Smooth $A_{\infty}$ form on a diffeological loop space

Abstract: To construct an $A_{\infty}$-form for a loop space in the category of diffeological spaces, we have two minor problems. Firstly, the concatenation of paths in the category of diffeological spaces needs a small technical trick (see P.~I-Zemmour \cite{MR3025051}), which apparently restricts the number of iterations of concatenations. Secondly, we do not know a natural smooth decomposition of an associahedron as a simplicial or a cubical complex. To resolve these difficulties, we introduce a notion of a $q$-cubic set which enjoys good properties on dimensions and representabilities, and show, using it, that the smooth loop space of a reflexive diffeological space is a h-unital smooth $A_{\infty}$-space. In appendix, we show an alternative solution by modifying the concatenation to be stable without assuming reflexivity for spaces nor stability for paths.