nLab effective Lie groupoid

Idea

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.)

Definition

Let $X_\bullet = (X_1 \stackrel{\longrightarrow}{\longrightarrow} Y)$ be a Lie groupoid. Equivalently, let $X$ be a differentiable stack equipped with an atlas $X_0 \to X$.

Then given any element $f$ in $X_1$, hence given a morphism $f \colon x \to y$, it induces a germ of a local diffeomorphism $\tilde f \colon (X_0,x) \to (X_0,y)$ as follows:

choose $U_x \subset X_0$ to be any neighbourhood of $x$ small enough such that the restricted source and target maps

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

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

Definition

The Lie groupoid $X_\bullet$ 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 $x$ on the germ at $x$ is faithful.

Properties

Equivalent characterizations

Proposition

The following are equivalent

1. $X_\bullet$ is an effective Lie groupoid, def. ;

2. the canonical smooth functor $X_\bullet \to \mathbb{H}(X_0)$ (see here) to the Haefliger groupoid of the manifold of objects is faithful, i.e. gives a representable morphism of stacks;

3. under the equivalence (here) between smooth étale stacks and stacks on the site $SmthMfd^{et}$ of smooth manifolds with local diffeomorphisms between them, $X$ corresponds to a sheaf (i.e. to a 0-truncated stack) on $SmthMfd^{et}$.

(Carchedi 12, theorem 4.1).

Related concepts

• proper Lie groupoid, étale groupoid

References

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

• David Carchedi, A quick note on étale stacks (pdf)

Discussion in a more general context of étale stacks is in

• David Carchedi, Étale Stacks as Prolongations (arXiv:1212.2282)