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  "body": "\n## Definition\n\nFor a [[Lie groupoid|HNVW0E732J]] $G$, write $G_0$ for its manifold of objects and $G_1$ for its manifold of arrows.\nWrite $s, t \\colon G_1 \\to G_0$ for the source and target maps.\nLet $\\phi = (\\phi_1, \\phi_0) \\colon K \\to G$ be a morphism of Lie groupoids.\nThe morphism $\\phi$ is a *weak equivalence* if both of the following hold.\n\n1. The map $G_1 \\times_{s, G_0, \\phi_0} K_0 \\to G_0$, $(g, y) \\mapsto t(g)$, is a surjective [[submersion|BMY55EW3RF]].\n2. The fibre product $(K_0 \\times K_0) \\times_{G_0 \\times G_0} G_1$, formed with $\\phi_0 \\times \\phi_0$ and $(t, s)$, is a [[smooth manifold|F45VQ47GPV]].\nThe map $k \\mapsto ((t(k), s(k)), \\phi_1(k))$ from $K_1$ to it is a diffeomorphism.\n\nTwo Lie groupoids $G$ and $H$ are *Morita equivalent* if there are a Lie groupoid $K$ and weak equivalences $K \\to G$ and $K \\to H$.\n\n## Remarks\n\nClause 1 says that every object of $G$ is isomorphic to an object in the image of $\\phi_0$, in a smooth way.\nClause 2 says that the arrows of $K$ between two objects are exactly the arrows of $G$ between their images.\nMorita equivalence is an equivalence relation.\nAn equivalent definition uses a principal bibundle: a manifold $P$ with commuting actions of $G$ and $H$ and surjective submersions as moment maps.\nEach action must be free and transitive on the fibres of the moment map of the other action.\n\nExample: let $\\pi \\colon U \\to M$ be a surjective submersion, and let $K$ be the groupoid with $K_0 = U$ and $K_1 = U \\times_M U$, where $(u, v)$ is an arrow from $v$ to $u$.\nThe map $\\pi$ gives a weak equivalence from $K$ to the groupoid $M$ that has only identity arrows.\nClause 1 holds because the map in question is $\\pi$ itself.\nClause 2 holds because $K_1$ is exactly the set of pairs of points of $U$ with the same image in $M$.\nSo $K$ and $M$ are Morita equivalent; taking $U$ to be the disjoint union of the sets of an open cover gives the usual Čech groupoid.\n\nNon-example: a nontrivial Lie group $G$, viewed as a groupoid with one object, is not Morita equivalent to the one-point groupoid with only the identity arrow.\nWeak equivalences induce isomorphisms of isotropy groups, and the isotropy groups here are $G$ and the trivial group.\n\n## Explanations\n\n### L1 human\n\nTwo Lie groupoids are Morita equivalent when one can pass between them by refining or coarsening the space of objects without changing the symmetries between objects.\n",
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