Schreiber flat differential cohomology

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.