ScalingStacks

Let $\xi:{\mathbf{R}}_{>0}\to Z$ be an injective morphism of curves, where ${\mathbf{R}}_{>0}$ is viewed as an unoriented curve. Let $M$ be a subset of $Z\setminus\xi({\mathbf{R}}_{\geq 1})$.

We say that $\xi$ is terminal for $(Z,M)$ if the following two conditions hold:

• •

  given an admissible homotopy class of paths $\zeta$ in $Z$ with endpoints in $M$, there is an admissible path $\gamma$ in $\zeta$ contained in $Z\setminus\xi({\mathbf{R}}_{\geq 1})$

• •

  there is no admissible path in $Z\setminus\{\xi(1)\}$ from a point of $M$ to $\xi(2)$.

Note that $\xi$ is terminal for $(Z,M)$ if and only if $\xi$ is terminal for $(Z(\xi),Z(\xi)\cap M)$, where $Z(\xi)$ is the component of $Z$ containing $\xi({\mathbf{R}}_{>0})$.

We say that $\xi$ is outgoing for $Z$ if $\xi({\mathbf{R}}_{\geq 1})$ is closed in $Z$. Note that if $\xi$ is outgoing for $Z$ then it is terminal for $(Z,M)$ for any $M\subset Z\setminus\xi({\mathbf{R}}_{\geq 1})$.

We assume now that $\xi$ is terminal for $(Z,M)$. Thanks to Lemma 7.4.36, we have a differential pointed functor

for $\alpha\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S^{\prime},S)$, $\beta\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(T,T^{\prime})$, $\sigma\in\operatorname{End}\nolimits_{{\mathcal{U}}^{\bullet}}(e^{n})$, and $f\in L^{\bullet}(T,S,n)$. We have used the strands realization of ${\mathcal{U}}^{\bullet}$ given by Theorem 7.4.34.

We put $L^{\bullet}(T,S)=L^{\bullet}(T,S,e)$. As usual, we put $L_{\xi}={\mathbf{F}}_{2}[L_{\xi}^{\bullet}]$.

The naturality in the next lemma is immediate as in Lemma 7.4.36.

Lemma 8.1.2 shows that there is an isomorphism of functors, functorial in $S$ and $T$

The functor $E=E_{\xi}=L^{\bullet}(-,-)$ gives a bimodule $2$-representation on ${\mathcal{S}}^{\bullet}_{M}(Z)$. The endomorphism $\tau$ of $L^{\bullet}(-,-,e^{2})$ is given by the non-identity non-zero braid $\{1,2\}\to\{1,2\}$.

We have obtained the following proposition.

Lemma 8.1.2 shows that $L_{\xi}(-,-)$ is left finite.

The next remark discusses the dependence of $L_{\xi}^{\bullet}$ on $\xi$.

Assume $\xi^{-1}(M)$ has no maximum. Fix an increasing sequence $m_{0},m_{1},\ldots$ of points of $(0,1)$ with $\xi(m_{i})\in M$ for all $i$ and with $\lim_{i}m_{i}>t$ for all $t\in\xi^{-1}(M)$.

Fix $n\geq 0$ and define the braid $\beta_{r}:\{m_{r},\ldots,m_{r+n-1}\}\to\{1,\ldots,n\}$ of ${\mathbf{R}}_{>0}$ by $(\beta_{r})_{m_{r+i}}=[m_{r+i}\to i+1]$.

Let $S$ and $T$ be two finite subsets of $M$. Consider $r$ such that $m_{r}>\xi^{-1}(t)$ for all $t\in T\cap\xi({\mathbf{R}}_{>0})$. There is an isomorphism

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

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 $(\theta_{r})_{m_{s}}=[m_{s}\to m_{s+1}]$.

We deduce that $T\mapsto(S\mapsto L^{\bullet}(T,S,e^{n}))$ is isomorphic to the functor

Let $f:Z\to Z^{\prime}$ be a morphism of curves. Assume $\xi$ is terminal for $(Z,M)$ and $f\circ\xi$ is terminal for $(Z^{\prime},f(M))$.

