Within the same point of view that let us to see a group as a category, a groupoid is a category like a group category, but with several objects, not only one. It's as if we were combining several independent groups...
A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group.
(From this talk about the leaf space of a foliation.)
Let $(M, F)$ be a foliated manifold. The holonomy groupoid $H = \text{Hol}(M, F)$ is a smooth groupoid with $M$ as the space of objects. If $x, y \in M$ are two points on different leaves, there are no arrows from $x$ to $y$ in $H$. If $x$ and $y$ lie on the same leaf $L$, an arrow $h : x \to y$ in $H$ (i.e., a point $h \in H_1$ with $s(h) = x$ and $t(h) = y$) is an equivalence class $h = [\alpha]$ of smooth paths $\alpha : [0, 1] \to L$ with $\alpha(0) = x$ and $\alpha(1) = y$.
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Author of the notes: Antonio J. Pan-Collantes
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