#
Reduced smooth stacks?

##
Giorgio Trentinaglia

An arbitrary Lie groupoid gives rise to a groupoid of germs of local
diffeomorphisms over its base manifold, known as its *effect*. The effect
of any bundle of Lie groups is trivial. All quotients of a given Lie groupoid
determine the same effect. It is natural to regard the effects of any two
Morita equivalent Lie groupoids as being ``equivalent''. In this paper we shall
describe a systematic way of comparing the effects of different Lie groupoids.
In particular, we shall rigorously define what it means for two arbitrary Lie
groupoids to give rise to ``equivalent'' effects. For effective orbifold
groupoids, the new notion of equivalence turns out to coincide with the
traditional notion of Morita equivalence. Our analysis is relevant to the
presentation theory of proper smooth stacks.

Keywords:
Lie groupoids, effective orbifolds, categories of fractions

2010 MSC:
Primary 58H05; Secondary 22A22, 18E35

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 31, pp 1032-1066.

Published 2015-07-31.

http://www.tac.mta.ca/tac/volumes/30/31/30-31.pdf

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