ScalingStacks

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.