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FUNCTORIAL ASPECTS OF THE RECONSTRUCTION OF LIE GROUPOIDS FROM THEIR BISECTIONS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 101 / Issue 2 / October 2016
- Published online by Cambridge University Press:
- 14 March 2016, pp. 253-276
- Print publication:
- October 2016
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- Article
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To a Lie groupoid over a compact base
$M$, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing 
$M$). Moreover, it gives rise to an adjunction between the category of Lie groupoids over 
$M$ and the category of Lie groups acting on 
$M$. In the last section we then show how to promote this adjunction to almost an equivalence of categories.
