Subsection 7.4.4. (0PHL) · ScalingStacks

7.4.4. Strand category

Let $Z$ be a curve.

We have a positivity result in the setting of Lemma 7.4.9.

Lemma 7.4.21. Let $\theta$ and $\theta^{\prime}$ be two braids such that $\theta\circ\theta^{\prime}$ is a braid. We have $\deg(\theta)\cdot\deg(\theta^{\prime})\leq\deg(\theta\circ\theta^{\prime})$ .

Given $D$ a subset of $T(Z)$ containing $Z_{exc}^{+}$ and such that $D\cap\iota(D)=\emptyset$ , the following assertions are equivalent:

•

$\deg(\theta)\cdot\deg(\theta^{\prime})=\deg(\theta\circ\theta^{\prime})$
•

$\deg(\theta)\cdot\deg(\theta^{\prime})$ and $\deg(\theta\circ\theta^{\prime})$ have the same image in $\Gamma(Z,D)$
•

$\deg(\theta)\cdot\deg(\theta^{\prime})$ and $\deg(\theta\circ\theta^{\prime})$ have the same image in $\bar{\Gamma}(Z,D)$ .

0PBF

Proof. Assume $Z=S^{1}$ unoriented. Let $M$ be a family as in §7.4.3. Assume $M$ contains $\theta_{s}(r)$ and $\theta^{\prime}_{s}(r)$ for $r\in\{0,1\}$ and all $s$ . Proposition 7.4.18 and Lemma 7.4.19 show that the inequality follows from the corresponding inequality for maps in ${\mathcal{S}}_{n}$ , which is given by Lemmas 6.2.1 and 6.2.5.

Given $Z$ a non-singular connected curve, there is an injective morphism of curves $Z\to S^{1}$ , and the lemma follows from Proposition 7.4.3 and Lemma 7.4.12. We deduce that the inequality holds for any non-singular curve $Z$ .

Consider now a general curve $Z$ and let $q:\hat{Z}\to Z$ be the non-singular cover. Since the functor $q^{\#}:\operatorname{add}\nolimits({\mathcal{P}}(Z))\to\operatorname{add}\nolimits({\mathcal{P}}(\hat{Z}))$ is compatible with degrees (Proposition 7.4.13), it follows that the inequality holds for $Z$ .

The equivalence of the three assertions follows from the fact that an element of $(\frac{1}{2}{\mathbf{Z}}_{\geq 0})^{\pi_{0}(Z)}\subset\Gamma(Z,Z_{exc}^{+})$ is zero if and only if its image in $\frac{1}{2}{\mathbf{Z}}_{\geq 0}\subset\bar{\Gamma}(Z,D)$ is zero. ∎

By Lemma 7.4.21, the degree function gives a $\Gamma(Z,Z_{exc}^{+})$ -filtration on the category ${\mathcal{P}}^{\bullet}(Z)$ .

0PBG

Definition 7.4.22. We define the strand category ${\mathcal{S}}^{\bullet}(Z)$ as the $\Gamma(Z,Z_{exc}^{+})$ -graded pointed category associated with the filtered pointed category ${\mathcal{P}}^{\bullet}(Z)$ (cf §2.3.3).

The category ${\mathcal{S}}^{\bullet}(Z)$ has the same objects and the same maps as the category ${\mathcal{P}}^{\bullet}(Z)$ . It is a pointed category with objects the finite subsets of $Z$ and with $\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(I,J)$ the set of braids $I\to J$ , together with a $0$ -element.

The product of two braids $\theta:I\to J$ with $\theta^{\prime}:J\to K$ is defined as follows:

\theta^{\prime}\cdot\theta=\begin{cases}\theta^{\prime}\circ\theta&\text{ if }\deg(\theta^{\prime}\circ\theta)=\deg(\theta^{\prime})\cdot\deg(\theta)\\ 0&\text{ otherwise.}\end{cases}

Note that the strand category decomposes as a disjoint union ${\mathcal{S}}^{\bullet}(Z)=\coprod_{n\geq 0}{\mathcal{S}}^{\bullet}(Z,n)$ , where ${\mathcal{S}}^{\bullet}(Z,n)$ is the full subcategory with objects subsets with $n$ elements.

It follows from Lemma 7.4.21 that given $D$ a subset of $T(Z)$ containing $Z_{exc}^{+}$ and such that $D\cap\iota(D)=\emptyset$ , the structure of $\Gamma(Z,D)$ -graded (resp. $\bar{\Gamma}(Z,D)$ -graded) category on ${\mathcal{S}}(Z)$ obtained from the quotient morphism $f:\Gamma(Z,Z_{exc}^{+})\to\Gamma(Z,D)$ (resp. $f:\Gamma(Z,Z_{exc}^{+})\to\bar{\Gamma}(Z,D)$ ) is the same as the graded category obtained from the structure of $\Gamma(Z,D)$ -filtered (resp. $\bar{\Gamma}(Z,D)$ -filtered) category on ${\mathcal{P}}^{\bullet}(Z)$ that is deduced from the structure of $\Gamma(Z,Z_{exc}^{+})$ -filtered category via $f$ .

0PBH

Remark 7.4.23. We leave to the reader to check the following alternate definition of the product in the strand category.

