Paths and cylinders
There are various simplicial dg-algebras that assign to the standard -simplex a kind of de Rham algebra on .
By the discussion at differential forms on presheaves, each such extends to a notion of differential forms on simplicial sets.
Smooth differential forms
For the smooth n-simplex is the smooth manifold with boundary and corners defined, up to isomorphism, as the following locus inside the Cartesian space :
For the function
which picks the th component in the above definition is called the th barycentric coordinate function.
a morphism of finite non-empty linear orders , let
be the smooth function defined by .
(smooth differential forms on the smooth -simplex)
For then a smooth differential k-form on the smooth -simplex (def. 1) is a smooth differential form in the sense of smooth manifolds with boundary and corners. Explicitly this means the following.
be the affine plane in that contains in its defining inclusion from def. 1. This is a smooth manifold diffeomorphic to the Cartesian space .
A smooth differential form on of degree k$ is a collection of linear functions
out of the -fold skew-symmetric tensor power of the tangent space of at some point to the real numbers, for all such that this extends to a smooth differential -form on .
Write for the graded real vector space defined this way. By definition there is then a canonical linear map
from the de Rham complex of and there is a unique structure of a differential graded-commutative algebra on that makes is a homomorphism of dg-algebras form the de Rham algebra of . This is the de Rham algebra of smooth differential forms on the smooth -simplex.
For a homomorphism of finite non-empty linear orders with 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 . 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 may be uniquely written
where is as above the barycentric coordinate function and each is a -function on .
With this representation the multiplication and differential are given by the usual formulae. The multiplication is defined by and extends linearly the product
on the generating forms. Now if is a differentiable function
Polynomial differential forms
for the quotient of the -graded symmetric algebra over the rational numbers on generators in degree 0 and generators 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 -simplex according to def. 2, obtained by regarding the generator as the th barycentric coordinate function.
Observe that the tensor product of the polynomial differential forms over these polynomial functions with the smooth functions on the -simplex, is canonically isomorphic to the space of smooth differential forms, according to def. 2:
where moreover the generators are identified with the de Rham differential of the th barycentric coordinate functions.
This defines a canonical inclusion
and there is uniquely the structure of a differential graded-commutative algebra on that makes this a homomorphism of dg-algebras. This is the dg-algebra of polynomial differential forms.
For 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
Piecewise polynomial differential forms
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 be a field of characteristic 0. Let be the left Kan extension of from above.
For , define a morphism of graded -vector spaces
from polynomial differential forms on simplices to cochains on simplicial sets by sending to the cochain that sends to
where on the right we have the ordinary integral of the -component of the restriction of to .
This is (Bousfield-Gugenheim, theorem 2.2, corollary 3.4).
The following is the central fact of the Sullivan approach to rational homotopy theory:
This is shown in (Bousfield-Gugenheim, section 8).
So in particular sends cofibrations of simplicial sets to fibrations of dg-algebras. Hence for a boundary inclusion the corresponding restriction
is degreewise surjective.
The functor is a lax monoidal functor whose lax monoidal structure map
is a quasi-isomorphism.
This is reviewed for instance in (Hess, page 12).
An original reference is
- Aldridge Bousfield and V. K. A. M. Gugenheim, §1 and §2_On PL De Rham Theory and Rational Homotopy Type_ , Memoirs of the A. M. S., vol. 179, 1976.
A standard textbook is
- Stephen Halperin, Lecture Notes on Minimal Models, Publications de l’U.E.R. Mathématiques Pures et Appliquées, Université des Sciences et techniques, Lille, Vol 3 (1981) Fasc.3.
This is based on
- Dennis Sullivan, Infinitesimal computations in topology, Publications Mathématiques de l’IHÉS, 47 (1977), p. 269-331 (numdam)
A useful survey is in