13
votes

Accepted

### What is the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces?

In place of a detailed answer, let me point to Internal categories, anafunctors and localisations, but more specific to your case is the diffeological groupoids in Smooth loop stacks of differentiable ...

8
votes

### Is every singular foliation induced by a Lie algebroid?

It is possible to define the Lie groupoid of a singular foliation and associates to it its Lie algebroid when it is smooth. This Lie algebroid satisfies the property 2.
Debord - Holonomy Groupoids of ...

8
votes

Accepted

### What are Lie groupoids intuitively?

Here is an expansion of my comment, by request. The 2-category of Lie groupoids $\mathrm{LieGpd}$ admits the category of manifolds $\mathrm{Mfld}$ as a full sub-2-category (i.e. $\mathrm{Mfld} \to \...

8
votes

### Morita equivalent Lie groupoids

Let me try to answer this part of your question:
"Does it imply some specific condition on the maps $\phi$ and $\varphi$?"
The quick answer: $\phi$ and $\varphi$ have to be surjective ...

7
votes

### What are Lie groupoids intuitively?

In the same way as groupoids $(\mathrm{src},\mathrm{trg}):X_1\rightrightarrows X_0$ are a symultaneous generalization of group actions
$$(\mathrm{pr}_1,\mathrm{act}):G\times X\rightrightarrows X$$
...

7
votes

### What is the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces?

Apologies for the late answer, I wish I'd found this earlier!
My MSc thesis was actually mainly devoted to developing a notion of Morita equivalence for diffeological groupoids! I don't think I'll ...

6
votes

### Is every singular foliation induced by a Lie algebroid?

$\newcommand\cF{\mathcal F}$For Stefan–Sussmann singular foliations, the answer is negative: See Prop. 1.3 in the following paper, for the construction of an explicit counterexample: Androulidakis and ...

6
votes

### Why study orbifolds?

I would like to propose an answer to this question, since 15 year ago I was asking it to myself and was thinking that orbiolds are useless. I read your question (maybe wrongly) as a question in ...

Community wiki

6
votes

Accepted

### Simplicial manifold associated to Lie groupoid

If the simplicial manifolds are isomorphic, then the groupoids are also isomorphic, since the nerve functor from groupoid objects in a category to simplicial objects in the same category is fully ...

6
votes

Accepted

### Notions of Lie 2-groupoids

For your first question:
They are essentially all the same thing: some globular, some simplicial (taking the nerve goes from the former to the latter). The only subtlety is perhaps in the requirement ...

6
votes

Accepted

### What is the natural Lie groupoid structure on the Atiyah Lie groupoid of a principal $G$-bundle?

Contrary to what is claimed in the comments, I would argue
that the definition given in nLab's Idea section is rigorous
enough to be an actual definition in a research-level paper,
possibly with an ...

6
votes

### Lie monoids as monoids internal to the category of smooth manifolds?

There is indeed a notion of "Lie category", introduced a 1959 paper of Charles Ehresmann: Catégories topologiques et categories différentiables. This is accessible in his OEuvres Complètes, ...

5
votes

Accepted

### Automorphisms of which structure form a Lie groupoid

Here is a construction due to Ehresmann, and covered in detail by Mackenzie in either of his Lie groupoids books.
Take a principal $G$-bundle, $\pi\colon P\to M$, everything here in smooth manifolds....

5
votes

Accepted

### Why study orbifolds?

I can not say why one studies orbifolds (or e.g. why one studies math at all). However, I can try the approach which might convince your funding agency: There are tons of interesting examples of how ...

5
votes

### What is the appropriate notion of weakly equivalent or Morita equivalent categories internal to a category of generalized smooth spaces?

I know this is a little late but I discuss this in the first two chapters of my thesis here:
https://arxiv.org/abs/1806.01939
Basically, as you mentioned, what you need is a notion of surjective ...

5
votes

### Applications of “Homotopical algebra” in the set up of Lie groupoids

There are many connections between Lie groupoids and homotopical algebra. In recent years, a particularly prominent connection is to the theory of simplicial presheaves, originally developed in the ...

