under construction
A differential function complex (HopkinsSinger) is a Kan complex of cocycle s for generalized differential cohomology, hence for differential refinements of generalized (Eilenberg-Steenrod) cohomology theories:
roughly, given a spectrum $E$ representing a given cohomology theory, its differential function complex over any given smooth manifold $U$ is the simplicial set whose $k$-simplices are triples consisting of
a continuous function $f : U \times \Delta^k \to E_{n}$;
a smooth differential form $\omega$ on $U \times \Delta^k$ whose corresponding real cohomology class (under the de Rham theorem) is that of the pullback of the real cohomology classes of $E$ along $f$;
an explicit coboundary in real cohomology exhibiting this fact.
(More precisely, in order for this construction to yield not just a single simplicial set (which will be a Kan complex) but a suitable spectrum object, there are conditons on the dependency of $\omega$ on the tangent vectors to the simplex.)
When applied to the Eilenberg-MacLane spectrum $K\mathbb{Z}$ this construction reproduces, on cohomology classes, ordinary differential cohomology. Applied to the classifying space $B U$ of topological K-theory it gives differential K-theory.
See also at differential cohomology diagram –Hopkins-Singer coefficients.
For the present purposes it will be convenient to collect cocycles of various degrees together to a single cocycle. For that purpose we make the following simple definition.
For $V = V^\bullet$ a graded vector space over the real numbers set
for $E$ a topological space:
and so on
(…)
For
$E$ a topological space;
$\iota \in Z^n(E,\mathbb{R})$ a cocycle on $E$ for real-valued singular cohomology on $E$,
a differential function on a smooth manifold $U$ with values in $(E,\iota)$ is a triple $(c,h,\omega)$ with
$c : U \to E$ a continuous map;
$\omega \in \Omega^n(S)$ a smooth differential form on $S$;
$h \in C^{n-1}(U,\mathbb{R})$ a cochain in real cohomology on (the topological space underlying) $U$;
such that in the abelian group $Z^n(S,\mathbb{R})$ of singular cochains the equation
holds, where
$\omega$ is here regarded as a singular cochain (that sends a chain to the integral of $\omega$ over it, as discussed at de Rham theorem),
$\delta$ denotes the coboundary operator,(the Moore complex differential of the singular simplicial complex).
This is (HopkinsSinger, def.4.1).
In words this is: a continuous map to the topological space together with a smooth refinement of the pullback of the chosen singular cochain.
For
$E$ be a topological space and
$\iota \in Z^n(E,\mathbb{R})$ a cocycle on $E$ for real-valued singular cohomology on $X$,
$U$ a smooth manifold,
the differential function complex
of all differential functions $S \to (X,\iota)$ is the simplicial set whose $k$-simplices are differential functions, def, 2
For applications one needs certain sub-complex of this, filtered by the number of legs that $\omega$ has along the simplices.
For $s \in \mathbb{N}$ write
$filt_s \Omega^\bullet(U \times \Delta^k)$
for the sub-simplicial set of differential forms that vanish when evaluated on more than $s$ vector fields tangent to the simplex;
$filt_s (X,\iota)^S \subset (X,\iota)^S$
for the sub-simplicial set of those differential functions whose differential form component is in $filt_s \Omega^\bullet(U \times \Delta^k)$.
This is (HopkinsSinger, def. 4.5).
The complex $filt_s (E,\iota)^U$ is (up to equivalence, of course) the homotopy pullback
in sSet (regarded as equipped with its standard model structure on simplicial sets).
Here $E^U$ is the internal hom in Top and $Sing(-)$ denotes the singular simplicial complex.
The following proposition gives the simplicial homotopy groups of these differential function complexes in dependence of the parameter $s$.
We have generally
(for instance by the Dold-Kan correspondence).
The simplicial homotopy groups of $filt_s \Omega^n_{cl}(S \times \Delta^\bullet_{Diff})$ are
This implies isomorphisms
This appears as HopkinsSinger, p. 36 and corollary D15.
Let $E_\bullet$ be an Omega-spectrum. Let $\iota_\bullet$ be the canonical Chern character class (…).
For $S$ a smooth manifold, and $s \in \mathbb{N}$, the sequence of differential function complexes, def. 3,
forms an Omega-spectrum.
