(also nonabelian homological algebra)
There are various simplicial dg-algebras that assign to the standard $n$-simplex a kind of de Rham algebra on $\Delta^n$.
By the discussion at differential forms on presheaves, each such extends to a notion of differential forms on simplicial sets.
(smooth $n$-simplex)
For $n \in \mathbb{N}$ the smooth n-simplex $\Delta^n_{smth}$ is the smooth manifold with boundary and corners defined, up to isomorphism, as the following locus inside the Cartesian space $\mathbb{R}^{n+1}$:
For $0 \leq i \leq n$ the function
which picks the $i$th component in the above definition is called the $i$th barycentric coordinate function.
For
a morphism of finite non-empty linear orders $[n] \coloneqq \{0 \lt 1 \lt \cdots \lt n\}$, let
be the smooth function defined by $x_i \mapsto x_{f(i)}$.
(smooth differential forms on the smooth $n$-simplex)
For $k \in \mathbb{N}$ then a smooth differential k-form on the smooth $n$-simplex (def. 1) is a smooth differential form in the sense of smooth manifolds with boundary and corners. Explicitly this means the following.
Let
be the affine plane in $\mathbb{R}^{n+1}$ that contains $\Delta^n_{smth}$ in its defining inclusion from def. 1. This is a smooth manifold diffeomorphic to the Cartesian space $\mathbb{R}^{n}$.
A smooth differential form on $\Delta^n_{smth}$ of degree k$ is a collection of linear functions
out of the $k$-fold skew-symmetric tensor power of the tangent space of $F^n$ at some point $x$ to the real numbers, for all $x \in \Delta^n_{smth}$ such that this extends to a smooth differential $k$-form on $F^n$.
Write $\Omega^\bullet(\Delta^n_{smth})$ for the graded real vector space defined this way. By definition there is then a canonical linear map
from the de Rham complex of $F^n$ and there is a unique structure of a differential graded-commutative algebra on $\Omega^\bullet(\Delta^n_{smth})$ that makes is a homomorphism of dg-algebras form the de Rham algebra of $F^n$. This is the de Rham algebra of smooth differential forms on the smooth $n$-simplex.
For $f \colon [n_1] \to [n_2]$ a homomorphism of finite non-empty linear orders with $\Delta_{smth}(f) \colon \Delta^{n_1}_{smth} \to \Delta^{n_2}_{smth}$ the corresponding smooth function according to def. 1, there is the induced homomorphism of differential graded-commutative algebras
induced from the usual pullback of differential forms on $F^n$. This makes smooth differential forms on smooth simplices be a simplicial object in differential graded-commutative algebras:
The standard proof of the Poincaré lemma applies to show that
Each element of $\Omega^p_{poly}(\Delta^n)$ may be uniquely written
where $b_j$ is as above the $j^{th}$ barycentric coordinate function and each $\Phi_{i_1\ldots i_p}$ is a $C^\infty$-function on $\mathbf{\Delta}^n$.
With this representation the multiplication and differential are given by the usual formulae. The multiplication is defined by $\Phi \wedge \Psi$ and extends linearly the product
on the generating forms. Now if $f$ is a differentiable function
so if
then
For $n \in \mathbb{N}$ write
for the quotient of the $\mathbb{Z}$-graded symmetric algebra over the rational numbers on $n+1$ generators $t_i$ in degree 0 and $n+1$ generators $d t_i$ of degree 1.
In particular in degree 0 this are called the polynomial functions
due to the canonical inclusion
into the smooth functions on the $n$-simplex according to def. 2, obtained by regarding the generator $t_i$ as the $i$th barycentric coordinate function.
Observe that the tensor product of the polynomial differential forms over these polynomial functions with the smooth functions on the $n$-simplex, is canonically isomorphic to the space $\Omega^\bullet(\Delta^n_{smth})$ of smooth differential forms, according to def. 2:
where moreover the generators $d t_i$ are identified with the de Rham differential of the $i$th barycentric coordinate functions.
This defines a canonical inclusion
and there is uniquely the structure of a differential graded-commutative algebra on $\Omega^\bullet_{poly}(\Delta^n)$ that makes this a homomorphism of dg-algebras. This is the dg-algebra of polynomial differential forms.
For $f \colon [n_1] \to [n_1]$ a morphism of finite non-empty linear orders, let
be the morphism of dg-algebras given on generators by
This yields a simplicial differential graded-commutative algebra
which is a sub-simplicial object of that of smooth differential form
By left Kan extension the functor of polynomial differential forms from def. 3 yields a functor on all simplicial sets
This is the left adjoint in a nerve and realization adjunction
Composing with the singular simplicial complex functor
on topological spaces, this yields a functor on topological spaces
which we may think of as assigning “piecewise polynomial” differential forms.
This is the starting point of the Sullivan approach to rational homotopy theory. See there for more
Let $k$ be a field of characteristic 0. Let $\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}$ be the left Kan extension of $\Omega^\bullet_{poly} : \Delta \to cdgAlg_k^{op}$ from above.
For $S \in sSet$, define a morphism of graded $k$-vector spaces
from polynomial differential forms on simplices to cochains on simplicial sets by sending $\omega \in \Omega^n_{poly}(K)$ to the cochain that sends $\sigma \in K_n$ to
where on the right we have the ordinary integral of the $1,\cdots,n$-component of the restriction of $f$ to $\sigma$.
The morphism $\int$ is a quasi-isomorphism of cochain complexes.
This is (Bousfield-Gugenheim, theorem 2.2, corollary 3.4).
The following is the central fact of the Sullivan approach to rational homotopy theory:
The functor $\Omega^\bullet_{poly}$ is the left adjoint of a Quillen adjunction
for the standard model structure on simplicial sets and the projective model structure on commutative dg-algebras.
This is shown in (Bousfield-Gugenheim, section 8).
So in particular $\Omega^\bullet_{poly}$ sends cofibrations of simplicial sets to fibrations of dg-algebras. Hence for $i : \partial \Delta[k] \hookrightarrow \Delta[k]$ a boundary inclusion the corresponding restriction
is degreewise surjective.
The functor $\Omega^\bullet_{poly}$ is a lax monoidal functor whose lax monoidal structure map
is a quasi-isomorphism.
This is reviewed for instance in (Hess, page 12).
Applications include
the sSet-enrichment of the model structure on dg-algebras over an operad.
higher Lie integration
An original reference is
A standard textbook is
This is based on
A useful survey is in