higher geometry / derived geometry
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∞-Lie theory (higher geometry)
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$\infty$-Lie groupoids
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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.)
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
are diffeomorphisms. Then define $\tilde f$ to be the germ $t \circ (s|_U)^{-1}$.
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.
The following are equivalent
$X_\bullet$ is an effective Lie groupoid, def. 1;
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;
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 $SmthMfd^{et}$.
A standard textbook aacount is in
Brief survey is in
Discussion in a more general context of étale stacks is in