Schreiber
twisted Bianchi identity

differential cohomology in an (∞,1)-topos

structures in an (∞,1)-topos

• shape

• cohomology

  • cocycle/characteristic class

  • principal ∞-bundle

  • ∞-vector bundle

• homotopy

  • covering ∞-bundles

  • Postnikov system

  • homotopy ∞-groupoid

  • geometric realization

  • Galois theory

  • internal homotopy ∞-groupoid

  • Whitehead system

• rational homotopy

  • ∞-Lie algebroid (internal)

  • ordinary rational homotopy

  • internal rational homotopy

  • Chern-character

• differential cohomology

  • flat differential cohomology

  • de Rham cohomology

  • de Rham theorem

  • Ehresmann ∞-connection

    • with groupal coefficients

    • with general coefficients

• Relative theory over a base-topos#OverBase)

  • relative homotopy theory-topos#OverBase)

  • infinitesimal homtopy theory

    • infinitesimal homotopy ∞-groupoid

    • ∞-Lie algebroid

    • ∞-Lie algebroid valued differential forms

    • invariant polynomial on ∞-Lie algebroid

    • Cartan-Ehresmann connection

Examples

• principal bundles with connection and differential K-theory

• principal 2-bundles with connection

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

• smooth (∞,1)-topos

• ∞-Lie groupoid

  • strict ∞-Lie groupoid

• ∞-Lie algebroid

  • Chevalley-Eilenberg algebra

• ∞-Lie differentiation and integration

  • path ∞-groupoid

  • infinitesimal path ∞-groupoid

  • integration of ∞-Lie algebroid valued differential forms

∞-Chern-Weil theory

• invariant polynomial on ∞-Lie algebroid

• theory of differential nonabelian cohomology

  • ∞-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…)