nLab
differential forms on presheaves

Contents

Idea

Let CC be any small category, write PSh(C)=[C op,Set]PSh(C) = [C^{op}, Set] for its category of presheaves and let

Ω C :C opdgAlg \Omega^\bullet_C : C^{op} \to dgAlg

be any functor to the category of dg-algebras. Following the logic of space and quantity, we may think of the objects of CC as being test spaces and the functor Ω C \Omega^\bullet_C as assigning to each test space its deRham dg-algebra.

An example of this construction that is natural from the point of view of differential geometry appears in the study of diffeological spaces, where CC is some subcategory of the category Diff of smooth manifolds, and Ω C \Omega^\bullet_C is the restriction of the ordinary assignment of differential forms to this. But in the application to topological spaces, in the following, we need a choice for CC and Ω C \Omega^\bullet_C that is non-standard from the point of view of differential geometry. Still, it follows the same general pattern.

After postcomposing with the forgetful functor that sends each dg-algebra to its underlying set, the functor Ω C \Omega^\bullet_C becomes itself a presheaf on CC. For XPSh(C)X \in PSh(C) any other presheaf, we extend the notation and write

Ω C (X):=Hom PSh(C)(X,Ω C ) \Omega^\bullet_C(X) := Hom_{PSh(C)}(X, \Omega^\bullet_C)

for the hom-set of presheaves. One checks that this set naturally inherits the structure of a dg-algebra itself, where all operations are given by applying “pointwise” for each p:UXp : U \to X with UCU \in C the operations in Ω C (U)\Omega^\bullet_C(U). This way we get a functor

Ω C :PSh(C)dgAlg op \Omega^\bullet_C : PSh(C) \to dgAlg^{op}

to the opposite category of that of dg-algebras. We may think of Ω C (X)\Omega^\bullet_C(X) as the deRham complex of the presheaf XX as seen by the functor Ω C :CdgAlg op\Omega^\bullet_C : C \to dgAlg^{op}.

By general abstract nonsense this functor has a right adjoint K C:dgAlg opPSh(C)K_C : dgAlg^{op} \to PSh(C), that sends a dg-algebra AA to the presheaf

K C(A):UHom dgAlg(Ω C (U),A). K_C(A) : U \mapsto Hom_{dgAlg}(\Omega^\bullet_C(U), A) \,.

The adjunction

Ω C :PSh(C):dgAlg op:K C \Omega^\bullet_C : PSh(C) \stackrel{\leftarrow}{\to} : dgAlg^{op} : K_C

is an example for the adjunction induced from a dualizing object.

Examples

Differential forms on simplicial sets

There are various variant of differential forms on simplices. Each gives rise to a notion of differential forms on simplicial sets. This is also known as the Sullivan construction in rational homotopy theory.

Revised on December 9, 2010 18:59:37 by Urs Schreiber (131.211.232.192)