nLab differential forms on presheaves

Idea

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

$$\Omega^\bullet_C : C^{op} \to dgAlg$$

be any functor to the category of dg-algebras. Following the logic of space and quantity, we may think of the objects of $C$ as being test spaces and the functor $\Omega^\bullet_C$ 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 $C$ is some subcategory of the category Diff of smooth manifolds, and $\Omega^\bullet_C$ 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 $C$ and $\Omega^\bullet_C$ 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 $\Omega^\bullet_C$ becomes itself a presheaf on $C$. For $X \in PSh(C)$ any other presheaf, we extend the notation and write

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

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 $p : U \to X$ with $U \in C$ the operations in $\Omega^\bullet_C(U)$. This way we get a functor

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

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

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

$$K_C(A) : U \mapsto Hom_{dgAlg}(\Omega^\bullet_C(U), A) \,.$$

The adjunction

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

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.