group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A differential characteristic class is a refinement of a characteristic class from ordinary cohomology to differential cohomology.
For characteristic classes of classifying spaces of Lie groups, the refinement to differential characteristic classes is the topic of Chern-Weil theory. In that context one traditionally speaks of secondary characteristic classes.
There is an unrefined and a refined version of differential characteristic classes. The unrefined version takes values in de Rham cohomology. The refined version lifts this to ordinary differential cohomology.
The following definition is in terms of the axiomatics of cohesive (∞,1)-toposes.
Let $\mathbf{H}$ be a cohesive (∞,1)-topos, $A \in \mathbf{H}$ any object and $K \in \mathbf{H}$ an abelian ∞-group object. Write $\mathbf{B}^n K$ for the $n$-fold delooping of $K$.
An ordinary characteristic class on $A$ of with coefficients in $K$ of degree $n$ is a morphism
or rather the class
that it represents. By general properties of cohesive (∞,1)-toposes there is a canonical morphism $curv : \mathbf{B}^n K \to \mathbf{\flat}_{dR} \mathbf{B}^{n+1}K$ to the de Rham coefficient object of $\mathbf{B}^n K$. This is the universal curvature characteristic class on $\mathbf{B}^n K$.
The (unrefined) differential characteristic class or curvature characteristic class lifting the characteristic class $\mathbf{c} : A \to \mathbf{B}^n K$ is the composite
or rather its class
that it represents.
Postcomposition with differential characteristic classes induces the (unrefined) abstract Chern-Weil homomorphism
For $G \in \mathbf{H}$ an ∞-group and $A = \mathbf{B}G$ its delooping, this morphism
sends $G$-principal ∞-bundles $P \to X$ to the curvature characteristic class $\mathbf{c}_{dR}(P)$ that represents the characteristic class $\mathbf{c}(P)$ in intrinsic de Rham cohomology.
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See the references at Chern-Weil theory and Chern-Weil theory in Smooth∞Grpd.
Lecture notes on secondary cohomology classes? in differential cohomology for flat connections is presented in