A comparison between two de Rham complexes in diffeology

There are two de Rham complexes in diffeology. The original one is due to Souriau and the other one is the singular de Rham complex defined by a simplicial differential graded algebra. We compare the first de Rham cohomology groups of the two complexes within the \v{C}ech–de Rham spectral sequence by making use of the {\it factor map} which connects the two de Rham complexes. As a consequence, it follows that the singular de Rham cohomology algebra of the irrational torus $T_{\theta }$ is isomorphic to the tensor product of the original de Rham cohomology and the exterior algebra generated by a non-trivial flow bundle over $T_{\theta }$.