Schreiber C-infinity differential forms

For $X \in Spaces = Sheaves(CartesianSpaces)$ the set (or smooth space) of smooth differential forms on $X$

$$\Omega^\bullet(X) := Hom_{Spaces}(X,\Omega^\bullet(-)) \,.$$

This should extend to a C-infinity qDGCA by setting

$$\Omega^\bullet(X) = \wedge^\bullet_{C^\infty(X)} ( \Omega^1(X) \oplus \Omega^2_{indec}(X) \oplus \Omega^3_{indec}(X) \oplus \cdots ) \,,$$

where $\Omega^\k_{indec}(X)$ is the C-infinity module over the C-infinity algebra $C^\infty(X)$ of indecomposable $k$-forms on $X$: those that cannot be written as sums of wedge products of lower degree forms.

Remarks

• recall that on generalized smooth spaces there may be higher forms which are indecomposable in that they are not sums of wedge products of lower degree forms. For instance on the generalized space $S(b \mathfrak{u}(1)) : U \mapsto \Omega^2(U)$ we have no 1-form, but a (hence indecomposable) 2-form.

• Notice that

  $$\begin{aligned} \Omega^0(X) &= Hom_{Spaces}(X,\Omega^0(-)) \\ & = Hom_{Spaces}(X, Hom_{CartesianSpaces}(-,\mathbb{R})) \\ &= Hom_{Spaces}(X,\mathbb{R}) \\ &= C^\infty(X)(\mathbb{R}) \end{aligned} \,.$$

Open issues

• think about the above definition of $\Omega^\bullet(X)$ as a C-infinity qDGCA.