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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" style="color:black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="tale_morphisms">Étale morphisms</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism">étale morphism</a></strong> (<a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>)</p> <p>generally in <strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_aaa8c720d35345b38f171647a1e2b8f35af14dd3_1"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale</a> (<a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+topos">étale topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+homotopy">étale homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+cohomology">étale cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%E2%88%9E-groupoid">étale ∞-groupoid</a></p> </li> </ul> <p>in <strong><a class="existingWikiWord" href="/nlab/show/topology">topology</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+homeomorphism">local homeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+space">étale space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+groupoid">étale groupoid</a></p> </li> </ul> <p>in <strong><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, (<a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/immersion+of+smooth+manifolds">immersion of smooth manifolds</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+Lie+groupoid">étale Lie groupoid</a></p> </li> </ul> <p>in <strong><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+morphism+of+schemes">étale morphism of schemes</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism+of+schemes">formally étale morphism of schemes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+site">étale site</a>, <a class="existingWikiWord" href="/nlab/show/pro-%C3%A9tale+site">pro-étale site</a>, <a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne-Mumford+stack">Deligne-Mumford stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%84%93-adic+cohomology">ℓ-adic cohomology</a></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='#GeneralAbstractNotion'>General abstract notion</a></li> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> </ul> <li><a href='#ConcreteNotion'>Concrete notions</a></li> <ul> <li><a href='#in_differential_geometry'>In differential geometry</a></li> <li><a href='#in_algebraic_geometry'>In algebraic geometry</a></li> <li><a href='#in_noncommutative_geometry'>In noncommutative geometry</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> </ul> </div> <p>A <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> of <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>p</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \overset{p}{\longrightarrow} Y</annotation></semantics></math> is called <em>formally étale</em> if it has a <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> as <a class="existingWikiWord" href="/nlab/show/%C3%A9tal%C3%A9+spaces">étalé spaces</a> do locally, but for <em><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesmal</a></em> extensions: If for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi><mover><mo>→</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">Z \overset{f}{\to} Y</annotation></semantics></math> any morphism and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\Re(Z) \to X</annotation></semantics></math> a lift of its restriction along its <a class="existingWikiWord" href="/nlab/show/reduction">reduction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo stretchy="false">(</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">\Re(Z) \to Z</annotation></semantics></math>, there is a unique extension to a complete lift.</p> <p>(If there exists at least one such infinitesimal extension, it is called a <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>. If there exists at most one such extension, it is called a <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>. The formally étale morphisms are precisely those that are both formally smooth and formally unramified.)</p> <p>Traditionally this has been considered in the context of <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> over formal duals of <a class="existingWikiWord" href="/nlab/show/ring">ring</a>s and <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>s. This we discuss in the section (<a href="#ConcreteNotion">Concrete notion</a>). But generally the notion makes sense in any context of <a href="http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos#InfinitesimalCohesion">differential cohesion</a>. This we discuss in the section <a href="#GeneralAbstractNotion">General abstract notion</a>.</p> <h2 id="GeneralAbstractNotion">General abstract notion</h2> <h3 id="definition">Definition</h3> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mover><mover><munder><mo>→</mo><mrow><msup><mi>u</mi> <mo>!</mo></msup></mrow></munder><mover><mo>←</mo><mrow><msub><mi>u</mi> <mo>*</mo></msub></mrow></mover></mover><mover><mo>↪</mo><mrow><msup><mi>u</mi> <mo>*</mo></msup></mrow></mover></mover><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex"> \mathbf{H} \stackrel{\overset{u^*}{\hookrightarrow}}{\stackrel{\overset{u_*}{\leftarrow}}{\underset{u^!}{\to}}} \mathbf{H}_{th} </annotation></semantics></math></div> <p>be an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> of <a class="existingWikiWord" href="/nlab/show/functor">functor</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">u^*</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a> that preserves the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>.</p> <p>We may think of this as exhibiting <a href="http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos#InfinitesimalCohesion">differential cohesion</a> (see there for details, but notice that in the notation used there we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup><mo>=</mo><msub><mi>i</mi> <mo>!</mo></msub></mrow><annotation encoding="application/x-tex">u^* = i_!