Differential forms on $C^\infty$-ringed spaces

TL;DR

This paper develops a functorial complex of differential forms on local $C^$-ringed spaces, extending classical differential geometry concepts to more general spaces like Sikorski differential spaces and $C^$-schemes, enabling integration and Stokes' theorem.

Contribution

It introduces a functorial construction of differential forms on local $C^$-ringed spaces, generalizing manifold theory to broader geometric contexts.

Findings

Forms can be integrated over simplices

Stokes' theorem holds in this setting

Construction applies to differential spaces and $C^$-schemes

Abstract

We construct a complex of differential forms on a local $C^\infty$-ringed space. The two main classes of spaces we have in mind are differential spaces in the sense of Sikorski and $C^\infty$-schemes. Just as in the case of manifolds the construction is functorial. Consequently forms can be integrated over simplices and Stokes' theorem holds.