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  "body": "\n## Definition\n\nA *Lie groupoid* $G_1 \\rightrightarrows G_0$ is a [[groupoid|FH3C0T643F]] with set of objects $G_0$ and set of arrows $G_1$, with source and target maps $s, t \\colon G_1 \\to G_0$, such that:\n\n1. $G_0$ and $G_1$ are [[smooth manifolds|F45VQ47GPV]];\n2. $s$ and $t$ are [[smooth|0V33BNKZ9S]] submersions;\n3. the composition $m \\colon G_1 \\times_{G_0} G_1 \\to G_1$, defined on pairs $(h, g)$ with $s(h) = t(g)$, is smooth;\n4. the unit map $u \\colon G_0 \\to G_1$, sending $x$ to the identity arrow at $x$, and the inversion $i \\colon G_1 \\to G_1$ are smooth.\n\nSince $s$ and $t$ are submersions, the set $G_1 \\times_{G_0} G_1 = \\set{(h, g) \\in G_1 \\times G_1 : s(h) = t(g)}$ is an embedded submanifold of $G_1 \\times G_1$, so clause 3 makes sense.\n\n## Remarks\n\nExample: a [[Lie group|7JBRGCVJRB]] $G$ is a Lie groupoid $G \\rightrightarrows \\set{*}$ with one object; the maps $s$ and $t$ to a point are submersions, and composition is the group product.\n\nExample: for a smooth [[action|96B4E2GPKJ]] of a Lie group $G$ on a manifold $M$, the action groupoid has $G_0 = M$, $G_1 = G \\times M$, $s(g, x) = x$, $t(g, x) = g \\cdot x$, and $(h, g \\cdot x)(g, x) = (hg, x)$.\n\nNon-example: the compact group $(\\Z/2)^{\\N}$, viewed as a groupoid with one object, is not a Lie groupoid.\nIts space of arrows is not a manifold: it is totally disconnected but has no isolated points, so it is not locally Euclidean of any dimension.\n\nSome authors allow $G_1$ to be a non-Hausdorff manifold, which is needed for groupoids arising from foliations.\n",
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