The smooth (∞,1)-topos $\mathbf{H}$ equipped with the path ∞-groupoid functor
and its right adjoint
induces a notion of flat differential cohomology. Flat cohomology with coefficients in $A$ is the ordinary cohomology with coefficients in the flat differential refinement $A_{flat}$ of $A$:
In as far as cohomology $H(X,A)$ classifies principal ∞-bundles, flat differential cohomology classifies principal ∞-bundles with flat connection.
As discussed at path ∞-groupoid there is a natural morphism $A_{flat} \to A$. The corresponding morphism $H_{flat}(X,A) \to H(X,A)$ sends a differential cocycle for an ∞-bundle with connection to the cocycle for the underlying bundle without connection.
Conversely, lifting an $A$-cocycle through $A_{flat} \to A$ means equipping a plain ∞-bundle with a flat connection.
This leads us to a first characterization of general differential cohomology
In an (∞,1)-topos $\mathbf{H}$ equipped with a path ∞-groupoid functor $\Pi(-)$ and the corresponding flat differential refinement functor $(-)_{flat} : \mathbf{H} \to \mathbf{H}$ we conceive differential cohomology with coefficients in $A \in \mathbf{H}$ as the obstruction theory of lifts of cocycles through the morphism
from $A$-valued cohomology to flat differential $A$-valued cohomology.
There is a good obstruction theory for lifts through a morphism $f : A' \to A$ when $f$ is the homotopy fiber of a morphism $h : A \to B$. In that case the obstruction to lifing a cocycle $g : X \to A$ to a cocycle $\hat g : X \to A'$ is the class of the $B$-valued cocycle $X \to A \to B$.
We exhibit such a fibration sequence that realizes $A_{flat} \to A$ as a homotoy fiber
for the case that $A$ is group-like in that its delooping $\mathbf{B}A$ exists. The object $\mathbf{B}A_{dR}$ here we interpret as the coefficient object for $A$-valued nonabelian deRham cohomology.
The induced morphism on cohomology
we interpret as a (possibly nonabelian refinement) of the assignment of curvature characteristic classes to the principal ∞-bundles classified by $H(X,A)$.
Every fibration sequence induces a notion of twisted cohomology. We conceive general (non-flat, nonabelian) differential cohomology as thetwisted cohomology whose twist is given by the curvature characteristic classes $char$.
This gives a precise and useful meaning to the following very plausible-sounding statement
Differential cocycles $\nabla$ with curvature characteristic class $[char(\nabla)]$ are cocycles in $[char(\nabla)]$-twisted flat differential cohomology.
In order to construct the objects $\mathbf{B}A_{dR}$ and the corresponding fibration sequence we use techniques from relative cohomology. In terms of that the nonabelian deRham object in question is, as smooth ∞-stacks
This is by construction the classifying object for those flat differential $\mathbf{B}A$-cocycles whose underlying plain $\mathbf{B}A$-cocycles are trivial.
In terms of principal ∞-bundles with connecton this means that
flat $A$-principal bundles with connection are the fundamental notion;
$\mathbf{B}A$-valued differential forms are identified with those flat $A$-principal bundles with connection whose underlying $A$-principal bundle is trivial.
So to summarize the situation for grouplike coefficients: following the general principle of twisted cohomology applied to the fibration sequence $A_{flat} \to A \to \mathbf{B}A_{dR}$ the differential cohomology of $X$ with coefficients in $A$ and with curvature characteristic class $P$ is the homotopy pullback
One finds that the cocycles $\nabla \in \mathbf{H}^{[P]}(X,A)$ are presented by diagrams in $\mathbf{H}$ of the form
It is in these diagrams that the notion of connection manifests itself: the connection is the choice of a morphism $\nabla : \Pi(X) \to \mathbf{E}A$ that interpolates between the plain $A$-valued cocycle $g : X \to A$ (a principal ∞-bundle with connection) and its curvature characteristic nonabelian deRham forms $P : \Pi(X) \to \mathbf{B}A$.
Every connection in this sense induces (nonabelian) deRham differential form data that yields a description of these connections in a way that generalizes the Ehresmann connection formulation of a connection on a principal bundle.
This leads to the description of differential cocycles in terms of
If $A$ is not grouplike, one can uses grouplike approximations to $A$ and proceed with these through the above constructions. One drastic but useful grouplike approximation to any $A$ is given by an integral Chern character morphism
From this one obtains along the above lines a coefficient object
equipped with a morphism $A \to \mathbf{B}A_{ch}$ such that the induced morphism on cohomology
generalizes the Chern character morphism in abelian cohomology.
Proceeding with this morphism as a substitute for the non-existing $A \to \mathbf{B}A_{dR}$ produces a the notion of differential nonabelian cohomology that measures obstructions to flatness only up to some approximation, but that is direct generalization of the defintition of classical abelian differential cohomology in terms of homotopy fibers of the Chern character map.
Fix a site $C$ as described at path ∞-groupoid.
Write $SPSh(C)^{loc}$ for the injective or projective local model structure on simplicial presheaves on $C$. This is a combinatorial simplicial model category.
Fix a cosimplicial object $\Delta_C : \Delta \to C$ as described at path ∞-groupoid at let $\Pi(-) : SPSh(C) \to SPSh(C)$ be the corresponding path $\infty$-groupoid functor and $(-)_{flat}$ its right adjoint.
Recall from path ∞-groupoid the definition of flat differential cohomology.
For $A \in SPSh(C)$ the flat differential cohomology with coefficients in $A$ is cohomology with coefficients in $A_{flat}$.
By the differential Quillen adjunction? a cocycle in $H_{flat}(X,A)$ is represented by a morphism $\Pi(X) \to A$.
Since we have the canonical natural inclusion $X \to \Pi(X)$ we may consider relative cohomology with respect to this.
At nonabelian deRham cohomology the following definition is discussed in detail
For $A \in SPSh(C)$ a pointed object with point $pt_A : {*} \to A$ define $A_{dR} \in SPSh(C)$ by
This we call the deRham differential refinement of $A$.
The cohomology with coefficients in $A_{dR}$
we call $A$-valued deRham cohomology
Here $[I,SPSh(C)]$ is the arrow category of $SPSh(C)$ as described at relative cohomology.
We conceive differential cohomology with coefficients in $A$ as being the obstruction classes to lifts from bare $A$-cohomology to flat differential $A$-cohomology through $\mathbf{H}(X,A_{flat}) \to \mathbf{H}(X,A)$.
There are two cases to be distinguished in such an obstruction problem:
in the case that $A_{diff} \to A$ is the homotopy fiber of some morphism $A \to Q$, the obstruction classes live precisely in the $A_{flat}$-twisted cohomology defined by that morphism;
in this case the obstuction classes in $Q$-cohomology are precise: they vanish if and only if the lift exists: this is just the restatement of the assumed homotopy pullback property of $A_{diff}$
in the case that $A_{flat} \to A$ is not the homotopy fiber of any morphism in that it does not fit into a square
that is a homotopy pullback square, we don’t have precise obstruction classes as above. But as a next best approximation we can still find necessary (but generally not sufficient) obstruction classes by finding any homotopy commutative square as above.
For any such square with some $Q$ we get the statement: for a bare $A$-cocycle $g : X \to A$ to have a lift $\hat g : X \to A_{flat}$ to a flat differential $A$-cocycle the $Q$-cocycle $X \stackrel{g}{\to} A \to Q$ necessarily has to be have trivial cohomology class. This is just what the homotopy commutativity of this square means.
This is discussed at differential cohomology - with general coefficients .