The simplicial de Rham complex of a simplicial manifold $X_\bullet$ is the analog of the de Rham complex of differential forms of an ordinary manifold: it is the complex whose elements in degree $k$ may be thought of as $k$-forms on $X_\bullet$.
One useful conceptual way to think of this is to notice that a a simplicial manifold may be thought of as a degreewise representable simplicial presheaf on Diff and then to realize that by way of the standard model structure on simplicial presheaves such a simplicial presheaf presents an ∞-stack on Diff which we may think of as a Lie ∞-groupoid. From that perspective we expect that
Slogan. The simplicial de Rham complex is the complex of differential forms on a geometric ∞-Lie groupoid.
We shall discuss this in more detail below.
There are several definitions that are quasi-isomorphic. The first one we give is the conceptually most straightforward one. The second one we give is sometimes more useful in computations.
Let $X_\bullet : \Delta^{op} \to Diff$ be a simplicial manifold.
(simplicial de Rahm complex, first version)
Write
for the cochain complex that is the total complex of the double complex on the degreewise de Rham complex with one differential the simplicial-degree-wise de Rham differential and the other one the de Rham-degree-wise alternating sum of pullback along the face maps
So an element $\omega \in \mathcal{A}(X_\bullet)$ in degree $n$ is a collection $(\omega^p_q \in \Omega^p(X_q))_{p+q = n}$ of ordinary differential forms.
(simplicial de Rahm complex, second version)
Write $\Delta^n_{Diff}$ for the standard $n$-simplex in its standard incarnations as a smooth manifold (with boundary). These arrange in the obvious way into the cosimplicial object $\Delta_{Diff} : \Delta \to Diff$.
Say that a differential form $\omega^p_q \in \Omega^\bullet(\Delta^p_{Diff}\times X_q)$ is compatible if for each face map $\delta_i$ we have
(some condition, need to look something up…)
There is a decomposition
This defines a bidegree and $A(X_\bullet)$ is the obvious total complex of the obvious double complex here
will polish this up later
The following proposition says that and how these two complexes are related.
(Dupont)
Consider the map between the two double complexes involved above which integrates each element in degree $(p,q)$ over $\Delta^p_{Diff}$. This induces a map on the corresponding total complexes
This is a morphism of cochain complexes which is a quasi-isomorphism.
A main application of this technology is to the simplicial manifold $\mathbf{B}G = (\cdots G \times G \stackrel{\stackrel{\to}{\to}}{\to} G \stackrel{\to}{\to} {*})$ that represents the smooth groupoid which is the delooping of a Lie group $G$.
Dupont exhibits a Chern-Weil homomorphism
and constructs a canonical connection on the universal G-principal-bundle
(which, in traditional simplicial group notation, reads $W G \to \bar W G$ – but recall that these are “Lie” simplicial groups here).
Urs Schreiber: This section is supposed to provide a useful reformulation of the simplicial de Rham complex in the context described at ∞-Lie theory.
In the language uses there, the statement we establish is the following:
Let $C =$ CartSp${}_{th}$ and $\mathbf{H} = (sPSh(C)_{proj}^{loc})^\circ$ the (∞,1)-category of (∞,1)-sheaves on CartSp, the (∞,1)-topos of Lie ∞-groupoids. This is a locally contractible (∞,1)-topos (as discussed there). Accordinly we have its path ∞-groupoid and infinitesimal path ∞-groupoid? $\mathbf{\Pi}_{inf}(-)$.
Then
the Chevalley-Eilenberg algebra of $\mathbf{\Pi}_{inf}(X)$ is quasi-isomorphic to the simplicial de Rham complex
The following discussion breaks this down and then describes the proof.
As a preparation, recall from the discussion at differential forms in synthetic differential geometry that if we pass from Diff to a smooth topos $(\mathcal{T},R)$ that models the axioms of synthetic differential geometry, then for sufficiently well-behaved objects $X \in \mathcal{T}$ there is the infinitesimal singular simplicial complex $X^{(\Delta^\bullet_{inf})} : \Delta^{op} \to \mathcal{T}$, the simplicial object that in degree $k$ is the space of infinitesimal $k$-simplices in $X$.
