flat differential cohomology

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

In a locally contractible (∞,1)-topos $\mathbf{H}$ with internal path ∞-groupoid functor $(\mathbf{\Pi} \dashv \mathbf{\flat})$, the **flat differential cohomology** of an object $X$ with coefficient in an object $A$ is the $A$-cohomology of the path ∞-groupoid $\mathbf{\Pi}(X)$:

$\mathbf{H}_{flat}(X,A) :=
\mathbf{H}(\mathbf{\Pi}(X),A)
\simeq
\mathbf{H}(X, \mathbf{\flat}(A))
\,.$

The constant path inclusion $X \to \mathbf{\Pi}(X)$ induces a morphism

$\mathbf{H}_{flat}(X,A) \to \mathbf{H}(X,A)$

which sends a flat differential cocycle to its *underlying* or *bare* cocycle.

The obstruction theory for lifts through this morphism is the differential cohomology of $X$.

See differential cohomology in an (∞,1)-topos for more details.

Revised on May 29, 2012 22:04:00
by Andrew Stacey
(129.241.15.200)