Subsection 8.1.1. (0PHX) · ScalingStacks

8.1.1. Definition

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 $M\subset Z\setminus\xi({\mathbf{R}}_{\geq 1})$ .

0PC5

Remark 8.1.1. Assume $\xi$ is not outgoing for $Z$ and let $z_{0}\in Z$ such that $\overline{\xi({\mathbf{R}}_{\geq 1})}\setminus\xi({\mathbf{R}}_{\geq 1})=\{z_{0}\}$ . Note that $\xi$ is outgoing for $Z\setminus\{z_{0}\}$ . The map $\xi$ is terminal for $(Z,M)$ if and only if $z_{0}{\not\in M}$ and the inclusion induces an isomorphism $\operatorname{Hom}\nolimits_{{\mathcal{A}}^{\bullet}(Z\setminus\{z_{0}\},1)}(m,z)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{{\mathcal{A}}^{\bullet}(Z,1)}(m,z)$ for all $m\in M$ and $z\in M\cup\{\xi(1)\}$ .

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

L^{\bullet}=L_{\xi}^{\bullet}:{\mathcal{S}}^{\bullet}_{M}(Z)\times{\mathcal{S}}^{\bullet}_{M}(Z)^{\operatorname{opp}\nolimits}\times{\mathcal{U}}^{\bullet}\to\operatorname{diff}\nolimits

L^{\bullet}(T,S,e^{n})=\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S,T\sqcup\{\xi(1),\ldots,\xi(n)\})

L^{\bullet}(\beta,\alpha,\sigma)(f)=(\beta\boxtimes\xi(\sigma))\cdot f\cdot\alpha\in L^{\bullet}(T^{\prime},S^{\prime},n)

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.

0PC6

Lemma 8.1.2. Given $S\subset M$ and $n\geq 0$ , there is an isomorphism of functors ${\mathcal{S}}^{\bullet}_{M}(Z)\to\operatorname{Sets}\nolimits^{\bullet}$ (forgetting the differential)

	$\displaystyle\bigvee_{\begin{subarray}{c}S^{\prime}\subset S\\ \|S^{\prime}\|=n\end{subarray}}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S^{\prime},\{\xi(1),\ldots,\xi(n)\})\wedge\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S\setminus S^{\prime},-)$	$\displaystyle\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{\bullet}(-,S,e^{n})$
	$\displaystyle(\alpha,\beta)$	$\displaystyle\mapsto\alpha\boxtimes\beta.$

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

	$\displaystyle L^{\bullet}(T,-,e^{n})\wedge L^{\bullet}(-,S,e^{m})$	$\displaystyle\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L^{\bullet}(T,S,e^{n+m})$
	$\displaystyle(\alpha,\beta)$	$\displaystyle\mapsto(\alpha\boxtimes\xi([r\to n+r]_{1\leq r\leq m}))\cdot\beta.$

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.

0PC7

Proposition 8.1.3. The bimodule $E$ and the endomorphism $\tau$ define a bimodule $2$ -representation on ${\mathcal{S}}^{\bullet}_{M}(Z)$ and on ${\mathcal{S}}_{M}(Z)$ .

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

0PC8

Remark 8.1.4. Proposition 8.1.3 generalizes and make more precise a result of Douglas and Manolescu [DouMa, §5.2].

Let $({\mathcal{Z}},{\mathbf{a}})$ be a chord diagram where ${\mathcal{Z}}=[0,1]$ . Let $\tilde{Z}^{\prime}=(0,\infty)$ , viewed as a curve with $\tilde{Z}^{\prime}_{o}=\tilde{Z}=(0,1)$ (with its usual orientation). We extend the equivalence relation from $\tilde{Z}$ to $\tilde{Z}^{\prime}$ by having all points of $[1,\infty)$ alone in their class. Let $Z^{\prime}=\tilde{Z}^{\prime}/\!\sim$ . We have $Z^{\prime}_{o}=Z_{o}$ . Let $M=Z^{\prime}_{exc}$ be the image of ${\mathbf{a}}$ in $Z^{\prime}$ . Let $\xi:{\mathbf{R}}_{>0}\to Z^{\prime},\ x\mapsto x+1$ . Note that $\xi$ is outgoing for $Z^{\prime}$ .

The lax $2$ -representation underlying the $2$ -representation on ${\mathcal{S}}_{M}(Z)={\mathcal{S}}_{M}(Z^{\prime})$ provided by Proposition 8.1.3 is the “bottom algebra module” constructed by Douglas and Manolescu, via the identification of §5.7.

0PC9

Example 8.1.5. The left picture below gives an example where $\xi$ is terminal for $(Z,M)$ but not outgoing for $Z$ . The right picture is an example where $\xi$ is outgoing for $Z$ .

The picture below considers the case of a curve quotient of the disjoint union of an interval and a circle, with an outgoing $\xi$ at an end of the interval. The middle picture describes an element of $L^{\bullet}_{\xi}(-,-,e^{2})$ . The rightmost picture provides a different graphical representation of that element: the interval $\xi({\mathbf{R}}_{\geq 1})$ has been moved to the bottom horizontal line.

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

0PCA

Remark 8.1.6. Assume $\xi$ is terminal for $(Z,M)$ . Consider $f:Z\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}Z$ an isomorphism of curves fixing $M$ . Note that $f\circ\xi$ is terminal for $(Z,M)$ and the map $f$ induces an isomorphism $L_{\xi}^{\bullet}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L_{f\circ\xi}^{\bullet}$ .

Consider now another injective morphism of curves $\xi^{\prime}:{\mathbf{R}}_{>0}\to Z$ such that $\xi^{\prime}$ is terminal for $(Z,M)$ . Assume there is a connected open subset $U$ of $Z_{u}$ containing $\overline{\xi({\mathbf{R}}_{>0})}$ and $\overline{\xi^{\prime}({\mathbf{R}}_{>0})}$ and assume the canonical orientations on $\xi({\mathbf{R}}_{>0})$ and $\xi^{\prime}({\mathbf{R}}_{>0})$ extend to an orientation of $U$ . There is an isomorphism of curves $f:Z\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}Z$ fixing $Z\setminus U$ such that $\xi^{\prime}=f\circ\xi$ . It induces an isomorphism $L_{\xi}^{\bullet}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L_{\xi^{\prime}}^{\bullet}$ , and that isomorphism does not depend on the choice of $f$ .

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2