nLab (infinity,1)-topos - contents

(∞,1)-topos theory

Background

• sheaf and topos theory

• (∞,1)-category

• (∞,1)-functor

• (∞,1)-presheaf

• (∞,1)-category of (∞,1)-presheaves

Definitions

• elementary (∞,1)-topos

• (∞,1)-site

• reflective sub-(∞,1)-category

  • localization of an (∞,1)-category

  • topological localization

  • hypercompletion

• (∞,1)-category of (∞,1)-sheaves

  • (∞,1)-sheaf/∞-stack/derived stack

• (∞,1)-topos

• (n,1)-topos, n-topos

  • n-truncated object

  • n-connected object

  • (1,1)-topos

    • presheaf

    • sheaf

  • (2,1)-topos, 2-topos

    • (2,1)-presheaf

• (∞,1)-quasitopos

  • separated (∞,1)-presheaf

  • quasitopos

    • separated presheaf

  • (2,1)-quasitopos?

    • separated (2,1)-presheaf

• (∞,2)-topos

• (∞,n)-topos

Characterization

• universal colimits

• object classifier

• groupoid object in an (∞,1)-topos

  • effective epimorphism

Morphisms

• (∞,1)-geometric morphism

• (∞,1)Topos

• Lawvere distribution

Extra stuff, structure and property

• hypercomplete (∞,1)-topos

  • hypercomplete object

  • Whitehead theorem

• over-(∞,1)-topos

• n-localic (∞,1)-topos

• locally n-connected (n,1)-topos

• structured (∞,1)-topos

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

• locally ∞-connected (∞,1)-topos, ∞-connected (∞,1)-topos

• local (∞,1)-topos

  • concrete (∞,1)-sheaf

• cohesive (∞,1)-topos

Models

• models for ∞-stack (∞,1)-toposes

  • model category

  • model structure on functors

  • model site/sSet-site

  • model structure on simplicial presheaves

  • descent for simplicial presheaves

  • descent for presheaves with values in strict ∞-groupoids

Constructions

structures in a cohesive (∞,1)-topos

• shape / coshape

• cohomology

• homotopy

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

  • categorical/geometric homotopy groups

  • Postnikov tower

  • Whitehead tower

• rational homotopy

• dimension

  • homotopy dimension

  • cohomological dimension

  • covering dimension

  • Heyting dimension

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