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flat differential cohomology

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

H flat(X,A):=H(Π(X),A)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 XΠ(X)X \to \mathbf{\Pi}(X) induces a morphism

H flat(X,A)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 XX.

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)