effective Lie groupoid
Formal Lie groupoids
A Lie groupoid is said to be effective if its morphisms act locally freely, in a sense.
Beware that this use of the term is entirely independent of “effective” in the sense of Giraud’s axuioms, as discussed are groupoid object in an (infinity,1)-category.
Given a Lie groupoid a morphism induces a germ of a local diffeomorphism : for that 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.
- Ieke Moerdijk, Janez Mrčun Introduction to Foliations and Lie Groupoids ,Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003)