4
votes

### What is the relation between the holonomy groupoid of a foliation and the corresponding Haefliger groupoid?

Write $F$ your foliation, $M$ its ambiant manifold, $q=dim(M)-dim(F)$ its codimension. The holonomy groupoid $H(F)$, if I'm correct, is the set of classes of triples $(x,\gamma,y)$ where $\gamma$ is a ...

4
votes

Accepted

### Gluing together together differentiable stacks

Try p. 17 of notes by Breen (http://math.uchicago.edu/~may/IMA/Breen.pdf) Notes on 1- and 2-gerbes

4
votes

Accepted

### Isotropy group of a Lie groupoid is a Lie group

There is also a rundown in the more modern book of Mackenzie: General Theory of Lie groupoids and Lie algebroids (p.26 Corollary 1.4.11)
Let me summarise the idea: Fix a Lie groupoid $G \...

4
votes

Accepted

### unit element under map of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$

This is more of a long comment, but since the question has been answered in the last edits I will post it as an answer.
There are multiple perspectives on the stack presented by a Lie groupoid $\...

4
votes

Accepted

### Isotropy subgroupoid of a regular Lie groupoid

This is not exactly an answer to your question, but I hope it helps.
If it would be fine to first pass to the connected components of the identity of each isotropy group, I believe you find the ...

4
votes

### Morita equivalent Lie groupoids

I will answer the new version of the question:
Does $\ker(\varphi_{*,a})=\ker (\phi_{*,a})$ for all $a\in P$,
where $\phi_{*,a}:T_aP\to T_{\phi(a)}X_0$ and $\varphi_{*,a}:T_aP\to T_{\varphi(a)}Y_0$ ...

4
votes

### Connection between Grothendieck's homotopy hypothesis and Lie's second and third theorems?

There are analogues of Lie's theorems in homotopy theory, primarily for rational and $p$-adic homotopy types, as well as Lie ∞-groupoids, which can be seen as smooth homotopy types.
In the rational ...

4
votes

Accepted

### Anafunctors vs the plus construction

The long-expected answer. $\DeclareMathOperator{\op}{op} \DeclareMathOperator{\Cat}{\mathbf{Cat}}\DeclareMathOperator{\Gpd}{\mathbf{Gpd}} \DeclareMathOperator{\disc}{disc}\DeclareMathOperator{\pr}{pr}$...

4
votes

Accepted

### What does it mean for a space to be a differentiable stack?

Stacks form an (∞,1)-category. The latter informal notion has many equivalent implementations: simplicial category, topological category, quasicategory (also known as ∞-category), Segal category, ...

3
votes

Accepted

### Fibered product of stacks comes from a Lie groupoid

Think of $BG$ and $BH$ as topological stacks, whereby one can calculate a topological groupoid presenting the stack $BG\times_{BH} BG$, namely the following: the object space is the space underlying $...

3
votes

### Fibered product of stacks comes from a Lie groupoid

Pullbacks of stacks coming from Lie groupoids are not always equivalent to Lie groupoids.
Take $G=H=\mathbb{R}$. Define $F(x)=0$ if $x\leq 0$ and $F(x)=exp(−1/x^2)$ if $x>0$.
The pullback is not ...

3
votes

### Morita equivalence of Lie groupoids

(Using the same notations as mentioned in the question.)
Let $g \in \mathcal{G}_0$. Then the first condition ensures the existence of a $(\gamma , x) \in \mathcal{G}_1 \times{_{s,\mathcal{G}_0,\phi
...

3
votes

### What are Lie groupoids intuitively?

Here is a somewhat trivial observation, but one that helped me to get an initial 'visual' feeling of the thing. I tried to make it a comment, but since I have just signed up, I didn't have enough ...

3
votes

Accepted

### Necessity/Motivation for generalised homomorpisms

Let me attempt a very simple-minded answer.
Say your objects of interest are orbifolds. You have an orbifold V and you want to describe it through a groupoid, usually an action groupoid $G\ltimes M \...

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