∞-Lie theory

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 $X$ a morphism $f:x\to y$ induces a germ of a local diffeomorphism $\stackrel{˜}{f}:\left({X}_{0},x\right)\to \left({X}_{0},y\right)$: for that choose $U\subset {X}_{0}$ to be any neighbourhood of $x$ small enough such that the restricted source and target maps

$\begin{array}{ccc}{X}_{1}{×}_{{X}_{0}}U& \to & {X}_{1}\\ {}^{s{\mid }_{U},t{\mid }_{U}}↓& & {↓}^{s,t}\\ U& \stackrel{}{↪}& {X}_{0}\end{array}$\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 $\stackrel{˜}{f}$ to be the germ $t\circ \left(s{\mid }_{U}{\right)}^{-1}$.

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 February 1, 2012 12:30:56 by Urs Schreiber (82.169.65.155)