nLab geometry

geometry $\leftarrow$ Isbell duality $\rightarrow$ algebra

See also topology.

Related entries

• geometric type theory, geometric homotopy type theory

• synthetic geometry

• differential geometry, differential topology, Diff, cobordism

• differential form, tangent space, tangent bundle, cotangent bundle, cotangent complex

• symplectic geometry, symplectic manifold, Poisson manifold, Lagrangian submanifold

• multisymplectic geometry, n-symplectic manifold, foliation, integrable distribution, G-structure

• fibre bundle, principal bundle, noncommutative principal bundle, vector bundle

• connection (and links therein), connection on a bundle

• Morse function, Morse lemma, Morse theory, perfect Morse function

• Casson invariant, Donaldson-Thomas invariant, Kähler manifold, mirror symmetry

• metric space, convex set, Riemannian manifold, geodesic flow, geodesic convexity, star-shaped

• differentiable stack, orbifold, stratifold, orientifold

• algebraic geometry, analytic geometry,

arithmetic geometry, GAGA, book entry EGA

• scheme, quasicompact, noetherian scheme, reduced scheme, integral scheme

• open subscheme, closed subscheme, configuration scheme

• formal scheme, formal group scheme, formal group law, algebraic group

• spectrum (geometry)

• algebraic variety, normal variety, hyperplane line bundle

• derived algebraic geometry, derived smooth manifold

• noncommutative geometry, derived noncommutative algebraic geometry

• noncommutative algebraic geometry, noncommutative scheme, noncommutative thin scheme

• hyperplane section theorem, BBDG decomposition theorem

• rational map, rational variety, unirational variety, birational map, birational geometry, image of a rational map

• algebraic space, algebraic stack, topological stack

• quasicoherent sheaf, coherent sheaf, relative scheme

• smooth scheme, smooth morphism of schemes, etale morphism, formally smooth morphism

• pure motive, Voevodsky motive

• D-module, local system, regular differential operator, holonomic D-module

• D-geometry, Grothendieck connection, crystal, diffiety

• flag variety, geometric quantization, coherent state, orbit, coadjoint orbit

• Lie group, Lie groupoid, Lie algebroid, Courant algebroid, Atiyah Lie groupoid

• Lie theory, Lie's three theorems, Erlangen program

• ringed space, ringed site, ringed topos, lined topos

• sheaf of ideals, defining sheaf, conormal sheaf, conormal bundle, subscheme of an Abelian category

• synthetic differential geometry, infinitesimal object, smooth topos, Kock-Lawvere axiom, infinitesimal singular simplicial complex, differential forms in synthetic differential geometry

  • book entry Models for Smooth Infinitesimal Analysis

• generalized smooth space:

  • Froelicher space, diffeological space, smooth space

• space and quantity

• Euclidean geometry, trigonometry

• D-brane geometry

There are many entries on sheaf, stack, site, locale and topos theory including

• germ, etale space, soft sheaf, flabby sheaf, fine sheaf, Grothendieck topos, Grothendieck topology, (infinity,1)-topos, smooth topos, (infinity,1)-category of (infinity,1)-sheaves, cover, hypercover, local object, local isomorphism, coverage, local epimorphism, hypercompletion, sieve, Verdier site, Grothendieck fibration, Pierce spectrum, Stone duality, point of a topos, geometric morphism…

and pages on various cohomologies, including sheaf cohomology, nonabelian cohomology, differential cohomology, Deligne cohomology, etale cohomology, equivariant cohomology, Bredon cohomology and their cocycle classes including torsors, gerbes, principal 2-bundles as well as the related picture of the descent theory (cf. oriental, descent for simplicial presheaves…). A modern systematic theory of cohomology and descent can be done using the language of $(\infinity,1)$-categories and abstract homotopy theory, say via Quillen model categories (e.g. of simplicial presheaves).

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$	$\phantom{A}$fin. gen.$\phantom{A}$ $\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\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}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\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 }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$