∞-Lie theory

# Contents

## 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 $X$ a morphism $f : x \to y$ induces a germ of a local diffeomorphism $\tilde f : (X_0,x) \to (X_0,y)$: for that choose $U \subset X_0$ to be any neighbourhood of $x$ small enough such that the restricted source and target maps

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

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

The Lie groupoid $X$ 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 October 9, 2014 20:46:04 by Anonymous Coward (130.126.108.190)