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I edited it because I realized that effectiveness implies the gluing-condition.
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zxcv
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Let $G_1\substack{\to \\ \to}G_0$ be an étale Lie groupoid, whose source- and target-maps are denoted by $s$ and $t$, respectively. Let \begin{equation} \Psi=\{(t|_U)\circ(s|_U)^{-1}:\text{$U$ is an open set of $G_1$, and $s$ and $t$ are injective on $U$}\}\text. \end{equation} A textbook I am reading (p.138 of "Introduction to foliations and Lie groupoids") claims that $\Psi$ is a pseudogroup on $G_0$. I checked everything except for the gluing-condition: For all diffeomorphism $f:V\to W$ between open sets of $G_0$, if there exists an open cover $(V_i)_{i\in I}$ of $V$ such that $f|_{V_i}\in\Psi$ for all $i\in I$, then $f\in\Psi$. How do you prove thisI checked that the gluing-condition is true if $G_1\substack{\to \\ \to}G_0$ is effective. Is it still true without effectiveness?

Let $G_1\substack{\to \\ \to}G_0$ be an étale Lie groupoid, whose source- and target-maps are denoted by $s$ and $t$, respectively. Let \begin{equation} \Psi=\{(t|_U)\circ(s|_U)^{-1}:\text{$U$ is an open set of $G_1$, and $s$ and $t$ are injective on $U$}\}\text. \end{equation} A textbook I am reading (p.138 of "Introduction to foliations and Lie groupoids") claims that $\Psi$ is a pseudogroup on $G_0$. I checked everything except for the gluing-condition: For all diffeomorphism $f:V\to W$ between open sets of $G_0$, if there exists an open cover $(V_i)_{i\in I}$ of $V$ such that $f|_{V_i}\in\Psi$ for all $i\in I$, then $f\in\Psi$. How do you prove this?

Let $G_1\substack{\to \\ \to}G_0$ be an étale Lie groupoid, whose source- and target-maps are denoted by $s$ and $t$, respectively. Let \begin{equation} \Psi=\{(t|_U)\circ(s|_U)^{-1}:\text{$U$ is an open set of $G_1$, and $s$ and $t$ are injective on $U$}\}\text. \end{equation} A textbook I am reading (p.138 of "Introduction to foliations and Lie groupoids") claims that $\Psi$ is a pseudogroup on $G_0$. I checked everything except for the gluing-condition: For all diffeomorphism $f:V\to W$ between open sets of $G_0$, if there exists an open cover $(V_i)_{i\in I}$ of $V$ such that $f|_{V_i}\in\Psi$ for all $i\in I$, then $f\in\Psi$. I checked that the gluing-condition is true if $G_1\substack{\to \\ \to}G_0$ is effective. Is it still true without effectiveness?

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zxcv
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How to prove the gluing-condition for a pseudogroup induced by an étale Lie groupoid?

Let $G_1\substack{\to \\ \to}G_0$ be an étale Lie groupoid, whose source- and target-maps are denoted by $s$ and $t$, respectively. Let \begin{equation} \Psi=\{(t|_U)\circ(s|_U)^{-1}:\text{$U$ is an open set of $G_1$, and $s$ and $t$ are injective on $U$}\}\text. \end{equation} A textbook I am reading (p.138 of "Introduction to foliations and Lie groupoids") claims that $\Psi$ is a pseudogroup on $G_0$. I checked everything except for the gluing-condition: For all diffeomorphism $f:V\to W$ between open sets of $G_0$, if there exists an open cover $(V_i)_{i\in I}$ of $V$ such that $f|_{V_i}\in\Psi$ for all $i\in I$, then $f\in\Psi$. How do you prove this?