ScalingStacks

Consider two injective morphisms of curves $\xi_{1}^{+}:{\mathbf{R}}_{>0}\to Z$ and $\xi_{2}^{-}:{\mathbf{R}}_{<0}\to Z$ where ${\mathbf{R}}_{<0}$ and ${\mathbf{R}}_{>0}$ are unoriented. We assume that $\xi_{1}^{+}$ is outgoing for $Z$, that $\xi_{2}^{-}$ is incoming for $Z$ and that $\xi_{1}^{+}({\mathbf{R}}_{>0})\cap\xi_{2}^{-}({\mathbf{R}}_{<0})=\emptyset$. We write $r$ instead of $\xi_{1}^{+}(r)$ and $-r$ instead of $\xi_{2}^{-}(-r)$, for $r\in{\mathbf{Z}}_{>0}$.

Let $M$ be a subset of $Z\setminus(\xi_{1}^{+}({\mathbf{R}}_{\geq 1})\sqcup\xi_{2}^{-}({\mathbf{R}}_{\leq-1}))$.

Fix an oriented diffeomorphism ${\mathbf{R}}_{>0}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathbf{R}}_{<-1}$ and let $i_{+}:{\mathbf{R}}_{>0}\to{\mathbf{R}}$ be its composition with the inclusion map. Similarly, fix an oriented diffeomorphism ${\mathbf{R}}_{<0}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathbf{R}}_{>1}$ and let $i_{-}:{\mathbf{R}}_{<0}\to{\mathbf{R}}$ be its composition with the inclusion map.

Consider $m,n\geq 0$. Let $E_{m,n}$ be the $({\mathcal{S}}_{M}^{\bullet}(Z),{\mathcal{S}}_{M}^{\bullet}(Z))$-bimodule given by

Note that $E_{0,1}=R^{\bullet}_{\xi_{2}^{-}}$ and $E_{1,0}=L^{\bullet}_{\xi_{1}^{+}}$, but $E_{m,n}$ is not isomorphic to $(R^{\bullet}_{\xi_{2}^{-}})^{n}(L^{\bullet}_{\xi_{1}^{+}})^{m}$ in general.

There is an action of $H_{m}^{\bullet}\wedge H_{n}^{\bullet}$ on $E_{m,n}$ given by

for $\sigma\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\sqcup(-n,-1),T\sqcup(1,m))$, $a\in{\mathfrak{S}}_{m}$ and $b\in{\mathfrak{S}}_{n}$.

There is a map $\ast:E_{m,n}E_{m^{\prime},n^{\prime}}\to E_{m+m^{\prime},n+n^{\prime}}$ given by

This map is compatible with the action of $(H_{m}^{\bullet}\wedge H_{n}^{\bullet})\wedge(H_{m^{\prime}}^{\bullet}\wedge H_{n^{\prime}}^{\bullet})$ via the canonical embeddings $H_{m}^{\bullet}H_{m^{\prime}}^{\bullet}\to H_{m+m^{\prime}}^{\bullet}$ and $H_{n}^{\bullet}H_{n^{\prime}}^{\bullet}\to H_{n+n^{\prime}}^{\bullet}$. We have $(\alpha\ast\beta)\ast\gamma=\alpha\ast(\beta\ast\gamma)$.

So, we have defined a bimodule lax bi-$2$-representation on ${\mathcal{S}}_{M}^{\bullet}(Z)$.

Let $Z_{\xi}=Z\sqcup_{{\mathbf{R}}_{>0}\sqcup{\mathbf{R}}_{<0}}{\mathbf{R}}$, where the gluing is done along the maps $\xi_{1}^{+}\sqcup\xi_{2}^{-}:{\mathbf{R}}_{>0}\sqcup{\mathbf{R}}_{<0}\to Z$ and $i_{+}\sqcup i_{-}:{\mathbf{R}}_{>0}\sqcup{\mathbf{R}}_{<0}\to{\mathbf{R}}$. Note that $Z_{\xi}$ is a $1$-dimensional space and it comes with an injective open morphism of $1$-dimensional spaces $\xi:{\mathbf{R}}\to Z_{\xi}$. We endow ${\mathbf{R}}$ with a curve structure by setting ${\mathbf{R}}_{u}={\mathbf{R}}_{\leq-1}\sqcup{\mathbf{R}}_{\geq 1}$ and by endowing $(-1,1)$ with its usual orientation. We extend the curve structure on $Z$ by endowing $\xi({\mathbf{R}})$ with the curve structure of ${\mathbf{R}}$. Note that $(Z_{\xi})_{u}=\overline{Z_{u}}$.

Given $\varepsilon,\varepsilon^{\prime}\in\{+,-\}$ and $a\in{\mathbf{R}}_{\varepsilon}$, $b\in{\mathbf{R}}_{\varepsilon^{\prime}}$, we put $[a\to b]=\xi([i_{\varepsilon}(a),i_{\varepsilon^{\prime}}(b)])$.

We consider the differential pointed category $T_{{\mathcal{S}}^{\bullet}_{M}(Z)}(R_{\xi_{2}^{-}}^{\bullet}L_{\xi_{1}^{+}}^{\bullet})$ with objects those of ${\mathcal{S}}^{\bullet}_{M}(Z)$ and with

We define a differential pointed functor $\tilde{\Xi}:T_{{\mathcal{S}}^{\bullet}_{M}(Z)}(R_{\xi_{2}^{-}}^{\bullet}L_{\xi_{1}^{+}}^{\bullet})\to{\mathcal{S}}^{\bullet}_{M}(Z_{\xi})$. It is the identity on objects and defined on maps by

The sections §8.2.2-8.2.4 below are devoted to the proof of Theorem 8.2.1.