# Questions tagged [lie-groupoids]

The lie-groupoids tag has no usage guidance.

**4**

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**1**answer

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### 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**

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**2**answers

307 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**

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**1**answer

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**

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**1**answer

357 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

**1**answer

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**

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**2**answers

192 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

**1**answer

202 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**

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**0**answers

271 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

**1**answer

183 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**

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**1**answer

193 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 ...