Differential forms on $C^{\mathrm{\infty }}$-ringed spaces

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