</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>u</mi> <mo>*</mo></msub><mo>=</mo><msup><mi>i</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">u_* = i^*</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>!</mo></msup><mo>=</mo><msub><mi>i</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">u^! = i_*</annotation></semantics></math>).</p> <p>We think of the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> as <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive space</a>s and of the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{th}</annotation></semantics></math> as such cohesive spaces possibly equipped with <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal extension</a>.</p> <p>As a class of examples that is useful to keep in mind consider a <a class="existingWikiWord" href="/nlab/show/Q-category">Q-category</a></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>cod</mi><mo>⊣</mo><mi>ϵ</mi><mo>⊣</mo><mi>dom</mi><mo stretchy="false">)</mo><mo>:</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> (cod \dashv \epsilon \dashv dom) : \bar A \to A </annotation></semantics></math></div> <p>of <a href="http://ncatlab.org/nlab/show/Q-category#InfinitesimalThickening">infinitesimal thickening of rings</a> and let</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><mo stretchy="false">(</mo><msup><mi>u</mi> <mo>*</mo></msup><mo>⊣</mo><msub><mi>u</mi> <mo>*</mo></msub><mo>⊣</mo><msup><mi>u</mi> <mo>!</mo></msup><mo stretchy="false">)</mo><mo>:</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mi>th</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>dom</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>⊣</mo><mo stretchy="false">[</mo><mi>ϵ</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>⊣</mo><mo stretchy="false">[</mo><mi>cod</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>:</mo><mo stretchy="false">[</mo><mover><mi>A</mi><mo stretchy="false">¯</mo></mover><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>Set</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> ((u^* \dashv u_* \dashv u^!) : \mathbf{H}_{th} \to \mathbf{H}) := ([dom,Set] \dashv [\epsilon, Set] \dashv [cod,Set] : [\bar A, Set] \to [A,Set]) </annotation></semantics></math></div> <p>be the corresponding <a href="http://ncatlab.org/nlab/show/Q-category#PresheafQCategory">Q-category of copresheaves</a>.</p> <p>For any such setup there is a canonical <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></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><msup><mi>u</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>u</mi> <mo>!</mo></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi : u^* \to u^! \,. </annotation></semantics></math></div> <p>Details of this are in the section <a href="http://ncatlab.org/nlab/show/cohesive%20topos#AdjointQuadruples">Adjoint quadruples</a> at <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a>.</p> <p>From this we get for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> a canonical morphism</p> <div class="maruku-equation" id="eq:MorphismIntoPullback"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>X</mi><mo>→</mo><msup><mi>u</mi> <mo>*</mo></msup><mi>Y</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><msup><mi>u</mi> <mo>!</mo></msup><mi>Y</mi></mrow></munder><msup><mi>u</mi> <mo>!</mo></msup><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> u^* X \to u^* Y \prod_{u^! Y} u^! X \,. </annotation></semantics></math></div> <div class="num_defn" id="AbstractFormallyEtaleMorphism"> <h6 id="definition_2">Definition</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is called <strong>formally étale</strong> if <a class="maruku-eqref" href="#eq:MorphismIntoPullback">(1)</a> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <p>This appears as (<a href="#KontsevichRosenbergSpaces">KontsevichRosenberg, def. 5.1, prop. 5.3.1.1</a>).</p> <p>In other words, <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 formally étale if the <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>-component naturality square</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><msup><mi>u</mi> <mo>*</mo></msup><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>f</mi></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>*</mo></msup><mi>Y</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϕ</mi> <mi>X</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ϕ</mi> <mi>Y</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>u</mi> <mo>!</mo></msup><mi>f</mi></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ u^* X &\stackrel{u^* f}{\to}& u^* Y \\ {}^{\mathllap{\phi_X}}\downarrow && \downarrow^{\mathrlap{\phi_Y}} \\ u^! X &\stackrel{u^! f}{\to}& u^! Y } </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a>.</p> <div class="num_remark" id="FormallySmooth"> <h6 id="remark">Remark</h6> <p>The partial notions of this condition are: if the above morphism is a <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> then <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 a <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>, if it is an <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a> then <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 a <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>.</p> </div> <div class="num_defn" id="AbstractFormallyEtaleObject"> <h6 id="definition_3">Definition</h6> <p>An object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> is called <strong>formally étale</strong> if the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">X \to *</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> is formally étale.</p> </div> <div class="num_prop" id="AbstractFormallyEtaleObjectDirect"> <h6 id="proposition">Proposition</h6> <p>The 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> is formally étale precisely if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>X</mi><mo>→</mo><msup><mi>u</mi> <mo>!