We have $\theta^{\prime}\cdot\theta\neq 0$ if and only if there are parametrized braids $\vartheta,\vartheta^{\prime}$ with $\theta=[\vartheta]$ and $\theta^{\prime}=[\vartheta^{\prime}]$ and there are $\alpha:I^{\prime}\to I$ and $\alpha^{\prime}:K\to K^{\prime}$ two parametrized braids with $I^{\prime},K^{\prime}\subset Z\setminus Z_{exc}$ such that $i(\alpha)=i(\alpha^{\prime})=0$ , $\alpha^{\prime}_{\vartheta^{\prime}\circ\vartheta\circ\alpha_{s}(1)}\circ\vartheta^{\prime}_{\vartheta\circ\alpha_{s}(1)}\circ\vartheta_{\alpha_{s}(1)}\circ\alpha_{s}$ is admissible for all $s\in I$ and $i(\alpha^{\prime}\circ\theta^{\prime}\circ\theta\circ\alpha)=i(\alpha^{\prime}\circ\theta^{\prime})+i(\theta\circ\alpha)$ .

Let ${\mathcal{S}}(Z)={\mathbf{F}}_{2}[{\mathcal{S}}^{\bullet}(Z)]$ , a $\Gamma(Z,Z_{exc}^{+})$ -graded ${\mathbf{F}}_{2}$ -linear category.

Let $f:Z\to Z^{\prime}$ be a morphism of curves.

Let ${\mathcal{S}}^{\bullet}_{f}(Z)$ be the full subcategory of ${\mathcal{S}}^{\bullet}(Z)$ with objects those finite subsets $I$ of $Z$ such that $|f(I)|=|I|$ . We deduce from Proposition 7.4.3 and Lemma 7.4.12 a faithful $\Gamma(Z^{\prime},Z_{exc}^{\prime+})$ -graded pointed functor $f:{\mathcal{S}}^{\bullet}_{f}(Z)\to{\mathcal{S}}^{\bullet}(Z^{\prime})$ . Here, the $\Gamma(Z^{\prime},Z_{exc}^{\prime+})$ -grading on ${\mathcal{S}}^{\bullet}_{f}(Z)$ comes from the $\Gamma(Z,f^{-1}(Z_{exc}^{\prime})^{+})$ -grading via the morphism $\Gamma(f)$ .

Assume $f$ is strict. Propositions 7.4.4 and 7.4.13 provide an additive ${\mathbf{F}}_{2}$ -linear $\Gamma(Z^{\prime},Z_{exc}^{\prime+})$ -graded functor $f^{\#}:\operatorname{add}\nolimits({\mathcal{S}}(Z^{\prime}))\to\operatorname{add}\nolimits({\mathcal{S}}_{f}(Z))$ , where the $\Gamma(Z^{\prime},Z_{exc}^{\prime+})$ -grading on ${\mathcal{S}}_{f}(Z)$ is deduced from the $\Gamma(Z,f^{-1}(Z_{exc}^{\prime})^{+})$ -grading via the morphism $\Gamma(f)$ .

If $f$ is a quotient morphism, it follows from Proposition 7.4.5 that $f^{\#}$ is faithful.

Given $M$ a subset of $Z$ , we denote by ${\mathcal{S}}^{\bullet}_{M}(Z)$ the full subcategory of ${\mathcal{S}}^{\bullet}(Z)$ whose objects are the finite subsets of $M$ . The $\Gamma(Z,Z_{exc}^{+})$ -grading on ${\mathcal{S}}^{\bullet}_{M}(Z)$ comes from a $\Gamma_{M}(Z,Z_{exc}^{+})$ -grading.

We denote by ${\mathcal{S}}_{M,f}^{\bullet}(Z)$ the full subcategory of ${\mathcal{S}}_{f}^{\bullet}(Z)$ with objects subsets contained in $M$ .

We put ${\mathcal{A}}^{\bullet}(Z)={\mathcal{S}}_{Z_{exc}}^{\bullet}(Z)$ and ${\mathcal{A}}^{\bullet}(Z,n)={\mathcal{S}}_{Z_{exc}}^{\bullet}(Z,n)$ . Let ${\mathcal{A}}(Z)={\mathbf{F}}_{2}[{\mathcal{A}}^{\bullet}(Z)]$ and ${\mathcal{A}}(Z,n)={\mathbf{F}}_{2}[{\mathcal{A}}^{\bullet}(Z,n)]$ .

Let $Z_{1},\ldots,Z_{r}$ be the connected components of $Z$ . The isomorphism (7.4.1) induces an isomorphism of $\Gamma(Z,Z_{exc}^{+})$ -graded pointed categories

(7.4.2)

{\mathcal{S}}^{\bullet}(Z_{1})\wedge\cdots\wedge{\mathcal{S}}^{\bullet}(Z_{r})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{S}}^{\bullet}(Z)

where the grading on the left hand term is deduced from the the $\bigl(\prod_{i=1}^{r}\Gamma(Z_{i},(Z_{i})_{exc}^{+})\bigr)$ -grading via (7.3.3) and an isomorphism of ${\mathbf{F}}_{2}$ -linear categories

(7.4.3)

{\mathcal{S}}(Z_{1})\otimes\cdots\otimes{\mathcal{S}}(Z_{r})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}{\mathcal{S}}(Z).

0PBI

Example 7.4.24. In the example below the first row is the product in ${\mathcal{P}}^{\bullet}(Z)$ , while the second row is the product in ${\mathcal{S}}^{\bullet}(Z)$ .

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2