Schreiber
flat differential cohomology

Idea

Flat differential cohomology of a space $X$ is the cohomology of its path ∞-groupoid $\Pi (X)$.

Infinitesimal flat differential cohomology of $X$ is the cohomology of the infinitesimal path ∞-groupoid ${\Pi }^{\inf }(X)$.

The obstruction theory for lifts from the cohomology of $X$ to the flat differential cohomology of $X$ the differential cohomology of $X$.

Applications

For infinitesimal flat differential cohomology, the differential Quillen adjunction

$${\Pi }^{\inf }(-):H\overleftarrow{\to }H:(-{)}_{\mathrm{flat}}^{\inf }$$\Pi^{inf}(-) : \mathbf{H} \stackrel{\leftarrow}{\to} \mathbf{H} : (-)_{flat}^{inf}

described at infinitesimal path ∞-groupoid restricts on coefficient objects $A$ being contractibel abelian Lie groups to the

• deRham theorem for ∞-Lie groupoids.