Subsection 7.4.10. (0PHS) · ScalingStacks

7.4.10. Subcurves

Let $Z$ be a curve.

Let $S$ and $T$ be two finite subsets of $Z$ . Let $S_{1}$ be a subset of $S$ and $S_{2}=S\setminus S_{1}$ . Let $T_{1}$ be a subset of $T$ and $T_{2}=T\setminus T_{1}$ . Let $\Phi_{i}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S_{i},T_{i})$ . We define $\Phi=\Phi_{1}\boxtimes\Phi_{2}\in\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S,T)$ by $\Phi_{s}=(\Phi_{i})_{s}$ when $s\in S_{i}$ . This gives an injective map of pointed sets

\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S_{1},T_{1})\wedge\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S_{2},T_{2})\hookrightarrow\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S,T).

Note that this is not compatible with composition in general. We obtain an isomorphism of pointed sets

\bigvee_{\begin{subarray}{c}T^{\prime}_{1}\subset T\\ |T^{\prime}_{1}|=|S_{1}|\end{subarray}}\bigl(\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S_{1},T^{\prime}_{1})\wedge\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S_{2},T\setminus T^{\prime}_{1})\bigr)\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S,T).

We have corresponding morphisms of ${\mathbf{F}}_{2}$ -modules between $\operatorname{Hom}\nolimits$ -spaces in ${\mathcal{S}}(Z)$ . Note these are not compatible with the differential.

Assume $S_{2}=T_{2}$ . The map $\Phi_{1}\mapsto\Phi_{1}\boxtimes\operatorname{id}\nolimits_{S_{2}}$ defines a canonical embedding of pointed sets (not compatible with the differential nor the multiplication in general)

\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S_{1},T_{1})\hookrightarrow\operatorname{Hom}\nolimits_{{\mathcal{S}}^{\bullet}(Z)}(S_{1}\sqcup S_{2},T_{1}\sqcup S_{2}).

Given $Z_{1}$ and $Z_{2}$ two disjoint closed subcurves of $Z$ , we obtain a faithful differential pointed functor

{\mathcal{S}}^{\bullet}(Z_{1})\wedge{\mathcal{S}}^{\bullet}(Z_{2})\to{\mathcal{S}}^{\bullet}(Z),\ (S_{1},S_{2})\mapsto S_{1}\sqcup S_{2}.

Let $Z_{1},\ldots,Z_{r}$ be the connected components of $Z$ . The construction above induces an isomorphism of differential pointed categories (cf (7.4.2))

(7.4.5)

{\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),\ (S_{1},\ldots,S_{r})\mapsto S_{1}\sqcup\cdots\sqcup S_{r}.

Let us record a case where the tensor product construction $\boxtimes$ is compatible with composition and the differential in the following immediate lemma.

0PC3

Lemma 7.4.36. Let $M$ be a subset of $Z$ and let $Z^{\prime}$ be a subcurve of $Z$ . Assume that given an admissible homotopy class of paths $\zeta$ in $Z$ with endpoints in $M$ , there is an admissible path $\gamma$ in $\zeta$ contained in $Z-Z^{\prime}$ . There is a faithful functor of differential pointed categories

	$\displaystyle{\mathcal{S}}^{\bullet}_{M}(Z)\wedge{\mathcal{S}}^{\bullet}(Z^{\prime})$	$\displaystyle\to{\mathcal{S}}^{\bullet}_{M\cup Z^{\prime}}(Z)$
	$\displaystyle(S,T)$	$\displaystyle\mapsto S\sqcup T$
	$\displaystyle(\alpha,\beta)$	$\displaystyle\mapsto\alpha\boxtimes\beta=(\alpha\boxtimes\operatorname{id}\nolimits)\cdot(\operatorname{id}\nolimits\boxtimes\beta)=(\operatorname{id}\nolimits\boxtimes\beta)\cdot(\alpha\boxtimes\operatorname{id}\nolimits).$

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2