nLab
effective Lie groupoid

Context

\infty-Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contemts

Idea

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.

Definition

Given a Lie groupoid XX a morphism f:xyf : x \to y induces a germ of a local diffeomorphism f˜:(X 0,x)(X 0,y)\tilde f : (X_0,x) \to (X_0,y): for that choose UX 0U \subset X_0 to be any neighbourhood of xx small enough such that the restricted source and target maps

X 1× X 0U X 1 s| U,t| U s,t U X 0 \array{ X_1 \times_{X_0} U&\to& X_1 \\ {}^{\mathllap{s|_U, t|_U}}\downarrow && \downarrow^{\mathrlap{s,t}} \\ U &\stackrel{}{\hookrightarrow} & X_0 }

are diffeomorphisms. Then define f˜\tilde f to be the germ t(s| U) 1t \circ (s|_U)^{-1}.

The Lie groupoid XX is called effective if this assignment of morphisms to germs of local diffeomorphisms is injective.

References

  • Ieke Moerdijk, Janez Mrčun Introduction to Foliations and Lie Groupoids ,Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003)
Revised on February 1, 2012 12:30:56 by Urs Schreiber (82.169.65.155)