ScalingStacks

We consider the unoriented curve ${\mathbf{R}}$. Let $M=\{\pm(1-\frac{1}{n})\}_{n\in{\mathbf{Z}}_{>0}}$.

Consider $S,T$ two finite subsets of ${\mathbf{R}}$ with $|S|=|T|=n$. Let $f_{S}:S\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\{1,\ldots,n\}$ and $f_{T}:T\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\{1,\ldots,n\}$ be the unique increasing bijections. We define

We define a functor $\Phi:{\mathcal{S}}^{\bullet}_{M}({\mathbf{R}})\to{\mathcal{U}}^{\bullet}$. We put $\Phi(S)=e^{|S|}$ and $\Phi(f)=\phi(S,T)(f)$ for $f\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}({\mathbf{R}})}(S,T)$.

The next proposition follows from Proposition 7.4.33.

Consider $\xi_{+}:{\mathbf{R}}_{>0}\to{\mathbf{R}}$ and $\xi_{-}:{\mathbf{R}}_{<0}\to{\mathbf{R}}$ the inclusion maps.

We define $\varphi_{\pm}:L_{\xi_{\pm}}(-,-,e^{n})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{\pm}(-,-,e^{n})\circ(\Phi\wedge\Phi)$ by

Similarly, we define $\varphi^{\prime}_{\pm}:R_{\xi_{\pm}}(-,-,e^{n})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}R^{\pm}(-,-,e^{n})\circ(\Phi\wedge\Phi)$ by