论文标题
平滑$ a _ {\ infty} $形式在差异循环空间上
Smooth $A_{\infty}$ form on a diffeological loop space
论文作者
论文摘要
要构建一个$ a _ {\ infty} $ - 为差异空间类别的环空间构建形式，我们有两个小问题。首先，差异空间类别中的路径的串联需要一个小的技术技巧（请参阅P.〜i-Zemmour \ cite {MR3025051}），这显然限制了连接的迭代次数。其次，我们不知道合作伙伴的天然平滑分解是简单的或立方体的复合物。为了解决这些困难，我们引入了$ Q $ - 皮基套件的概念，该套件在维度和代表性上享有良好的特性，并显示出来，表明反射性差异空间的平滑环空间是H-功能平滑的$ A _ _ {\ infty} $ - 空间。在附录中，我们通过修改串联的稳定性来显示替代解决方案，而不假设空间反射性或对路径的稳定性。
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.
