255 \end{equation*}

   255 \end{equation*}

   256 \end{itemize}

   256 \end{itemize}

   257 \end{property}

   257 \end{property}

   258 

   258 

   259 Finally, we state two more properties, which we will not prove in this paper.

   259 Finally, we state two more properties, which we will not prove in this paper.

   260 

   261 

   261 \begin{property}[Mapping spaces]

   262 \begin{property}[Mapping spaces]

   262 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps 

   263 Let $\pi^\infty_{\le n}(T)$ denote the $A_\infty$ $n$-category based on maps 

   263 $B^n \to T$.

   264 $B^n \to T$.

   264 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.)

   265 (The case $n=1$ is the usual $A_\infty$-category of paths in $T$.)