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class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="geometry">Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong> / <strong><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <blockquote> <p>“Geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.” [<a class="existingWikiWord" href="/nlab/show/Philosophi%C3%A6+naturalis+principia+mathematica">Newton (1687)</a>, <a href="https://en.wikisource.org/wiki/Page:Newton%27s_Principia_(1846).djvu/73">preface</a>]</p> </blockquote> <p>In its historical and etymological origins, “geometry” is the science of <em>measuring the earth</em> and here specifically of measuring <em><a class="existingWikiWord" href="/nlab/show/distances">distances</a></em> and <em><a class="existingWikiWord" href="/nlab/show/angles">angles</a></em>, first in the guise of <a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a> of the ancients and then eventually as the <em><a class="existingWikiWord" href="/nlab/show/differential+geometry+of+curves+and+surfaces">differential geometry of curves and surfaces</a></em> due to <a class="existingWikiWord" href="/nlab/show/Carl+Friedrich+Gauss">Gauss</a> and others.</p> <p>But in modern <a class="existingWikiWord" href="/nlab/show/mathematics">mathematical</a> jargon, the term <em>geometry</em> is used in much greater generality for the study of <em><a class="existingWikiWord" href="/nlab/show/spaces">spaces</a></em> equipped with <a class="existingWikiWord" href="/nlab/show/extra+structure">extra</a> “geometrical” <a class="existingWikiWord" href="/nlab/show/structure">structure</a> of a large variety of sorts, foremost in the guise of <em><a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></em> but subsuming also a wealth of variant notions of “geometries” which the ancient would not have recognized as such, for instance <em><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></em> and other (<a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion</a>-free) <em><a class="existingWikiWord" href="/nlab/show/G-structure"><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>-structured</a></em> <a class="existingWikiWord" href="/nlab/show/differentiable+manifolds">differentiable manifolds</a> (<em><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential</a> <a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a></em>), or (<a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher</a>) <a class="existingWikiWord" href="/nlab/show/topos">topos</a>-theoretic notions (cf. “<a class="existingWikiWord" href="/nlab/show/geometric+logic">geometric logic</a>”) of (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher</a>) <em><a class="existingWikiWord" href="/nlab/show/functorial+geometry">functorial geometry</a></em> famously including <em><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a></em> or <em><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></em> but also more exotic variants such as <em><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a></em> or <em><a class="existingWikiWord" href="/nlab/show/absolute+geometry">absolute geometry</a></em>. Finally, taking the <a class="existingWikiWord" href="/nlab/show/duality+between+algebra+and+geometry">duality between algebra and geometry</a> to the extreme yields notions of <em><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></em> and/or <em><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></em> whose underlying “spaces” are only indirectly conceived as whatever it is that given (<a class="existingWikiWord" href="/nlab/show/higher+algebra">higher</a>) <a class="existingWikiWord" href="/nlab/show/algebras">algebras</a> would see (measure!) if they are interpreted as <a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a> on a would-be space — which on the one hand is far remote from what Newton must have imagined geometry to be about, but on the other hand is quite in the spirit of geometry being about (maybe not space as such but) <em><a class="existingWikiWord" href="/nlab/show/measurement">measuring</a></em> space.</p> <p>Beware that bare <em><a class="existingWikiWord" href="/nlab/show/topology">topology</a></em> is sometimes regarded as a rudimentary kind of geometry (as reflected for instance in the common terminology <em><a class="existingWikiWord" href="/nlab/show/geometric+realization">geometric realization</a></em> for an operation that really is <em><a class="existingWikiWord" href="/nlab/show/topological+realization">topological realization</a></em>), but more often than not a “geometric space” is meant to be a <em><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a></em> equipped with <a class="existingWikiWord" href="/nlab/show/extra+structure">extra</a> geometrical <a class="existingWikiWord" href="/nlab/show/structure">structure</a>, of sorts.</p> <p>Even a plain <a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a> as considered in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> is often not quite regarded as <em>geometric</em> in itself unless equipped with further structure, as reflected for instance in the terminology <em><a class="existingWikiWord" href="/nlab/show/topological+field+theory">topological field theory</a></em> for those <a class="existingWikiWord" href="/nlab/show/functorial+field+theories">functorial field theories</a> which depend on <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a>-structure of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> but not on further (notably <a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+manifold">pseudo</a>-<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian</a>) geometric structure.