+stack">geometric stack</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">\mathcal{X}</annotation></semantics></math>, an atlas is a <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>𝒰</mi></mrow><annotation encoding="application/x-tex">\mathcal{U}</annotation></semantics></math> (for <a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a>) or <a class="existingWikiWord" href="/nlab/show/scheme">scheme</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">\mathcal{U}</annotation></semantics></math> (for <a class="existingWikiWord" href="/nlab/show/algebraic+stacks">algebraic stacks</a>) or similar, equipped with a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒰</mi><mo>⟶</mo><mi>𝒳</mi></mrow><annotation encoding="application/x-tex"> \mathcal{U} \longrightarrow \mathcal{X} </annotation></semantics></math></div> <p>that is an <a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a> and <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a> in the corresponding <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos">higher topos</a> (for instance in that of <a class="existingWikiWord" href="/nlab/show/formal+smooth+infinity-groupoids">formal smooth infinity-groupoids</a>).</p> <p>Here the terminology has a bifurcation:</p> <ol> <li> <p>In the general context of <a class="existingWikiWord" href="/nlab/show/geometric+stacks">geometric stacks</a> one typically drops the second condition and calls any <a class="existingWikiWord" href="/nlab/show/effective+epimorphism">effective epimorphism</a> from a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> or <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> to a <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a> or <a class="existingWikiWord" href="/nlab/show/algebraic+stack">algebraic stack</a>, respectively, an <em>atlas</em> (e.g. <a href="#Leman10">Leman 10, 4.4</a>).</p> </li> <li> <p>If in addition the condition is imposed that such an effective epimorphism exists which is also <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale</a>, then the <a class="existingWikiWord" href="/nlab/show/geometric+stack">geometric stack</a> is called an <em><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></em> or <em><a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stack">Deligne-Mumford stack</a></em> (often with various further conditions imposed).</p> </li> </ol> <p>From here, the terminology generalizes to <a class="existingWikiWord" href="/nlab/show/infinity-stacks"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-stacks</a> in general <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-toposes"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-toposes</a>, see <a href="groupoid+objects+in+an+infinity1-topos+are+effective#InterpretationInTermsOfInfinityStacksWithAtlases">this Remark</a> at <em><a class="existingWikiWord" href="/nlab/show/groupoid+objects+in+an+%28%E2%88%9E%2C1%29-topos+are+effective">groupoid objects in an (∞,1)-topos are effective</a></em>.</p> <p id="AtlasForACategory"> Yet more generally, the notion generalizes to <a class="existingWikiWord" href="/nlab/show/2-topos+theory">2-topos theory</a> and higher. Over the point this yields the notion of <em><a class="existingWikiWord" href="/nlab/show/category+with+an+atlas">category with an atlas</a></em> and <em><a class="existingWikiWord" href="/nlab/show/flagged+categories">flagged categories</a></em> (depending on the truncation of the atlas) and relates to the notions of <em>flagged higher categories</em> (<a href="category+with+an+atlas#AyalaFrancis18">Ayala & Francis 2018</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+an+atlas">category with an atlas</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-types+cover">n-types cover</a></p> </li> </ul> <h2 id="references">References</h2> <p>Review of the classical concept of atlases for geometric stacks:</p> <ul> <li id="Leman10"><a class="existingWikiWord" href="/nlab/show/Eugene+Lerman">Eugene Lerman</a>, Section 4.4 of: <em>Orbifolds as stacks?</em>, L’Enseign. Math. (2) 56 (2010), no. 3-4, 315–363 (<a href="https://arxiv.org/abs/0806.4160">arXiv:0806.4160</a>)</li> </ul> <p>Formalization in <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive homotopy theory</a> and <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive</a>/<a class="existingWikiWord" href="/nlab/show/modal+homotopy+type+theory">modal homotopy type theory</a>:</p> <ul> <li id="KhavkineSchreiber17"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Khavkine">Igor Khavkine</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em>Synthetic geometry of differential equations</em> (<a href="https://arxiv.org/abs/1701.06238">arXiv:1701.06238</a>)</p> </li> <li id="Wellen18"> <p><a class="existingWikiWord" href="/nlab/show/Felix+Wellen">Felix Wellen</a>, <em><a class="existingWikiWord" href="/schreiber/show/thesis+Wellen">Formalizing Cartan Geometry in Modal Homotopy Type Theory</a></em> (<a href="https://arxiv.org/abs/1806.05966">arXiv:1806.05966</a>)</p> </li> <li id="SatiSchreiber20"> <p><a class="existingWikiWord" href="/nlab/show/Hisham+Sati">Hisham Sati</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, p. 27 of: <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>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 18, 2022 at 08:14:25. 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