Smooth 𝐴_{∞}-form on a diffeological loop space

Abstract

To construct an A ∞ 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, which apparently restricts the number of iterations of concatenations. Secondly, we do not know a smooth decomposition of an associahedron as a simplicial or a cubical complex. To resolve these difficulties, we introduce a notion of a q 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 ∞ A_{\infty } -space. In appendix, we show an alternative solution by modifying the concatenation to be stable without assuming reflexivity on spaces nor stability on paths.