Schreiber
flat differential cohomology

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

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

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

$$H_{\mathrm{flat}}(X,A)\to H(X,A)$$\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.