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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 ...
David Roberts's user avatar
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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 ...
Tsemo Aristide's user avatar
8 votes
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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 \...
David Roberts's user avatar
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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 ...
Nesta's user avatar
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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 $$(\mathrm{pr}_1,\mathrm{act}):G\times X\rightrightarrows X$$ ...
Qfwfq's user avatar
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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 ...
Nesta's user avatar
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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 ...
Iakovos's user avatar
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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 ...
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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 ...
David Roberts's user avatar
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6 votes
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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 ...
David Roberts's user avatar
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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 ...
Dmitri Pavlov's user avatar
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, ...
David Roberts's user avatar
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5 votes
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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....
David Roberts's user avatar
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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 ...
Alexander Schmeding's user avatar
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 ...
Joel Villatoro's user avatar
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 ...
Dmitri Pavlov's user avatar
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 ...
Gael Meigniez's user avatar
4 votes
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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
Eugene Lerman's user avatar
4 votes
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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 \...
Alexander Schmeding's user avatar
4 votes
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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 $\...
Bertram Arnold's user avatar
4 votes
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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 ...
João Nuno Mestre's user avatar
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 $\varphi_{*,a}:T_aP\to T_{\varphi(a)}Y_0$ ...
Dmitri Pavlov's user avatar
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 ...
Dmitri Pavlov's user avatar
4 votes
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Anafunctors vs the plus construction

The long-expected answer. $\DeclareMathOperator{\op}{op} \DeclareMathOperator{\Cat}{\mathbf{Cat}}\DeclareMathOperator{\Gpd}{\mathbf{Gpd}} \DeclareMathOperator{\disc}{disc}\DeclareMathOperator{\pr}{pr}$...
David Roberts's user avatar
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4 votes
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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, ...
Dmitri Pavlov's user avatar
3 votes
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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 $...
David Roberts's user avatar
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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 ...
Dmitri Pavlov's user avatar
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 ...
Adittya Chaudhuri's user avatar
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 ...
Dry Bones's user avatar
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3 votes
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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 \...
Nicola Ciccoli's user avatar

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