Assume that $|f^{-1}(f(z))|=1$ for all $z\in M$. Let $M_{f}$ be the $({\mathcal{S}}^{\bullet}_{M}(Z),{\mathcal{S}}^{\bullet}_{f(M)}(Z^{\prime}))$-bimodule corresponding to $f$, i.e. given by $M_{f}(S,S^{\prime})=\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z^{\prime})}(S^{\prime},f(S))$. There is a morphism of functors $E_{\xi}\wedge_{{\mathcal{S}}^{\bullet}_{M}(Z)}M_{f}\to M_{f}\wedge_{{\mathcal{S}}^{\bullet}_{f(M)}(Z^{\prime})}E_{f\circ\xi}$ defined as making the following diagram commutative

The following lemma is a consequence of (8.1.1).

We consider now an arbitrary $M$ but we assume that $f$ is strict. Let $M_{f^{\#}}$ be the $({\mathcal{S}}_{f(M)}(Z^{\prime}),{\mathcal{S}}_{M}(Z))$-bimodule corresponding to $f^{\#}$, i.e. given by

There is a morphism of functors $E_{f\circ\xi}\otimes_{{\mathcal{S}}_{f(M)}(Z^{\prime})}M_{f^{\#}}\to M_{f^{\#}}\otimes_{{\mathcal{S}}_{M}(Z)}E_{\xi}$ defined as making the following diagram commutative

The following lemma is a consequence of (8.1.1).

We explain how to obtain a version of Lemma 8.1.2 with a differential.

We say that a homotopy class of path in $Z(\xi)$ is positive if it has the same orientation as $\xi([1\to 2])$.

Fix a finite subset $S$ of $M$ and $n\geq 0$.

Let $S^{\prime\prime}$ be a subset of $S$ with $n$ elements, let $s^{\prime}\in S\setminus S^{\prime\prime}$ and $s^{\prime\prime}\in S^{\prime\prime}$. Let $\zeta:s^{\prime\prime}\to s^{\prime}$ be a positive smooth homotopy class of paths in $Z$. We put $S^{\prime}=(S^{\prime\prime}\setminus\{s^{\prime\prime}\})\sqcup\{s^{\prime}\}$.

We define a map

We put

if

• •

  $\alpha_{s^{\prime\prime}}\circ\zeta^{-1}$ is smooth

• •

  and given $s\in S^{\prime\prime}\setminus\{s^{\prime\prime}\}$ and $\zeta^{\prime}:s\to s^{\prime}$ and $\zeta^{\prime\prime}:s^{\prime\prime}\to s$ smooth positive with $\zeta=\zeta^{\prime}\circ\zeta^{\prime\prime}$ and with $\alpha_{s^{\prime\prime}}\circ\zeta^{\prime\prime-1}$ smooth, then $\alpha_{s^{\prime\prime}}\circ\zeta^{\prime\prime-1}\circ\alpha_{s}^{-1}$ is negative.

We put $g_{S^{\prime\prime},\zeta}(\alpha)=0$ otherwise.

We obtain a map $f_{S^{\prime\prime},\zeta}:\alpha\wedge\beta\mapsto(\operatorname{id}\nolimits\wedge\beta)\circ g_{S^{\prime\prime},\zeta}(\alpha)$

Let $r(S^{\prime\prime})$ be the number of pairs $(s^{\prime\prime},s)\in S^{\prime\prime}\times(S\setminus S^{\prime\prime})$ such that there exists a positive path $s^{\prime\prime}\to s$.