</mo></msup><mi>X</mi></mrow><annotation encoding="application/x-tex"> u^* X \to u^! X </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>.</p> </div> <p>This appears as (<a href="#KontsevichRosenbergSpaces">KontsevichRosenberg, def. 5.3.2</a>).</p> <h3 id="properties">Properties</h3> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Formally étale morphisms are closed under composition.</p> </div> <p>This appears as (<a href="#KontsevichRosenbergSpaces">KontsevichRosenberg, prop. 5.4</a>).</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>This follows by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> for pullbacks: let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>Y</mi><mo>→</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">g : Y \to Z</annotation></semantics></math> be two formally étale morphisms. Then by definition both of the small squares in</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><msup><mi>u</mi> <mo>*</mo></msup><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>f</mi></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>*</mo></msup><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>g</mi></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>*</mo></msup><mi>Z</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>u</mi> <mo>!</mo></msup><mi>f</mi></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow><msup><mi>u</mi> <mo>!</mo></msup><mi>g</mi></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ u^* X &\stackrel{u^* f }{\to}& u^* Y &\stackrel{u^* g}{\to}& u^* Z \\ \downarrow && \downarrow && \downarrow \\ u^! X &\stackrel{u^! f }{\to}& u^! Y &\stackrel{u^! g}{\to}& u^! Z } </annotation></semantics></math></div> <p>are pullback squares. Hence so is the total outer square.</p> </div> <p>Using also the other case of the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>, the above proof shows more:</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>If</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>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>h</mi></mover></mtd> <mtd></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && Y \\ & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}} \\ X &&\stackrel{h}{\to}&& Z } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> such that <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi></mrow><annotation encoding="application/x-tex">h</annotation></semantics></math> are formally étale, then also <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 formally étale.</p> </div> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>Formally étale morphisms are closed under <a class="existingWikiWord" href="/nlab/show/retract">retract</a>s.</p> </div> <p>This means that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> is formally étale and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\to & X &\to& A \\ \downarrow^{\mathrlap{p}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{p}} \\ B &\to& Y &\to& B } </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> such that the two horizontal composites are <a class="existingWikiWord" href="/nlab/show/identities">identities</a>, then also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is formally étale.</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>By applying the <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msup><mi>u</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>u</mi> <mo>!</mo></msup></mrow><annotation encoding="application/x-tex"> \phi : u^* \to u^!</annotation></semantics></math> to this diagram we obtain a retract diagram in the category of squares, given by the naturality squares of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> on <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> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, where the middle square is a pullback square. By <a href="http://ncatlab.org/nlab/show/retract#RetractsOfLimits">this proposition</a> at <em><a class="existingWikiWord" href="/nlab/show/retract">retract</a></em> this implies that also the retracting square is a pullback, which means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is formally étale.</p> </div> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">u^*</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>, then formally étale morphisms are stable under pullback.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>Consider a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>p</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A \times_Y X &\to& X \\ {}^{\mathllap{p}}\downarrow && \downarrow^{\mathrlap{f}} \\ A &\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 formally étale.</p> <p>Applying the <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>:</mo><msup><mi>u</mi> <mo>*</mo></msup><mo>→</mo><msup><mi>u</mi> <mo>!</mo></msup></mrow><annotation encoding="application/x-tex">\phi : u^* \to u^!</annotation></semantics></math> to this yields a square of squares. Two sides of this are the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> 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><msup><mi>u</mi> <mo>*</mo></msup><mi>A</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>u</mi> <mo>*</mo></msup><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>X</mi></msub></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>u</mi> <mo>*</mo></msup><mi>f</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>u</mi> <mo>!</mo></msup><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>u</mi> <mo>*</mo></msup><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>u</mi> <mo>*</mo></msup><mi>Y</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>Y</mi></msub></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ u^* A \times_Y X &\to& u^* X &\stackrel{\phi_X}{\to}& u^! X \\ \downarrow && \downarrow^{\mathrlap{u^* f}} && \downarrow^{\mathrlap{u^! f}} \\ u^* A &\to& u^* Y &\stackrel{\phi_Y}{\to}& u^! Y } </annotation></semantics></math></div> <p>and the other two sides are the pasting 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><msup><mi>u</mi> <mo>*</mo></msup><mi>A</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mrow><mi>A</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>X</mi></mrow></msub></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>A</mi><msub><mo>×</mo> <mi>Y</mi></msub><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow></mrow></msup></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>u</mi> <mo>!