Schreiber C-infinity differential forms

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

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

This should extend to a C-infinity qDGCA by setting

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

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


  • 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(b𝔲(1)):UΩ 2(U)S(b \mathfrak{u}(1)) : U \mapsto \Omega^2(U) we have no 1-form, but a (hence indecomposable) 2-form.

  • Notice that

    Ω 0(X) =Hom Spaces(X,Ω 0()) =Hom Spaces(X,Hom CartesianSpaces(,)) =Hom Spaces(X,) =C (X)(). \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 Ω (X)\Omega^\bullet(X) as a C-infinity qDGCA.

Last revised on December 10, 2008 at 13:11:35. See the history of this page for a list of all contributions to it.