Groupoid

Definition

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.

The holonomy groupoid of a foliation

(From this talk about the leaf space of a foliation.)
Let $(M,F)$ be a foliated manifold. The holonomy of a foliation, $H=\text{Hol}(M,F)$, 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 $\alpha (1)=y$.

Two such paths $\alpha$ and $\beta$ from $x$ to $y$ are considered equivalent (i.e., they define the same arrow $h$) if they induce the same holonomy. Intuitively, this means that if you take a small transversal submanifold through $x$ (a slice that cuts across the leaves) and slide it point by point along $\alpha$ to a transversal through $y$, you get the same local diffeomorphism between these transversals as you would by sliding along $\beta$. In other words, the two paths have the exact same effect on how nearby leaves are "permuted" or "wrapped" around the leaf $L$.