</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+type+theory">geometric type theory</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+type+theory">geometric homotopy type theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+geometry">synthetic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a>, <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>, <a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a>, <a class="existingWikiWord" href="/nlab/show/tangent+space">tangent space</a>, <a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/cotangent+complex">cotangent complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a>, <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian+submanifold">Lagrangian submanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a>, <a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a>, <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a>, <a class="existingWikiWord" href="/nlab/show/integrable+distribution">integrable distribution</a>, <a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibre+bundle">fibre bundle</a>, <a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+principal+bundle">noncommutative principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/connection">connection</a> (and links therein), <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morse+function">Morse function</a>, <a class="existingWikiWord" href="/nlab/show/Morse+lemma">Morse lemma</a>, <a class="existingWikiWord" href="/nlab/show/Morse+theory">Morse theory</a>, <a class="existingWikiWord" href="/nlab/show/perfect+Morse+function">perfect Morse function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Casson+invariant">Casson invariant</a>, <a class="existingWikiWord" href="/nlab/show/Donaldson-Thomas+invariant">Donaldson-Thomas invariant</a>, <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a>, <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/metric+space">metric space</a>, <a class="existingWikiWord" href="/nlab/show/convex+set">convex set</a>, <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a>, <a class="existingWikiWord" href="/nlab/show/geodesic+flow">geodesic flow</a>, <a class="existingWikiWord" href="/nlab/show/geodesic+convexity">geodesic convexity</a>, <a class="existingWikiWord" href="/nlab/show/star-shaped">star-shaped</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>, <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a>, <a class="existingWikiWord" href="/nlab/show/stratifold">stratifold</a>, <a class="existingWikiWord" href="/nlab/show/orientifold">orientifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>, <a class="existingWikiWord" href="/nlab/show/analytic+geometry">analytic geometry</a>,</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/arithmetic+geometry">arithmetic geometry</a>, <a class="existingWikiWord" href="/nlab/show/GAGA">GAGA</a>, book entry <a class="existingWikiWord" href="/nlab/show/EGA">EGA</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/scheme">scheme</a>, <a class="existingWikiWord" href="/nlab/show/quasicompact">quasicompact</a>, <a class="existingWikiWord" href="/nlab/show/noetherian+scheme">noetherian scheme</a>, <a class="existingWikiWord" href="/nlab/show/reduced+scheme">reduced scheme</a>, <a class="existingWikiWord" href="/nlab/show/integral+scheme">integral scheme</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/open+subscheme">open subscheme</a>, <a class="existingWikiWord" href="/nlab/show/closed+subscheme">closed subscheme</a>, <a class="existingWikiWord" href="/nlab/show/configuration+scheme">configuration scheme</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+scheme">formal scheme</a>, <a class="existingWikiWord" href="/nlab/show/formal+group+scheme">formal group scheme</a>, <a class="existingWikiWord" href="/nlab/show/formal+group+law">formal group law</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum+%28geometry%29">spectrum (geometry)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+variety">algebraic variety</a>, <a class="existingWikiWord" href="/nlab/show/normal+variety">normal variety</a>, <a class="existingWikiWord" href="/nlab/show/hyperplane+line+bundle">hyperplane line bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a>, <a class="existingWikiWord" href="/nlab/show/derived+noncommutative+algebraic+geometry">derived noncommutative algebraic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncommutative algebraic geometry</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+scheme">noncommutative scheme</a>, <a class="existingWikiWord" href="/nlab/show/noncommutative+thin+scheme">noncommutative thin scheme</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hyperplane+section+theorem">hyperplane section theorem</a>, <a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rational+map">rational map</a>, <a class="existingWikiWord" href="/nlab/show/rational+variety">rational variety</a>, <a class="existingWikiWord" href="/nlab/show/unirational+variety">unirational variety</a>, <a class="existingWikiWord" href="/nlab/show/birational+map">birational map</a>, <a class="existingWikiWord" href="/nlab/show/birational+geometry">birational geometry</a>, <a class="existingWikiWord" href="/nlab/show/image+of+a+rational+map">image of a rational map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+space">algebraic space</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+stack">algebraic stack</a>, <a class="existingWikiWord" href="/nlab/show/topological+stack">topological stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaf">quasicoherent sheaf</a>, <a class="existingWikiWord" href="/nlab/show/coherent+sheaf">coherent sheaf</a>, <a class="existingWikiWord" href="/nlab/show/relative+scheme">relative scheme</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+scheme">smooth scheme</a>, <a class="existingWikiWord" href="/nlab/show/smooth+morphism+of+schemes">smooth morphism of schemes</a>, <a class="existingWikiWord" href="/nlab/show/etale+morphism">etale morphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pure+motive">pure motive</a>, <a class="existingWikiWord" href="/nlab/show/Voevodsky+motive">Voevodsky motive</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>, <a class="existingWikiWord" href="/nlab/show/local+system">local system</a>, <a class="existingWikiWord" href="/nlab/show/regular+differential+operator">regular differential operator</a>, <a class="existingWikiWord" href="/nlab/show/holonomic+D-module">holonomic D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+connection">Grothendieck connection</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a>, <a