nLab
effective Lie groupoid

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

$$\begin{array}{ccc}{X}_1{\times }_{X_0}U& \to & X_1\\ {}^{s{\mid }_U,t{\mid }_U}\downarrow & & {\downarrow }^{s,t}\\ U& \stackrel{}{\hookrightarrow }& X_0\end{array}$$\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 $t\circ (s{\mid }_U{)}^{-1}$.

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

Related concepts

• proper Lie groupoid, étale groupoid

References

• Ieke Moerdijk, Janez Mrčun Introduction to Foliations and Lie Groupoids ,Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, Cambridge, (2003)