ScalingStacks

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 $T$

Let us finally consider functoriality as in §8.1.3. Let $f:Z\to Z^{\prime}$ be a morphism of curves and assume $f\circ\xi^{\prime}$ is initial for $(Z^{\prime},f(M))$.

The functor $f:{\mathcal{S}}_{f,M}^{\bullet}(Z)\to{\mathcal{S}}^{\bullet}_{f(M)}(Z^{\prime})$ induces a morphism of bimodule $2$-representations $R_{f\circ\xi^{\prime}}^{\bullet}\to R_{\xi^{\prime}}^{\bullet}$, when $|f^{-1}(f(z))|=1$ for all $z\in M$.

If $f$ is strict, then the functor $f^{\#}:\operatorname{add}\nolimits({\mathcal{S}}_{f(M)}(Z^{\prime}))\to\operatorname{add}\nolimits({\mathcal{S}}_{M}(Z))$ induces a morphism of bimodule $2$-representations $R_{\xi^{\prime}}\to R_{f\circ\xi^{\prime}}$.