Subsection 7.1.3. (0PH0) · ScalingStacks

0P8I

Definition 7.1.12. We say that $\sim$ is a finite relation if the set of points that are not alone in their equivalence class is finite.

0P8J

Lemma 7.1.13. Let $f:X\to X^{\prime}$ be a morphism of $1$ -dimensional spaces. Define an equivalence relation on $X$ by $x_{1}\sim x_{2}$ if $f(x_{1})=f(x_{2})$ . This defines a finite relation on $X$ and $f$ factors uniquely as a composition $f=\bar{f}\circ q$ where $\bar{f}:X/\!\!\sim\ \to X^{\prime}$ is a morphism of $1$ -dimensional spaces and $q:X\to X/\!\!\sim$ is the quotient map.

0P8K

Lemma 7.1.14. Given $X$ a $1$ -dimensional space, there is a $1$ -dimensional manifold $\hat{X}$ with a finite relation $\sim$ and an isomorphism $f:\hat{X}/\!\!\sim\ \mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}X$ such that $f(\hat{X}_{f})=X_{exc}$ .

0P8L

Proof. Fix, for every $x\in X_{exc}$ , a small open neighbourhood $U_{x}$ of $x$ and a homeomorphism $f_{x}:U_{x}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\mathrm{St}(E_{x})$ , where $E_{x}$ is a finite subset of $S^{1}$ . We choose now an equivalence relation on $E_{x}$ whose classes have cardinality at most $2$ . Note that $f_{x}$ induces a bijection between $C(x)$ and $E_{x}$ , hence the equivalence relation can be viewed on $C(x)$ .

Define $\hat{U}_{x}=\coprod_{E^{\prime}\in E_{x}/\!\sim}\operatorname{St}\nolimits(E^{\prime})$ . The map $f_{x}$ provides an open embedding

U_{x}-\{x\}\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\mathrm{St}^{\circ}(E_{x})\mathrel{\mathop{\kern 0.0pt\to}\limits^{\sim}}\coprod_{E^{\prime}\in E_{x}/\!\sim}\operatorname{St}\nolimits^{\circ}(E^{\prime})\hookrightarrow\hat{U}_{x}.

We put

\hat{X}=(X-X_{exc})\coprod_{(\coprod_{x\in X_{exc}}(U_{x}-\{x\}))}\bigl(\coprod_{x\in X_{exc}}\hat{U}_{x}\bigr).

Note that $\hat{X}$ is a $1$ -dimensional manifold. Let $q:\hat{X}\to X$ be the canonical map: it identifies $X$ with the quotient of $\hat{X}$ by the equivalence relation given by $\hat{x}_{1}\sim\hat{x}_{2}$ if $q(\hat{x}_{1})=q(\hat{x}_{2})$ . Up to isomorphism, $\hat{X}$ depends only on the choice of an equivalence relation on $C(x)$ for $x\in X_{exc}$ . ∎

7.1.3. Quotients