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

étale morphism (submersion, immersion)

generally in differential cohesion

• reduction modality$\dashv$ infinitesimal shape modality

• formally étale (formally smooth, formally unramified)

• étale topos

• étale homotopy

• étale cohomology

• étale ∞-groupoid

in topology

• local homeomorphism, étale space

• étale groupoid

in differential geometry

• local diffeomorphism, (submersion, immersion of smooth manifolds)

• étale Lie groupoid

in algebraic geometry

• étale morphism of schemes, formally étale morphism of schemes

• étale site, pro-étale site, étale (∞,1)-site

• Deligne-Mumford stack

• ℓ-adic cohomology