| 1. | Thus any groupoid is equivalent to a multiset of unrelated groups.
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| 2. | The Lie algebroid of this Lie groupoid is the Atiyah algebroid.
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| 3. | The groupoid condition on those is fulfilled, in that homeomorphisms
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| 4. | Thus we have a groupoid in the algebraic sense.
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| 5. | This leads to the idea of using multiple groupoid objects in homotopy theory.
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| 6. | As an example consider the Lie groupoid cohomology.
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| 7. | First delete the commutativity restriction to obtain the concept of a magma or a groupoid.
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| 8. | Also there is proved there a nice normal form for the elements of the fundamental groupoid.
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| 9. | Just as every Lie group has a Lie algebra, every Lie groupoid has a Lie algebroid.
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| 10. | Transitivity is an interesting construction in the category of groupoid principal bundles and left to the reader.
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