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Questions tagged [lie-groupoids]

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4
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1answer
166 views

Proper and etale groupoid is locally a translation groupoid

I am reading Orbifolds as Groupoids: an Introduction by Ieke Moerdijk. In page $8$ when explaining local charts, it says the following : Let $\mathcal{G}$ be a Lie groupoid. For an open set $U\...
1
vote
2answers
302 views

Fibered product of stacks comes from a Lie groupoid

I am adding some context here. I am reading Introduction to Differentiable Stacks by Gregory Ginot. In page no $7$, just before the remark $2.2$ he says the following. One shall be careful that ...
14
votes
1answer
1k views

Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids

When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie ...
7
votes
1answer
349 views

Integrating representations of Lie algebroids

If $A \to M$ is a Lie algebroid over a smooth manifold $M$ then a representation of $A$ is a vector bundle $E \to M$ with a flat $A$-connection $$ \nabla : \Gamma(E) \to \Gamma(E\otimes A^*). $$ If $G$...
4
votes
1answer
459 views

Intrinsic Characterization of when an orbifold (or more general stack) is effective?

Recall that an orbifold is an etale and proper differentiable stack $X$. Etale means that it admits an etale atlas $M \to X$ from a manifold $M$ (which is to say it is represented by an etale Lie ...
2
votes
2answers
190 views

Necessity/Motivation for generalised homomorpisms

I am reading Ieke Moerdijk's article "Orbifolds as Groupoids : an Introduction". In that notes author defines a notion of generalized map between Lie groupoids. Let $\mathcal{G}$ and $\mathcal{H}$...
4
votes
1answer
201 views

In what sense bibundles are called as generalized morphisms

Definition : Let $\mathcal{G}$ and $\mathcal{H}$ be Lie groupoids. A bibundle from $\mathcal{G}$ to $\mathcal{H}$ is a manifold $P$ together with two maps $a_L:P\rightarrow \mathcal{G}_0,a_R:P\...
3
votes
0answers
264 views

Morita Equivalence of Lie groupoids

I am trying to understand what exactly is the Morita equivalence of Lie groupoids. I am reading Ieke Moerdijk’s notes Orbifolds as Groupoids. A homomorphism $\phi:\mathcal{H}\rightarrow \mathcal{G}...
3
votes
1answer
176 views

Isotropy group of a Lie groupoid is a Lie group

I am trying to see that Isotropy group/object group/vertex group of a Lie groupoid is a Lie group. Let $\mathcal{G}$ be a Lie groupoid and $x$ be an object in $\mathcal{G}$ i.e., $x\in \mathcal{G}_0$...
1
vote
2answers
190 views

Composition of bibundles

I am reading Orbifolds as stacks? Given Lie groupoids $\mathcal{G}$ and $\mathcal{H}$ there is a notion of what is called a bibundle from $\mathcal{G}$ to $\mathcal{H}$ which is supposed to be a ...