The lie-groupoids tag has no wiki summary.

**1**

vote

**0**answers

87 views

### adjoint representation of 2-Lie groups

Baez and Crans in their paper on Lie 2-algebras refer to adjoint representations of Lie 2-groups but don't say much, as far as I can tell, except to say that such a representation acts on a 2-Lie ...

**8**

votes

**1**answer

163 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 ...

**2**

votes

**0**answers

61 views

### Properties of the induced map between inertia stacks

Let $\mathcal X$ and $\mathcal Y$ be (separated) Deligne-Mumford stacks. A morphism of stacks $f:\mathcal X \to \mathcal Y$ induces a morphism between inertia stacks $\tilde f:I\mathcal X \to ...

**6**

votes

**1**answer

104 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 ...

**4**

votes

**0**answers

82 views

### Is this groupoid a model for the derived fixed-point locus of the free loop space?

In this paper, John Baez and Urs Schreiber define (see Definition 2.16) a Lie groupoid (there called a '2-space') associated to any manifold $M$. In fact it is a bundle of Lie groups over $M$ thought ...

**0**

votes

**1**answer

99 views

### étalé space of sheaves on a differentiable stack

If $F$ is a sheaf on a topological space $X$, the well-known étalé space
contruction gives rise to a bundle $\Gamma F$ on $X$ such that $F$ is
isomorphic to the sheaf of sections of $\Gamma F$.
On ...

**5**

votes

**0**answers

97 views

### KK-witnesses of Gysin maps between differentiable stacks

In 1982 Alain Connes gave the construction of a KK-element $f! \in KK(C(X), C(Y))$ that "witnesses" the fiber integration/Gysin/Umkehr/wrong-way map on topological $K$-theory along a K-orientable map ...

**1**

vote

**1**answer

299 views

### When are $k$-sectors of a Lie groupoid a manifold?

Let ${\mathcal{G} = \lbrace s,t:G_1 \to G_0 \rbrace}$ be a Lie groupoid. Define
$$(\mathcal{G}^k)_0:=\lbrace (a_1,\dots,a_k) \in G_1^k\mid s(a_1)=t(a_1)=\dots=s(a_k)=t(a_k) \rbrace$$
(This is the ...

**2**

votes

**2**answers

292 views

### What are the possible symplectic structures on a given Lie groupoid?

Recall the definition of a symplectic groupoid. Roughly this is a Lie groupoid such that the object manifold is Poisson, and the arrow manifold is symplectic such that the symplectic form is ...

**5**

votes

**2**answers

481 views

### Internal equivalence implies weak equivalence for Frechet Lie groupoids?

It is a known theorem that an internal equivalence of Lie groupoids (finite dimensional manifolds!) - that is an equivalence in the 2-category of Lie groupoids, smooth functors and transformations - ...

**1**

vote

**0**answers

202 views

### Which makes Lie groupoids so nice?

This is a continuation of my previous question.
A) Morphisms in (1') are basically internal anafunctors, their compositions heavily use (and only) pullback/limit.
B) Bibundles in (2) are basically ...

**5**

votes

**1**answer

357 views

### Compare three 2-categories of (Lie) groupoids

Lie groupoids are groupoids with smooth structures. There is a nature 2-category of Lie groupoids: Lie groupoids, smooth functors of Lie groupoids, smooth natural transformations of smooth functors. ...

**1**

vote

**1**answer

297 views

### Why is the base manifold of a Lie groupoid required to be second-countable?

I wonder why one requires that the base manifold of a Lie groupoid is second-countable?

**4**

votes

**2**answers

465 views

### Is there a “geometric” language that describes the equivalence groupoid of a foliated manifold?

Sitting on the couch in my office is a certain groupoid. It's waiting for me to say something to it. My problem is that I don't know its language. My question here is for some suggestions.
Here, ...

**4**

votes

**1**answer

405 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 ...

**14**

votes

**4**answers

590 views

### What is the 2-category whose 0-objects are Lie algebroids?

Recall the notion of Lie algebroid (n Lab, Wikipedia). One motivation for studying Lie algebroids is that they are infinitesimal versions of Lie groupoids, and Lie groupoids present stacks. In ...

**15**

votes

**2**answers

812 views

### Applications of topological and diferentiable stacks

What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well ...