class="existingWikiWord" href="/nlab/show/diffiety">diffiety</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flag+variety">flag variety</a>, <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a>, <a class="existingWikiWord" href="/nlab/show/orbit">orbit</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a></p> </li> <li> <p><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/Lie+algebroid">Lie algebroid</a>, <a class="existingWikiWord" href="/nlab/show/Courant+algebroid">Courant algebroid</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a>, <a class="existingWikiWord" href="/nlab/show/Erlangen+program">Erlangen program</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ringed+space">ringed space</a>, <a class="existingWikiWord" href="/nlab/show/ringed+site">ringed site</a>, <a class="existingWikiWord" href="/nlab/show/ringed+topos">ringed topos</a>, <a class="existingWikiWord" href="/nlab/show/lined+topos">lined topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+of+ideals">sheaf of ideals</a>, <a class="existingWikiWord" href="/nlab/show/defining+sheaf">defining sheaf</a>, <a class="existingWikiWord" href="/nlab/show/conormal+sheaf">conormal sheaf</a>, <a class="existingWikiWord" href="/nlab/show/conormal+bundle">conormal bundle</a>, <a class="existingWikiWord" href="/nlab/show/subscheme+of+an+Abelian+category">subscheme of an Abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal object</a>, <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimal+singular+simplicial+complex">infinitesimal singular simplicial complex</a>, <a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms in synthetic differential geometry</a></p> <ul> <li>book entry <a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Froelicher+space">Froelicher space</a>, <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space+and+quantity">space and quantity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/trigonometry">trigonometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D-brane+geometry">D-brane geometry</a></p> </li> </ul> <p>There are many entries on <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a>, <a class="existingWikiWord" href="/nlab/show/stack">stack</a>, <a class="existingWikiWord" href="/nlab/show/site">site</a>, <a class="existingWikiWord" href="/nlab/show/locale">locale</a> and <a class="existingWikiWord" href="/nlab/show/topos">topos</a> theory including</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/germ">germ</a>, <a class="existingWikiWord" href="/nlab/show/etale+space">etale space</a>, <a class="existingWikiWord" href="/nlab/show/soft+sheaf">soft sheaf</a>, <a class="existingWikiWord" href="/nlab/show/flabby+sheaf">flabby sheaf</a>, <a class="existingWikiWord" href="/nlab/show/fine+sheaf">fine sheaf</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topos">Grothendieck topos</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+topology">Grothendieck topology</a>, <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos">(infinity,1)-topos</a>, <a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category+of+%28infinity%2C1%29-sheaves">(infinity,1)-category of (infinity,1)-sheaves</a>, <a class="existingWikiWord" href="/nlab/show/cover">cover</a>, <a class="existingWikiWord" href="/nlab/show/hypercover">hypercover</a>, <a class="existingWikiWord" href="/nlab/show/local+object">local object</a>, <a class="existingWikiWord" href="/nlab/show/local+isomorphism">local isomorphism</a>, <a class="existingWikiWord" href="/nlab/show/coverage">coverage</a>, <a class="existingWikiWord" href="/nlab/show/local+epimorphism">local epimorphism</a>, <a class="existingWikiWord" href="/nlab/show/hypercompletion">hypercompletion</a>, <a class="existingWikiWord" href="/nlab/show/sieve">sieve</a>, <a class="existingWikiWord" href="/nlab/show/Verdier+site">Verdier site</a>, <a class="existingWikiWord" href="/nlab/show/Grothendieck+fibration">Grothendieck fibration</a>, <a class="existingWikiWord" href="/nlab/show/Pierce+spectrum">Pierce spectrum</a>, <a class="existingWikiWord" href="/nlab/show/Stone+duality">Stone duality</a>, <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">point of a topos</a>, <a class="existingWikiWord" href="/nlab/show/geometric+morphism">geometric morphism</a>…</li> </ul> <p>and pages on various <a class="existingWikiWord" href="/nlab/show/cohomology">cohomologies</a>, including <a class="existingWikiWord" href="/nlab/show/sheaf+cohomology">sheaf cohomology</a>, <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a>, <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+cohomology">Deligne cohomology</a>, <a class="existingWikiWord" href="/nlab/show/etale+cohomology">etale cohomology</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Bredon+cohomology">Bredon cohomology</a> and their cocycle classes including <a class="existingWikiWord" href="/nlab/show/torsors">torsors</a>, <a class="existingWikiWord" href="/nlab/show/gerbes">gerbes</a>, <a class="existingWikiWord" href="/nlab/show/principal+2-bundles">principal 2-bundles</a> as well as the related picture of the <a class="existingWikiWord" href="/nlab/show/descent">descent</a> theory (cf. <a class="existingWikiWord" href="/nlab/show/oriental">oriental</a>, <a class="existingWikiWord" href="/nlab/show/descent+for+simplicial+presheaves">descent for simplicial presheaves</a>…). A modern systematic theory of cohomology and descent can be done using the language of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infinity,1)</annotation></semantics></math>-categories and abstract <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, say via Quillen <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a> (e.g. of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a>).