Schreiber
twisted Bianchi identity

differential cohomology in an (∞,1)-topos -- survey

structures in an (∞,1)-topos

• shape

• cohomology

  • cocycle/characteristic class

  • twisted cohomology

  • principal ∞-bundle

  • ∞-vector bundle

• homotopy

  • covering ∞-bundles

  • Postnikov system

  • path ∞-groupoid

  • geometric realization

  • Galois theory

  • internal homotopy ∞-groupoid?

  • Whitehead system

• rational homotopy

  • ∞-Lie algebroid

  • ordinary rational homotopy

  • internal rational homotopy

  • Chern-character

• differential cohomology

  • flat differential cohomology

  • de Rham cohomology

  • de Rham theorem

• relative theory over a base

  • relative homotopy theory

  • Lie theory

Examples

(…)

Applications

• Background fields in twisted differential nonabelian cohomology

• Differential twisted String and Fivebrane structures

• D'Auria-Fre formulation of supergravity

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∞-Lie theory

∞-Lie groupoids and -algebroids

• ∞-Lie groupoid

• ∞-Lie algebroid

  • Chevalley-Eilenberg algebra

  • Weil algebra

• ∞-Lie differentiation and integration

  • path ∞-groupoid

  • infinitesimal path ∞-groupoid?

  • integration of ∞-Lie algebroid valued differential forms

∞-Chern-Weil theory

• invariant polynomial

• differential cohomology in an (∞,1)-topos

  • ∞-Lie algebroid valued differential forms

    • curvature?

    • integration

  • Cartan-Ehresmann ∞-connection

symplectic ∞-geometry

• symplectic ∞-Lie algebroid

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(…the Bianchi identity on curvature forms in differential twisted cohomology…)