This is the differential function spectrum for $E$, $S$, $s$.
This is ([HopkinsSinger, section 4.6]).
The differential $E$-cohomology group of the smooth manifold $S$ in degree $n$ is
This is (HopkinsSinger, def. 4.34).
For reference, we repeat from above the central statements about the homotopy types of the differential function complexes, def. 3.
For $E$ an Omega-spectrum, $S$ a smooth manifold, we have for all $s,n \in \mathbb{N}$, a weak homotopy equivalence
identifying the loop space object (at the canonical base point) of the differential function complex of $E_{n}$ at filtration level $s+1$ with that differential function complex of $E_{n-1}$ at filtration level $s$.
The following is a simple corollary or slight rephrasing of some of the above constructions, which may serve to show how differential function complexes present differential cohomology in the cohesive (∞,1)-topos of smooth ∞-groupoids.
For $E_\bullet$ a spectrum as above,
we have an (∞,1)-pullback square
By prop. 2 we have that
$filt_\infty (E; \iota_n)^S \simeq Sing X^S$;
$filt_0 \Omega_{cl}(S \times \Delta^\bullet) \simeq \Omega_{cl}(S)$.
The statement then follows with the pasting law for homotopy pullbacks
(…)
For $E = H \mathbb{Z}$ the Eilenberg-MacLane spectrum, $H_{diff}^n(-,E)$ is ordinary differential cohomology.
For $E = K U$ the K-theory spectrum, $H_{diff}^n(-,E)$ is differential K-theory.
For $E = M O, M U$ the Thom spectrum, $H_{diff}^n(-,E)$ is differential cobordism cohomology;
For $E = tmf$ the tmf spectrum, $H_{diff}^n(-,E)$ is differential cohomology;
Let $X = \mathcal{B} U(1) \simeq K(\mathbb{Z},2)$ be the Eilenberg-MacLane space that is the classifying space for $U(1)$-principal bundles. It carries the canonical cocycle $\iota := Id : \mathcal{B}U(1) \to \mathcal{B}U(1) \simeq K(\mathbb{Z},2)$ representing in $H^2(X,\mathbb{Z})$ the class of the universal complex line bundle $L \to X$ on $X$.
Accordingly, for $c : S\to \mathcal{B}U(1)$ a continuous map, we have the corresponding line bundle $c^* L$ on $S$.
One checks (…details…Example 2.7 in HopSin) that a refinement of $c$ to a differential function $(c,\omega,h)$ corresponds to equipping $c^* L$ with a smooth connection.
Now consider $((c,\omega,h) \to (c',\omega', h')) \in filt_0 (\mathcal{B}U(1),Id)^S$ a morphism between two such $(\mathcal{B}U(1),Id)$-differential functions. By definition this is now a $U(1)$-principal bundle $\hat L$ with connection on $S \times \Delta^i_{Diff}$, whose curvature form $\hat \omega \in \Omega^2(S \times \Delta^1_{Diff})$ is of the form $g \cdot \tilde \omega$, where $\tilde \omega$ is a 2-form on $S$ and $g$ is a smooth function on $\Delta^1_{Diff}$, both pulled back to $S \times \Delta^1_{Diff}$ and multiplied there.
But since $\hat \omega$ is necessarily closed it follows with $d (g \wedge \tilde \omega) = d t \frac{\partial g}{\partial t} \wedge \tilde \omega + g \wedge d_{S} \tilde \omega$ that $g$ is actually constant.
This means that that the parallel transoport of the connection $\hat \nabla$ on $S \times \Delta^1_{Diff}$ induces a insomorphism between the two line bundles on $S$ over the endpoints of $S \times \Delta^1_{Diff}$ that respects the connections.
…
(…)
For $E = H \mathbb{Z}$ the Eilenberg-MacLane spectrum, prop. 4 states that $filt_{1}(H \mathbb{Z}^{n+1}; \iota_{n})^S$ is an n-groupoid such that the automorphisms of the 0-object form ordinary differential cohomology in degree $n$.
Example 1 for $n = 4$ plays a central role in the description of T-duality by twisted differential K-theory in (KahleValentino).
Differential function complexes were introduced and studied in
For further references see differential cohomology.