# Schreiber flat differential cohomology

differential cohomology in an (∞,1)-topos -- survey

structures in an (∞,1)-topos

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## Applications

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

${H}_{\mathrm{flat}}\left(X,A\right):=H\left(\Pi \left(X\right),A\right)\simeq H\left(X,♭\left(A\right)\right)\phantom{\rule{thinmathspace}{0ex}}.$\mathbf{H}_{flat}(X,A) := \mathbf{H}(\mathbf{\Pi}(X),A) \simeq \mathbf{H}(X, \mathbf{\flat}(A)) \,.

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

${H}_{\mathrm{flat}}\left(X,A\right)\to H\left(X,A\right)$\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)