Questions tagged [lie-groupoids]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3 votes
0 answers
73 views

Example of a groupoid internal to the category of smooth manifolds that is not a Lie groupoid

This questions is about the distinction between: Lie groupoids: we require source and target maps to be submersions. This implies that the domain of the composition map, $G_1 \;{}_s\!\times_t G_1$, ...
1 vote
0 answers
28 views

How to prove the gluing-condition for a pseudogroup induced by an étale Lie groupoid?

Let $G_1\substack{\to \\ \to}G_0$ be an étale Lie groupoid, whose source- and target-maps are denoted by $s$ and $t$, respectively. Let \begin{equation} \Psi=\{(t|_U)\circ(s|_U)^{-1}:\text{$U$ is ...
2 votes
1 answer
380 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 ...
6 votes
1 answer
363 views

Anafunctors vs the plus construction

Given a Lie groupoid $G$, we can view it as representing a prestack on $\text{Mfld}$ by sending and manfold $M$ to the groupoid of smooth functors and smooth natural transformations $$G(M) := \text{...
2 votes
1 answer
218 views

references to learn the general theory Lie $\infty$-groupoids and Lie $\infty$-algebroids

Kirill Mackenzie has a book on the general theory of Lie groupoids and Lie algebroids. Is there such a reference for the general theory of Lie $\infty$-groupoids and Lie $\infty$-algebroids; that ...
3 votes
2 answers
475 views

Lie algebroid associated to a vector bundle

Let $E\rightarrow M$ be a vector bundle. Kirill Mackenzie in the book General theory of Lie groupoids and Lie algebroids associates a Lie algebroid to $E\rightarrow M$ in the following steps: talk ...
16 votes
3 answers
1k views

Is every singular foliation induced by a Lie algebroid?

Let $M$ be a smooth manifold. A smooth distribution $D$ on $M$ is the union of a family $\{D_p \leq T_p M : p\in M\}$ of vector spaces such that there is a family $\mathcal C $ of smooth vector fields ...
4 votes
0 answers
115 views

Averaging over a Weinstein groupoid?

(Not sure if this question belongs here or on m.SE) For a Lie group, $G$ (of dimension $n$), one can average over the group: $$ \Gamma = \int_{G} d\mu(g) ~g $$ (where $d\mu(g)$ is the left-Haar ...
3 votes
3 answers
475 views

Lie groupoids in practice

I am familiar with the notion of Lie groupoids. But, only easy examples of Lie groupoids I am familiar with are the following: Lie groupoids coming from manifolds; that are of the form $(M\...
1 vote
1 answer
271 views

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

The question is as in the title. (What are some of the) are there any applications of Homotopical algebra (in the context of Quillen’s book “Homotopical algebra”) in better understanding (or ...
3 votes
0 answers
172 views

Lie group (topological group) action on differentiable stack (topological stack)

Let $G$ be a Lie group and $\mathcal{D}$ be a differentiable stack (I am also ok to start with a topological group and topological stack). I have seen someone mentioning somewhere that the notion of ...
2 votes
1 answer
203 views

Gauge groupoid of Lorentz group & complexification

I'm learning about Lie groupoids and was inspired (by Mackenzie's book) to consider the following problem. Consider first a principal bundle $P\xrightarrow G M$; we can construct the quotient manifold ...
7 votes
3 answers
803 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, ...
5 votes
1 answer
1k 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
157 views

Models for computing cohomology of Lie groupoids

Given a Lie groupoid $\mathcal{G}=[\mathcal{G}_1\rightrightarrows \mathcal{G}_0]$, let $\mathcal{G}_\bullet$ be the associated simplicial manifold. Let $\Omega^\bullet(\mathcal{G}_\bullet)$ be the ...
8 votes
1 answer
399 views

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

I'm not an expert on homotopy theory, but I speculated about this in my thesis, so I figured I'd ask about it here. As I understand it, the homotopy hypothesis says that $\infty$-groupoids, with $\...
4 votes
1 answer
184 views

Conformal groupoid

I asked this over on Math.SE but it remained completely silent for over a week so I've deleted it and am reposting it here (I'm not really sure which site it fits better). The question itself is ...
0 votes
0 answers
105 views

Is there a classifying space for transitive Lie algebroids? If so, what is it?

Let $M$ be a manifold. The data of a Lie groupoid over $M$ is equivalent to the data of a singular foliation $M=\sqcup\mathcal{F}_i$ and, for each $i$, a map (mod homotopy) $f_i:F_i\to BG_i$ (where $...
2 votes
0 answers
125 views

Does the convolution $C^*$-algebra of locally compact Hausdorff groupoids recover back the respective groupoid?