As discussed there, this is such that under the Dold-Kan correspondence the cosimplicial algebra $Hom( X^{\Delta^\bullet_{inf}}, R )$ maps to the de Rham complex (and under the monoidal Dold-Kan correspondence to the full de Rham dg-algebra):
It would be nice to have an analog of this statement for simplicial objects and the simplicial de Rham complex. I am thinking that the answer should be the following:
Let $X_\bullet : \Delta^{op} \to \mathcal{T}$ be a simplicial object that is degreewise of the sort such that the infinitesimal singular simplicial complex $(X_n)^{\Delta^\bullet_{inf}}$ exists. Use that $\mathcal{T}$ is canonically tensored over $Set$ to find that simplicial objects in $\mathcal{T}$ are canonically tensored over simplicial sets. Then consider the realization
where
$\Delta[n]$ is the standard simplicial $n$-simplex
the integrand is the tensor operatoin of simplicial objects by simplicial sets
the integral sign denotes the coend.
By the lemma expression in terms of simplicial realization at infinitesimal path ∞-groupoid? this is the same as $\mathbf{\Pi}_{inf}(X)$.
The above proposition now reads in pedestrian terms:
The Moore cochain complex of the cosimplicial algebra $C^\infty(\mathbf{\Pi}_{inf}(X_\bullet)) := Hom_{\mathcal{T}}(\mathbf{\Pi}_{inf}(X_\bullet),R)$ is quasi-isomorphic to the simplicial de Rham complex of $X_\bullet$.
We use the cosimplicial and the simplicial version of the Eilenberg-Zilber theorem together with the fact that for a bisimplicial set the diagonal is given by the realization (as discussed there) $Diag F_{\bullet,\bullet} \simeq \int^{[n] \in \Delta} \Delta[n] \cdot F_{n,\bullet}$ to compute
Here in the last step we used the following reasoning on the bisimplicial object $([p],[q]) \mapsto (X_p)^{(\Delta^q_{inf})}$. We know that
for fixed $p$, the normalized Moore complex of the cosimplicial algebra $Hom((X_p)^{(\Delta^\bullet_{inf})},R)$ is the de Rham complex of $X_p$ – this is the statement about combinatorial differential forms in synthetic differential geometry.
for fixed $q$ the Moore complex of the cosimplicial algebra $Hom((X_\bullet)^{((\Delta^q_{inf}))}, R)$ is the complex of functions on simplices whose differential is the alternating sum of pullbacks along face maps – by the very definition of the Moore complex.
This means that the simplicial de Rham complex is (quasi-isomorphic to) the total complex of the bi-cosimplicial algebra
So it remains to show that this total complex is also (quasi-isomorphic to) the Moore complex of $Hom(\mathbf{\Pi}_{inf}(X_\bullet),R)$. For this use exercise 1.6 here which says (transported from Set to $\mathcal{T}$) that this is the diagonal simplicial object of our bisimplicial object
This implies that the Moore complex of $Hom(\mathbf{\Pi}_{inf}(X_\bullet),R)$ is the Moore complex of the diagonal of the bisimplicial algebra $Hom(X_\bullet^{(\Delta^\bullet_{inf})}),R)$.
This way the desired statement recudes to the quasi-isomorphism
But this (even their chain-homotopy equivalence) is the content of the generalized Eilenberg-Zilber theorem.
Canonical references on simplicial de Rham cohomology are by Johan Louis Dupont. For instance
Johan Louis Dupont, Simplicial de Rham Cohomology and characteristic classes of flat bundles Topology 15 (1976)
Johan Louis Dupont, A dual simplicial de Rham complex, Lecture notes in Mathematics 1318 (1988) (journal)
chaper 3 in Curvature and Characteristic Classes
(I am still looking for the best survey reference…)
When restricted to low degree this is closely related to or synonymous to considerations of de Rham cohomology in the context of differentiable stacks.
(need to dig out references)
A de Rham theorem for simplicial manifolds was proven in the classical
In principle closely related is the discussion of a de Rham theorem for ∞-stacks as discussed in
though a simplicial de Rham complex is only somewhat implicit in that article.