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