</mo></msup><mi>f</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>u</mi> <mo>*</mo></msup><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>ϕ</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>u</mi> <mo>!</mo></msup><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ u^* A \times_Y X &\stackrel{\phi_{A \times_Y X}}{\to}& u^! A \times_Y A &\stackrel{}{\to}& u^! X \\ \downarrow^{} && \downarrow && \downarrow^{\mathrlap{u^! f}} \\ u^* A &\stackrel{\phi_A}{\to}& u^! A &\to& u^! Y } \,. </annotation></semantics></math></div> <p>Counting left to right and top to bottom, we have that</p> <ul> <li> <p>the first square is a pullback by assumption on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">u^*</annotation></semantics></math>;</p> </li> <li> <p>the second square is a pullback, since <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 formally étale.</p> </li> <li> <p>the fourth square is a pullback since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mo>!</mo></msup></mrow><annotation encoding="application/x-tex">u^!</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> and so also preserves pullbacks;</p> </li> <li> <p>also the total bottom rectangle is a pullback, since it is equal to the bottom total rectangle;</p> </li> <li> <p>therefore finally the third square is a pullback, by the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a>, hence also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is formally étale.</p> </li> </ul> </div> <h2 id="ConcreteNotion">Concrete notions</h2> <p>We discuss realizations of the above general abstract definition in concrete models of the axioms.</p> <p>See also the concrete notions of <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a> and <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>.</p> <h3 id="in_differential_geometry">In differential geometry</h3> <p>The <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/SmoothMfd">SmoothMfd</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>s may naturally be thought of as sitting inside the more general context of the <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>s. This is canonically equipped with a notion of <a class="existingWikiWord" href="/nlab/show/differential+cohesion">infinitesimal cohesion</a> exhibited by its inclusion into <a class="existingWikiWord" href="/nlab/show/SynthDiff%E2%88%9EGrpd">SynthDiff∞Grpd</a>. This implies that there is an intrinsic notion of <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a>s of smooth <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>-groupoids in general and of smooth manifolds in particular</p> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> is a formally étale morphism in this sense precisely if it is a <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a> in the traditional sense.</p> </div> <p>See <a href="http://ncatlab.org/nlab/show/synthetic+differential+infinity-groupoid#StructureSheaves">this section</a> for more details.</p> <h3 id="in_algebraic_geometry">In algebraic geometry</h3> <ul> <li><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism+of+schemes">formally étale morphism of schemes</a></li> </ul> <h3 id="in_noncommutative_geometry">In noncommutative geometry</h3> <p>See (<a href="#RosenbergKontsevich">RosenbergKontsevich, section 5.8</a>)</p> <h2 id="related_concepts">Related concepts</h2> <p><a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a> and <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⇒</mo></mrow><annotation encoding="application/x-tex">\Rightarrow</annotation></semantics></math> <strong>formally étale morphism</strong></p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a></p> </li> <li> <p><a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a></strong></p> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> <h2 id="References">References</h2> <p>The idea of defining étale morphisms <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> as those that get send to a pullback square by a natural transformation goes back to lectures by <a class="existingWikiWord" href="/nlab/show/Andr%C3%A9+Joyal">André Joyal</a> in the 1970s.</p> <p>See the introduction and see section 4 of</p> <ul id="Dubuc"> <li><a class="existingWikiWord" href="/nlab/show/Eduardo+Dubuc">Eduardo Dubuc</a>, <em>Axiomatic etal maps and a theory of spectrum</em>, Journal of pure and applied algebra, 149 (2000)</li> </ul> <p>The identification of the natural transformation in question with that induced by an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> (“<a class="existingWikiWord" href="/nlab/show/Q-categories">Q-categories</a>”) and the relation to <em>formal</em> étaleness is observed (apparently independently?) in</p> <ul id="KontsevichRosenbergSpaces"> <li><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Rosenberg">Alexander Rosenberg</a>, <em>Noncommutative spaces</em>, preprint MPI-2004-35 (<a class="existingWikiWord" href="/nlab/files/KontsevichRosenbergNCSpaces.pdf" title="pdf">pdf</a>, <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=2331">ps</a>, <a href="http://www.mpim-bonn.mpg.de/preprints/send?bid=2303">dvi</a>)</li> </ul> <p>Formalization and discussion in the context of <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-toposes">cohesive (∞,1)-toposes</a> is in section 2.5.3 (and defn 5.3.19) of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">differential cohomology in a cohesive topos</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 18, 2018 at 17:52:34. 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