We show a de Rham theorem for cubical manifolds, and study rational homotopy type of the classifying spaces of smooth quandles. We also show that secondary characteristic classes in [8, 9] produce cocycles of quandles.
"de Rham theory and cocycles of cubical sets from smooth quandles." Kodai Math. J. 42 (1) 111 - 129, March 2019. https://doi.org/10.2996/kmj/1552982509