E-∞ scheme in nLab

higher geometry / derived geometry

Ingredients

• higher topos theory

• higher algebra

Concepts

• geometric little (∞,1)-toposes

  • structured (∞,1)-topos

  • geometry (for structured (∞,1)-toposes)

  • generalized scheme

• geometric big (∞,1)-toposes

  • cohesive (∞,1)-topos

• function algebras on ∞-stacks

  • geometric ∞-stacks

Constructions

• loop space object, free loop space object

• fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos / of a locally ∞-connected (∞,1)-topos

Examples

• derived algebraic geometry

  • étale (∞,1)-site, Hochschild cohomology of dg-algebras

  • dg-geometry

    • dg-scheme

  • schematic homotopy type

• derived noncommutative geometry

  • noncommutative geometry

• derived smooth geometry

  • differential geometry, differential topology

  • derived smooth manifold, dg-manifold

  • smooth ∞-groupoid, ∞-Lie algebroid

• higher symplectic geometry

• higher Klein geometry

• higher Cartan geometry

Theorems

• Isbell duality

• Jones' theorem, Deligne-Kontsevich conjecture

• Tannaka duality for geometric stacks

higher algebra

universal algebra

Algebraic theories

• algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory

• monad / (∞,1)-monad

• operad / (∞,1)-operad

Algebras and modules

• algebra over a monad

  ∞-algebra over an (∞,1)-monad

• algebra over an algebraic theory

  ∞-algebra over an (∞,1)-algebraic theory

• algebra over an operad

  ∞-algebra over an (∞,1)-operad

• action, ∞-action

• representation, ∞-representation

• module, ∞-module

• associated bundle, associated ∞-bundle

Higher algebras

• monoidal (∞,1)-category

  • symmetric monoidal (∞,1)-category

  • monoid in an (∞,1)-category

  • commutative monoid in an (∞,1)-category

• symmetric monoidal (∞,1)-category of spectra

  • smash product of spectra

  • symmetric monoidal smash product of spectra

  • ring spectrum, module spectrum, algebra spectrum

• A-∞ algebra

  • A-∞ ring, A-∞ space

• C-∞ algebra

• E-∞ ring, E-∞ algebra

  • ∞-module, (∞,1)-module bundle

  • multiplicative cohomology theory

• L-∞ algebra

  • deformation theory

Model category presentations

• model structure on simplicial T-algebras / homotopy T-algebra

• model structure on operads

  model structure on algebras over an operad

Geometry on formal duals of algebras

• Isbell duality

• derived geometry

Theorems

• Deligne conjecture

• delooping hypothesis

• monoidal Dold-Kan correspondence

number theory

• arithmetic

• arithmetic geometry, arithmetic topology

• higher arithmetic geometry, E-∞ arithmetic geometry

number

• natural number, integer number, rational number, real number, irrational number, complex number, quaternion, octonion, adic number, cardinal number, ordinal number, surreal number

arithmetic

• Peano arithmetic, second-order arithmetic

• transfinite arithmetic, cardinal arithmetic, ordinal arithmetic

• prime field, p-adic integer, p-adic rational number, p-adic complex number

arithmetic geometry, function field analogy

• arithmetic scheme

  • arithmetic curve, elliptic curve

  • arithmetic genus

• arithmetic Chern-Simons theory

• arithmetic Chow group

• Weil-étale topology for arithmetic schemes

• absolute cohomology

• Weil conjecture on Tamagawa numbers

• Borger's absolute geometry

• Iwasawa-Tate theory

  • arithmetic jet space

• adelic integration

• shtuka

• Frobenioid

Arakelov geometry

• arithmetic Riemann-Roch theorem

• differential algebraic K-theory