De Rham cohomology of diffeological spaces and foliations

Let (M,F)(M,F) be a foliated manifold. We prove that there is a canonical isomorphism between the complex of base-like forms Ωb∗(M,F) of the foliation and the “De Rham complex” of the space of leaves M/FM/F when considered as a “diffeological” quotient. Consequently, the corresponding cohomology groups Hb∗(M,F) and H∗(M/F)H∗(M/F) are isomorphic.