We define now

Given $r^{\prime}<r^{\prime\prime}$, define $f_{r^{\prime},r^{\prime\prime}}=\sum_{S^{\prime\prime},\zeta}f_{S^{\prime\prime},\zeta}$, where

• •

  $S^{\prime\prime}$ is a subset of $S$ with $|S^{\prime\prime}|=n$ and $r(S^{\prime\prime})=r^{\prime\prime}$

• •

  $\zeta$ is a positive admissible homotopy class of paths in $Z$ with $\zeta(0)\in S^{\prime\prime}$ and $\zeta(1)\in S\setminus S^{\prime\prime}$

such that $\operatorname{supp}\nolimits(\zeta)\cap S^{\prime\prime}=\{s^{\prime\prime}\}$ and $r((S^{\prime\prime}\setminus\{\zeta(0)\})\sqcup\{\zeta(1)\})=r^{\prime}$.

Let $V=V_{n}(S)=\bigoplus_{r}V_{r}$ and let $d_{V}=\sum_{r}d_{V_{r}}+\sum_{r^{\prime},r^{\prime\prime}}f_{r^{\prime},r^{\prime\prime}}$. We will show below (Proposition 8.1.10) that $d_{V}^{2}=0$, i.e. $V$ is the object of ${\mathcal{S}}_{M}(Z)\operatorname{\!-diff}\nolimits$ corresponding to the twisted object $[\bigoplus V_{r},(f_{r^{\prime},r^{\prime\prime}})]$.

Consider now $\xi^{\prime}:{\mathbf{R}}_{<0}\to Z$ an injective morphism of curves, where ${\mathbf{R}}_{<0}$ is unoriented. Identifying $({\mathbf{R}}_{<0})^{\operatorname{opp}\nolimits}$ with ${\mathbf{R}}_{>0}$ by $x\mapsto-x$, we obtain a morphism of curves $\xi:{\mathbf{R}}_{>0}\to Z^{\operatorname{opp}\nolimits}$. Let $M$ be a subset of $Z\setminus\xi^{\prime}({\mathbf{R}}_{\leq-1})$.

We say that $\xi^{\prime}$ is initial for $(Z,M)$ if $\xi$ is terminal for $(Z^{\operatorname{opp}\nolimits},M)$ and that $\xi^{\prime}$ is incoming for $Z$ if $\xi^{\prime}({\mathbf{R}}_{\leq-1})$ is closed in $Z$.

Assume $\xi^{\prime}$ is initial for $(Z,M)$. As in the left action case, we define a differential functor

for $\alpha\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(T^{\prime},T)$, $\beta\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S,S^{\prime})$ and $\sigma\in\operatorname{End}\nolimits_{{\mathcal{U}}^{\bullet}}(e^{n})$, and $f\in R^{\bullet}(S,T,n)$.

We put $R_{\xi}^{\bullet}(S,T)=R_{\xi}^{\bullet}(S,T,e)$ and $R_{\xi}={\mathbf{F}}_{2}[R_{\xi}^{\bullet}]$.

Recall that the isomorphism (7.4.4) of differential categories ${\mathcal{S}}^{\bullet}_{M}(Z)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{S}}^{\bullet}_{M}(Z^{\operatorname{opp}\nolimits})^{\operatorname{opp}\nolimits}$. This isomorphism provides an isomorphism $R^{\bullet}_{\xi^{\prime}}(S,T,e^{n})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{\bullet}_{\xi}(T,S,e^{n})$ functorial in $S$, $T$ and $e^{n}$.

In particular, $R^{\bullet}$ provides a “right” $2$-representation on ${\mathcal{S}}^{\bullet}_{M}(Z)$ and all results of §8.1.1–8.1.4 have counterparts for $R^{\bullet}$.

Given $S\subset M$ and $n\geq 0$, there is an isomorphism of functors

There is an isomorphism of functors, functorial in $S$ and $T$

Assume there is a decreasing sequence $m_{0},m_{-1},\ldots$ of points of $\xi^{\prime-1}(M)$ with $\lim_{i}m_{i}<t$ for all $t\in\xi^{\prime-1}(M)$.

We obtain as in (8.1.1) isomorphisms functorial in $S$ and $T$