structure sheaf in nLab

topos theory

• Toposes

Background

• category theory

  • category

  • functor

Toposes

• (0,1)-topos, Heyting algebra, locale

• pretopos

• topos

  • Grothendieck topos

    • category of presheaves

      • presheaf

      • representable presheaf

    • category of sheaves

      • site

        • sieve

        • coverage, pretopology, topology

      • sheaf

      • sheafification

• quasitopos

• base topos, indexed topos

Internal Logic

• categorical semantics

• internal logic

  • subobject classifier

  • natural numbers object

Topos morphisms

• logical morphism

• geometric morphism

  • direct image/inverse image

  • global sections

  • geometric embedding

  • surjective geometric morphism

  • essential geometric morphism

    • locally connected geometric morphism

    • connected geometric morphism

    • totally connected geometric morphism

  • étale geometric morphism

  • open geometric morphism

  • proper geometric morphism, compact topos

  • separated geometric morphism, Hausdorff topos

  • local geometric morphism

  • bounded geometric morphism

  • base change

  • localic geometric morphism

  • hyperconnected geometric morphism

  • atomic geometric morphism

• topological locale

• localic topos

• petit topos/gros topos

• locally connected topos, connected topos, totally connected topos, strongly connected topos

• local topos

• cohesive topos

• classifying topos

• smooth topos

Cohomology and homotopy

• cohomology

• homotopy

• abelian sheaf cohomology

• model structure on simplicial presheaves

In higher category theory

• higher topos theory

• (0,1)-topos

  • (0,1)-site

• 2-topos

  • 2-site

  • 2-sheaf, stack

• (∞,1)-topos

  • (∞,1)-site

  • (∞,1)-sheaf, ∞-stack, derived stack

Theorems

• Diaconescu's theorem

• Barr's theorem

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