nLab
effective Lie groupoid

Context

Higher geometry

Higher 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

Contents

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 =(X 1Y)X_\bullet = (X_1 \stackrel{\longrightarrow}{\longrightarrow} Y) be a Lie groupoid. Equivalently, let XX be a differentiable stack equipped with an atlas X 0XX_0 \to X.

Then given any element ff in X 1X_1, hence given a morphism f:xyf \colon x \to y, it induces a germ of a local diffeomorphism f˜:(X 0,x)(X 0,y)\tilde f \colon (X_0,x) \to (X_0,y) as follows:

choose U xX 0U_x \subset X_0 to be any neighbourhood of xx small enough such that the restricted source and target maps

X 1× X 0U x X 1 s| U,t| U s,t U x X 0 \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 f˜\tilde f to be the germ t(s| U) 1t \circ (s|_U)^{-1}.

Definition

The Lie groupoid X 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 xx on the germ at xx is faithful.

Properties

Equivalent characterizations

Proposition

The following are equivalent

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

  2. the canonical smooth functor X (X 0)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 etSmthMfd^{et} of smooth manifolds with local diffeomorphisms between them, XX corresponds to a sheaf (i.e. to a 0-truncated stack) on SmthMfd etSmthMfd^{et}.

(Carchedi 12, theorem 4.1).

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

Discussion in a more general context of étale stacks is in

Revised on December 27, 2014 17:27:44 by Urs Schreiber (127.0.0.1)