ScalingStacks

It follows that there are isomorphisms functorial in $S$ and $T$

(8.1.1)

$$\operatorname{colim}\nolimits_{r\to\infty}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}_{M}(Z)}(S,T\sqcup\xi(\{m_{r},\ldots,m_{r+n-1}\}))\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{\bullet}(T,S,e^{n}).$$

Here, the colimit is taken over the invertible maps $\xi(\theta_{r})$, where $\theta_{r}:\{m_{r},\ldots,m_{r+n-1}\}\to\{m_{r+1},\ldots,m_{r+n}\}$ is the braid in ${\mathbf{R}}_{>0}$ given by $(\theta_{r})_{m_{s}}=[m_{s}\to m_{s+1}]$.