effective Lie groupoid
Higher Lie theory
Formal Lie groupoids
A Lie groupoid is said to be effective if its morphisms act locally freely on germs, in a sense.
(Beware that this use of the term is entirely independent of “effective” in the sense of Giraud's axioms, as discussed are groupoid object in an (infinity,1)-category.)
Let be a Lie groupoid. Equivalently, let be a differentiable stack equipped with an atlas .
Then given any element in , hence given a morphism , it induces a germ of a local diffeomorphism as follows:
choose to be any neighbourhood of small enough such that the restricted source and target maps
are diffeomorphisms. Then define to be the germ .
The Lie groupoid is called effective if this assignment of morphisms to germs of local diffeomorphisms is injective.
Similarly a differentiable stack is called an effective étale stack if it is represented by an effective étale Lie groupoid.
This means that the action of the automorphism group at any point on the germ at is faithful.
(Carchedi 12, theorem 4.1).
A standard textbook aacount is in
- Ieke Moerdijk, Janez Mrčun Introduction to Foliations and Lie Groupoids ,Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003)
Brief survey is in
Discussion in a more general context of étale stacks is in
Revised on December 27, 2014 17:27:44
by Urs Schreiber