nLab geometry

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

Idea

“Geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring.” [Newton (1687), preface]

In its historical and etymological origins, “geometry” is the science of measuring the earth and here specifically of measuring distances and angles, first in the guise of Euclidean geometry of the ancients and then eventually as the differential geometry of curves and surfaces due to Gauss and others.

But in modern mathematical jargon, the term geometry is used in much greater generality for the study of spaces equipped with extra “geometrical” structure of a large variety of sorts, foremost in the guise of Riemannian geometry but subsuming also a wealth of variant notions of “geometries” which the ancient would not have recognized as such, for instance symplectic geometry and other (torsion-free) $G$-structured differentiable manifolds (differential Cartan geometry), or (higher) topos-theoretic notions (cf. “geometric logic”) of (higher) functorial geometry famously including algebraic geometry or supergeometry but also more exotic variants such as arithmetic geometry or absolute geometry. Finally, taking the duality between algebra and geometry to the extreme yields notions of noncommutative geometry and/or derived geometry whose underlying “spaces” are only indirectly conceived as whatever it is that given (higher) algebras would see (measure!) if they are interpreted as algebras of functions 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) measuring space.

Beware that bare topology is sometimes regarded as a rudimentary kind of geometry (as reflected for instance in the common terminology geometric realization for an operation that really is topological realization), but more often than not a “geometric space” is meant to be a topological space equipped with extra geometrical structure, of sorts.

Even a plain differentiable manifold as considered in differential geometry is often not quite regarded as geometric in itself unless equipped with further structure, as reflected for instance in the terminology topological field theory for those functorial field theories which depend on diffeomorphism-structure of spacetime but not on further (notably pseudo-Riemannian) geometric structure.

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).

References

• Wikipedia History of geometry

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}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$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$\phantom{A}$