nLab geometry


geometry \leftarrow Isbell duality \rightarrow algebra



“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 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.

arithmetic geometry, GAGA, book entry EGA

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

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 (,1)(\infinity,1)-categories and abstract homotopy theory, say via Quillen model categories (e.g. of simplicial presheaves).


duality between \;algebra and geometry

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\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 }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}

Last revised on July 8, 2023 at 12:22:47. See the history of this page for a list of all contributions to it.