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coreduced object in nLab
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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="cohesion">Cohesion</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></strong></p> <p><strong>Backround</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+cohesive+toposes">motivation for cohesive toposes</a></p> </li> </ul> <p><strong>Definition</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">strongly ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">totally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a> / <a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> / <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <p><strong>Presentation over a site</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+site">locally connected site</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+site">locally ∞-connected site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+site">connected site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+site">∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+site">strongly connected site</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+site">strongly ∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+site">totally connected site</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+site">totally ∞-connected site</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+site">local site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-local+site">∞-local site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-cohesive+site">∞-cohesive site</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, <a class="existingWikiWord" href="/nlab/show/discrete+groupoid">discrete groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-topological+%E2%88%9E-groupoid">D-topological ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/topological+groupoid">topological groupoid</a>, <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>, <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-groupoid">Lie 2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+super+%E2%88%9E-groupoid">smooth super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a>, <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>, <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a>, <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/synthetic+differential+super+%E2%88%9E-groupoid">synthetic differential super ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion+of+global-+over+G-equivariant+homotopy+theory">cohesion of global- over G-equivariant homotopy theory</a></strong></p> </li> </ul> </div></div> <h4 id="discrete_and_concrete_objects">Discrete and concrete objects</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/discrete+objects">discrete objects</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/codiscrete+objects">codiscrete objects</a></strong> (<a class="existingWikiWord" href="/nlab/show/chaos">chaos</a>)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/codiscrete+space">codiscrete space</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+groupoid">codiscrete groupoid</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/concrete+objects">concrete objects</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/concrete+sheaf">concrete sheaf</a>, <a class="existingWikiWord" href="/nlab/show/concrete+%28%E2%88%9E%2C1%29-sheaf">concrete (∞,1)-sheaf</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/co-concrete+object">co-concrete object</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/ionad">ionad</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a></strong></p> <p><strong>typical contexts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a>, <a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/discrete+and+concrete+objects+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="modalities_closure_and_reflection">Modalities, Closure and Reflection</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/modal+type+theory">modal type theory</a></strong>, <a class="existingWikiWord" href="/nlab/show/modal+logic">modal logic</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/closure+operator">closure operator</a></strong>, <a class="existingWikiWord" href="/nlab/show/universal+closure+operator">universal closure operator</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent monad</a>, <a class="existingWikiWord" href="/nlab/show/comonad">comonad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modal+type">modal type</a>, <a class="existingWikiWord" href="/nlab/show/local+object">local object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a>, <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Moore+closure">Moore closure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+modality">geometric modality</a>/<a class="existingWikiWord" href="/nlab/show/Lawvere-Tierney+topology">Lawvere-Tierney topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/S4+modal+logic">S4 modal logic</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-truncated+object+in+an+%28infinity%2C1%29-category">n-truncation</a></p> </li> <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> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In a context of <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>/<a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> a <em>coreduced object</em> is one all whose <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal</a> paths are constant. Compare the <a class="existingWikiWord" href="/nlab/show/discrete+object">discrete objects</a>, in which all paths are constant, meaning all discrete objects are also coreduced.</p> <h2 id="definition">Definition</h2> <p>A context of <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> is determined by the existence of an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> of <a class="existingWikiWord" href="/nlab/show/modalities">modalities</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \Re \dashv \Im \dashv \& \,, </annotation></semantics></math></div> <p>where <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">\Re</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>&</mi></mrow><annotation encoding="application/x-tex">\&</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a> <a class="existingWikiWord" href="/nlab/show/comonads">comonads</a> 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">\Im</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent monad</a>.</p> <p>A <strong>coreduced object</strong> or <strong>coreduced type</strong> is one in the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> defined by the <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><mi>ℑ</mi></mrow><annotation encoding="application/x-tex">\Im</annotation></semantics></math> or equivalently the <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>&</mi></mrow><annotation encoding="application/x-tex">\&</annotation></semantics></math>.</p> <p>Note that 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> being coreduced is the same as it being <a class="existingWikiWord" href="/nlab/show/formally+etale">formally etale</a>.</p> <h2 id="examples">Examples</h2> <ul> <li>A coreduced <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> is also called a <a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>. The <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> over such is <a class="existingWikiWord" href="/nlab/show/crystalline+cohomology">crystalline cohomology</a>, the <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent sheaves</a> over these define <a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>.</li> <li>In a <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometric</a> setting, <a class="existingWikiWord" href="/nlab/show/formally+etale+morphism">formally etale maps</a> are the <a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphisms</a>, and the coreduced objects are those for which the terminal map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">X \to 1</annotation></semantics></math> is a local diffeomorphism.</li> </ul> <h2 id="related_concepts">Related concepts</h2> <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> <ul> <li><a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+etale+morphism">formally etale morphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 29, 2017 at 16:19:24. 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