</p> <h2 id="references">References</h2> <ul> <li>Wikipedia <em><a href="https://en.wikipedia.org/wiki/History_of_geometry">History of geometry</a></em></li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/duality+between+algebra+and+geometry">duality between</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></strong></p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/category">category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dual+category">dual category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a></mtext></mover><msubsup><mi>Alg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/topology">topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>TopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}TopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a></mtext></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup><mo>,</mo><mi>comm</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+C%2A-algebra">comm. C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+topology">noncomm. topology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCTopSpaces</mi> <mrow><mi>H</mi><mo>,</mo><mi>cpt</mi></mrow></msub></mrow><annotation encoding="application/x-tex">NCTopSpaces_{H,cpt}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><msubsup><mi>TopAlg</mi> <mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>general <a class="existingWikiWord" href="/nlab/show/C-star-algebra">C-star-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>NC</mi></mphantom><msub><mi>Schemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">\phantom{NC}Schemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≃</mo><mtext><a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op} </annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A} \phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">noncomm. algebraic</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/noncommutative+algebraic+geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>NCSchemes</mi> <mi>Aff</mi></msub></mrow><annotation encoding="application/x-tex">NCSchemes_{Aff}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>≔</mo><mphantom><mtext>Gelfand duality</mtext></mphantom></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mrow><mi>fin</mi><mo>,</mo><mi>red</mi></mrow> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/finitely+generated+algebra">fin. gen.</a> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SmoothManifolds</mi></mrow><annotation encoding="application/x-tex">SmoothManifolds</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mtext><a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a></mtext></mover><mphantom><mi>Top</mi></mphantom><msubsup><mi>Alg</mi> <mi>comm</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>SuperSpaces</mi> <mi>Cart</mi></msub></mtd></mtr> <mtr><mtd></mtd></mtr> <mtr><mtd><msup><mi>ℝ</mi> <mrow><mi>n</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mphantom><mtext>Milnor's exercise</mtext></mphantom></mover></mtd> <mtd><msubsup><mi>Alg</mi> <mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mphantom><mi>AAAA</mi></mphantom></mrow> <mi>op</mi></msubsup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>q</mi></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex">\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/formal+moduli+problem">formal</a> <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>(<a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super Lie theory</a>)<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mi>Super</mi><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>fin</mi></msub></mtd></mtr> <mtr><mtd><mi>𝔤</mi></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom><mrow><mtable><mtr><mtd><mover><mo>↪</mo><mrow><mphantom><mi>A</mi></mphantom><mtext><a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a></mtext><mphantom><mi>A</mi></mphantom></mrow></mover></mtd> <mtd><msup><mi>sdgcAlg</mi> <mi>op</mi></msup></mtd></mtr> <mtr><mtd><mo>↦</mo></mtd> <mtd><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">FDAs</a>”)</td></tr> </tbody></table> <p><strong>in <a class="existingWikiWord" href="/nlab/show/physics">physics</a></strong>:</p> <table style="margin:auto"><thead><tr><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+algebra">Poisson algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/algebra+of+observables">algebra of observables</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/space+of+states">space of states</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Heisenberg+picture">Heisenberg picture</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Schr%C3%B6dinger+picture">Schrödinger picture</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/AQFT">AQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/FQFT">FQFT</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Poisson+n-algebra">Poisson n-algebra</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/n-plectic+manifold">n-plectic manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/En-algebras">En-algebras</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Beilinson-Drinfeld+algebra">BD</a>-<a class="existingWikiWord" href="/nlab/show/BV+quantization">BV quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/higher+geometric+quantization">higher geometric quantization</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+algebra+of+observables">factorization algebra of observables</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/extended+quantum+field+theory">extended quantum field theory</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/factorization+homology">factorization homology</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism representation</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div></body></html> </div> <div class="revisedby"> <p> Last revised on July 8, 2023 at 12:22:47. See the <a href="/nlab/history/geometry" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/geometry" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/16838/#Item_1">Discuss</a><span class="backintime"><a href="/nlab/revision/geometry/44" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/geometry" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/geometry" accesskey="S" class="navlink" id="history" rel="nofollow">History (44 revisions)</a> <a href="/nlab/show/geometry/cite" style="color: black">Cite</a> <a href="/nlab/print/geometry" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/geometry" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>