Schreiber C-infinity module

Recalling that a C-infty algebra $A$ is a copresheaf $A \in Quantities = CoPrSh(CartesianSpaces)$ which becomes a copresheaf with values in algebras when restricted along $FinSet \hookrightarrow CartesianSpaces$, the idea is to define a smooth module over a C-infinity algebra $A$ to be a copresheaf $N \in Quantities$ which becomes a copresheaf of modules over the copresheaf of algebras $A$ when restricted along $FinSet \to CartesianSpaces$.

Definition

A $C^\infty$-module over a C-infinity algebra $A$ is a co-presheaf $N \in Quantities = CoPr(CratesianSpaces)$ such that restricted along $FinSet \hookrightarrow CartesianSpaces$ it becomes a copresheaf of modules over the copresheaf (on $FinSet$) of algebras $A$.

Examples

• Differential forms on a generalized smooth space.

References

• Cartesian Spaces

(…expand on this….)

Open issues

We need to think about

• the details of the definition of tensor product of $C^\infty$-modules over $C^\infty$-algebras;

• the details of the definition of dual modules. I.e. with $N$ a $$C^\infty$-module over the $C^\infty$-algebras $A$, what can we say about

  $$N^* := Hom_{C^\infty Mod}(N,A) \,.$$

  For instance: what notion of hom do we use, how do we equip the extra structure?