  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.
