Subsection 7.3.4. (0PHD) · ScalingStacks

7.3.4. Functoriality

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

Lemma 7.3.13. Let $\zeta$ be a homotopy class of paths in $Z$ . The class $f(\zeta)$ is smooth if and only if $\zeta$ is smooth. If $\zeta$ is admissible, then $f(\zeta)$ is admissible.

0PA1

Proof. Given $\gamma$ an oriented path in $Z$ , the path $f(\gamma)$ is oriented. It is smooth if and only $f(\gamma)$ is smooth. This shows that if $\zeta$ is a smooth (resp. admissible) homotopy class of paths in $Z$ , then $f(\zeta)$ is smooth (resp. admissible).

Consider now $\zeta$ a homotopy class of paths in $Z$ such that $f(\zeta)$ is smooth. Given $\gamma$ a minimal path in $\zeta$ , then $f(\gamma)$ is minimal (Lemma 7.1.20). Since $f(\zeta)$ is smooth, it follows that $f(\gamma)$ is smooth (Properties 7.3.5(4)), hence $\gamma$ is smooth and finally $\zeta$ is smooth. ∎

It follows from the previous lemma that the morphism $f$ induces a functor $f:{\mathcal{S}}^{\bullet}(Z,1)\to{\mathcal{S}}^{\bullet}(Z^{\prime},1)$ . We have constructed a functor ${\mathcal{S}}^{\bullet}(-,1)$ from the category of curves to the category of pointed categories.

Let us state a version of Lemma 7.1.20 for morphisms of curves.

0PA2

Lemma 7.3.14. Let $\gamma,\gamma^{\prime}$ be two admissible paths in $Z$ . If $[f(\gamma)]=[f(\gamma^{\prime})]\neq\operatorname{id}\nolimits$ , then $[\gamma]=[\gamma^{\prime}]$ . The functor $f:{\mathcal{S}}^{\bullet}(Z,1)\to{\mathcal{S}}^{\bullet}(Z^{\prime},1)$ is faithful.

Note that $f$ induces an injective morphism of groups $f:L(Z)\to L(Z^{\prime})$ and a map $f:\pi_{0}(Z)\to\pi_{0}(Z^{\prime})$ .

The next lemma is an immediate consequence of Lemma 7.1.24.

0PA3

Lemma 7.3.15. Given $\alpha,\beta\in R(Z)$ , we have $\langle f(\alpha),f(\beta)\rangle=f(\langle\alpha,\beta\rangle)$ .

It follows from Lemmas 7.3.15 and 7.1.25 that we have a morphism of groups $f:\Gamma(Z)\to\Gamma(Z^{\prime}),\ (r,(m,\alpha))\mapsto\bigl(f(r),(f(m),f(\alpha))\bigr)$ which restricts to an injective morphism of groups $\Gamma^{\prime}(Z)\to\Gamma^{\prime}(Z^{\prime})$ .

Let $D$ be a subset of $T(Z)$ such that given $z\in\operatorname{pt}\nolimits(D)$ , the composition $D\cap\operatorname{pt}\nolimits^{-1}(z)\to C(z)\to C(z)/\iota$ is bijective. The morphism $f:\Gamma(Z)\to\Gamma(Z^{\prime})$ induces a morphism $f:\Gamma(Z,D)\to\Gamma(Z^{\prime},f(D))$ . Let $g,h\in\Gamma(Z,D)$ . If $g<h$ , then $f(g)<f(h)$ . If $f:\pi_{0}(Z)\to\pi_{0}(Z^{\prime})$ is injective and $f(g)<f(h)$ , then $g<h$ .

Finally, the morphism $f:\Gamma(Z,D)\to\Gamma(Z^{\prime},f(D))$ induces a morphism $f:\bar{\Gamma}(Z,D)\to\bar{\Gamma}(Z^{\prime},f(D))$ . Given $g,h\in\bar{\Gamma}(Z,D)$ , we have $g<h$ if and only if $f(g)<f(h)$ .

Let $Z_{1},\ldots,Z_{r}$ be the connected components of $Z$ . There are isomorphisms of groups $R(Z_{1})\times\cdots\times R(Z_{r})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}R(Z)$ and $L(Z_{1})\times\cdots\times L(Z_{r})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}L(Z)$ given by the inclusions $Z_{i}\hookrightarrow Z$ . They induce an isomorphism of groups

(7.3.3)

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

The inclusions $Z_{i}\hookrightarrow Z$ induce pointed functors ${\mathcal{S}}^{\bullet}(Z_{i},1)\to{\mathcal{S}}^{\bullet}(Z,1)$ and give rise to an isomorphism of pointed categories

(7.3.4)

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

Andrew Manion, Raphael Rouquier

Original source: arXiv:2009.09627v2