ScalingStacks

Let $Z$ be a curve. Let $I$ and $J$ be two finite subsets of $Z$.

The objects of this pointed category are the finite subsets of $Z$ and $\operatorname{Hom}\nolimits_{{\mathcal{P}}^{\bullet}(Z)}(I,J)$ is the set of braids $I\to J$, together with a $0$-element. Given $\theta:I\to J$ and $\theta^{\prime}:J\to K$ two braids, we have $\theta^{\prime}\circ\theta=(\theta^{\prime}_{\theta_{s}(1)}\circ\theta_{s})_{s\in I}$ if $\theta^{\prime}_{\theta_{s}(1)}\circ\theta_{s}$ is admissible for all $s\in I$, and we have $\theta^{\prime}\circ\theta=0$ otherwise. If $\theta^{\prime}\circ\theta\neq 0$, we have $\chi(\theta^{\prime}\circ\theta)=\chi(\theta^{\prime})\circ\chi(\theta)$.

We put ${\mathcal{P}}(Z)={\mathbf{F}}_{2}[{\mathcal{P}}^{\bullet}(Z)]$.

Note that there is a decomposition ${\mathcal{P}}^{\bullet}(Z)=\bigvee_{n\geq 0}{\mathcal{P}}^{\bullet}(Z,n)$, where ${\mathcal{P}}^{\bullet}(Z,n)$ is the full subcategory of ${\mathcal{P}}^{\bullet}(Z)$ with objects subsets with $n$ elements. We have ${\mathcal{P}}^{\bullet}(Z,1)={\mathcal{S}}^{\bullet}(Z,1)$.

Given $M$ a subset of $Z$, we denote by ${\mathcal{P}}^{\bullet}_{M}(Z)$ the full subcategory of ${\mathcal{P}}^{\bullet}(Z)$ with objects the finite subsets of $M$.

Given $\theta:I\to J$ a braid and $I^{\prime}$ a subset of $I$, we denote by $\theta_{|I^{\prime}}$ the braid $(\theta_{s})_{s\in I^{\prime}}$.

Let $f:Z\to Z^{\prime}$ be a morphism of curves. We denote by ${\mathcal{P}}^{\bullet}_{f}(Z)$ the full subcategory of ${\mathcal{P}}^{\bullet}(Z)$ with objects those finite subsets $I$ of $Z$ such that $|f(I)|=|I|$.

The next proposition follows immediately from Lemma 7.3.14 and §2.4.

We define a non-multiplicative $f^{\#}:\operatorname{add}\nolimits({\mathcal{P}}(Z^{\prime}))\to\operatorname{add}\nolimits({\mathcal{P}}(Z))$ that commutes with coproduct. Given $I^{\prime}$ a finite subset of $Z^{\prime}$, we put

Consider now $\theta^{\prime}\in\operatorname{Hom}\nolimits_{{\mathcal{P}}^{\bullet}(Z^{\prime})}(I^{\prime},J^{\prime})$ non-zero. Given $s^{\prime}\in I^{\prime}$, we have a decomposition $f^{\#}(\theta^{\prime}_{s^{\prime}})=\sum_{s\in f^{-1}(s^{\prime})}f^{\#}(\theta^{\prime}_{s^{\prime}})_{s}$ along the decomposition $f^{\#}(s^{\prime})=\bigoplus_{s\in f^{-1}(s^{\prime})}s$ (cf §7.3.4). Given $p:I^{\prime}\to Z$ with $fp=\operatorname{id}\nolimits_{I^{\prime}}$, we put $f^{\#}_{p}(\theta^{\prime})=\bigl(f^{\#}(\theta^{\prime}_{f(s)})_{s}\bigr)_{s\in p(I^{\prime})}$, a map in ${\mathcal{P}}(Z)$ with source $p(I^{\prime})$.

We define

Note that $f^{\#}(\theta^{\prime})=\sum_{\theta\in f^{-1}(\theta^{\prime})}\theta$, where $f^{-1}(\theta^{\prime})$ is the set of braids in $Z$ lifting $\theta$.

Given $f^{\prime}:Z^{\prime}\to Z^{\prime\prime}$ a morphism of curves, we have $(f^{\prime}f)^{\#}=f^{\#}f^{\prime\#}$.

The next two propositions are immediate consequences of Propositions 7.3.18 and 7.3.19 (cf §2.4).

Note that the construction $Z\mapsto\operatorname{add}\nolimits({\mathcal{P}}(Z))$ and $f\mapsto f^{\#}$ defines a contravariant functor from the category of curves with strict morphisms to the category of ${\mathbf{F}}_{2}$-linear categories.

Let $Z_{1},\ldots,Z_{r}$ be the connected components of $Z$. The isomorphism (7.3.4) induces an isomorphism of pointed categories

Note that the inverse functor sends a braid $\theta:I\to J$ in $Z$ to $(\theta_{1},\ldots,\theta_{r})$, where $\theta_{i}$ is the restriction of $\theta$ to $I\cap Z_{i}$.