The nth Deligne cohomology is defined as cohomology with coefficients
in a truncated chain complex of sheaves of U(1)-valued differential forms:
U(1)→Ω^1→Ω^2→⋯→Ω^n for some n≥0.
Thus starting with an arbitrary Lie group G one can take the truncated
simplicial object of sheaves of groups of G-valued differential forms (defined using crystals, for example),
and then take the sheaf cohomology with values in this simplicial presheaf.
This defines a nonabelian analog of the Deligne cohomology
for any choice of the underlying site: smooth, holomorphic, algebraic.
This post imported from StackExchange MathOverflow at 2017-09-14 13:31 (UTC), posted by SE-user Dmitri Pavlov