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  "body": "\n## Summary\n\nTwo [[Lie groupoids|HNVW0E732J]] are Morita equivalent when they describe the same transverse geometry, even if their spaces of objects differ.\nFormally, they are joined by a third Lie groupoid that maps to both by weak equivalences, or equivalently by a principal bibundle.\nMorita equivalent groupoids have homeomorphic orbit spaces, isomorphic isotropy groups along corresponding orbits and the same invariants such as cohomology and K-theory.\nThis notion makes Lie groupoids a model for singular spaces such as orbifolds, leaf spaces of foliations and quotients by group actions.\n\n## Explanations\n\n### L1 human\n\nA groupoid can be thought of as a space together with rules saying which points count as the same and in how many ways.\nTwo groupoids can present the same \"quotient\" in different ways, for example by using a different set of charts.\nMorita equivalence says exactly when two groupoids present the same quotient, with the same symmetries at each point.\nFor example, let a Lie group act freely and properly on a manifold.\nThen the action groupoid is Morita equivalent to the quotient manifold, viewed as a groupoid with only identity arrows.\nThe idea parallels [[Morita equivalence|RD5WKA1VK7]] of rings, where two rings count as equivalent when their modules behave the same.\n",
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