First of all, my knowledge of operator algebras (and functional analysis) is very superficial, so sorry if the answer is actually well-known. Let $X$ be a locally compact Hausdorff groupoid (or Lie ...
4 votes
1 answer
100 views

Extension of an orbifold structure from punctured balls to balls

Let $\hat{D} := D \backslash \{0\}$ be a ball in $R^n$ with the origin $\{0\}$ removed. Assume that $\hat{D}$ has a structure as an orbifold (may be distinct from its standard manifold structure). Is ...
1 vote
0 answers
64 views

Is there an inverse image functor for sheaves on stacks?

I'm interested specifically in an inverse image functor between differentiable stacks, ie. stacks coming from Lie groupoids. Specifically, if I have a morphism of Lie groupoids $H\to G$ and I have a ...
2 votes
1 answer
115 views

Special cases of Lie II for groupoids using elementary techniques

I asked a similar question on math.stackexchange but did not get any responses, so I thought I'd kick it up to mathoverflow. In Crainic and Fernandes's "Integrability of Lie Brackets" (and ...
2 votes
1 answer
149 views

Necessary and sufficient conditions for a Lie groupoid to present a stack

Let $\mathcal{G} = G_1 \rightrightarrows G_0$ be a Lie Groupoid (although I am also interested in groupoids internal to other sites), the stack associated to $\mathcal{G}$, which is sometimes denoted $...
6 votes
3 answers
608 views

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

Let $G$ and $H$ be Lie groupoids. We know that there are two notions of equivalences of Lie groupoids: Strongly equivalent Lie groupoids: (My terminology) A homomorphism $\phi:G \rightarrow H$ of ...
7 votes
2 answers
319 views

Is there any Lie groupoid structure on $Hom(\mathcal{G}, \mathcal{H})$ where $\mathcal{G}$ and $\mathcal{H}$ are Lie groupoids?

We know that in general, there is no smooth manifold structure on $Hom(X,\, Y)$ where $X$ and $Y$ are smooth manifolds, but under certain nice conditions (see https://ncatlab.org/nlab/show/manifold+...
5 votes
1 answer
173 views

Equivalence of definitions of equivalence of étale Lie groupoids

I've come across two definitions of an equivalence of étale Lie groupoids, and I'd like to know whether they are equivalent. Let $\mathcal{G}$ be an étale Lie groupoid with space of objects $\mathcal{...
3 votes
2 answers
342 views

Morita equivalent Lie groupoids

Suppose $[X_1\rightrightarrows X_0]$ and $[Y_1\rightrightarrows Y_0]$ are Morita equivalent Lie groupoids. This means, there exists another Lie groupoid $[Z_1\rightrightarrows Z_0]$ and Morita ...
3 votes
0 answers
82 views

Degeneration of spectral sequence computing Hochschild cohomology of enveloping algebra of Lie algebroid

Let $L$ be a Lie algebroid on a smooth affine $k$-scheme $X=spec(R)$. Recall that by definition $L$ is a locally free sheaf with the structure of a sheaf of $k$ Lie algebras, so that there exists a ...
2 votes
1 answer
109 views

"Lie theory" for anchored bundles and reflexive graphs

Perhaps Lie theory is not the correct term, but I'm thinking of the intermediate result in the Lie groupoid to Lie algebroid correspondence. Given a Lie groupoid $G$ over $M$, we may construct the Lie ...
3 votes
0 answers
83 views

Couniversality of Lie integration in different categories of manifolds/smooth spaces

A fairly reasonable interpretation of Lie II and Lie III seems to be that the category of Lie algebras is a coreflective subcategory of the category of Lie groups, so that the Lie group integrating a ...
4 votes
0 answers
180 views

Do we have classification (upto Morita equivalence) of Lie groupoids?

Vague question is the following: Is there a classifcation of Lie groupoids? Slightly less vague question is the following: Is there a (short?) list of "types" of Lie groupoids such that ...
2 votes
1 answer
137 views

Identifying Lie groupoids among smooth groupoids

I have been approaching groupoids in the category of smooth manifolds using methods from essentially algebraic theories/limit sketches. Are there any results that identify Lie groupoids amongst ...
7 votes
1 answer
223 views

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

This question can be thought as a complement to this one. Lie groups can be defined as groups internal to the category of smooth manifolds. Lie monoids, however, as a particular case of Lie semigroups,...
1 vote
1 answer
262 views

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

$\DeclareMathOperator\At{At}\DeclareMathOperator\Obj{Obj}\DeclareMathOperator\Mor{Mor}$According to https://ncatlab.org/nlab/show/Atiyah+Lie+groupoid#idea the Atiyah Lie groupoid $\At(P)$ of a ...
7 votes
5 answers
2k views

What are Lie groupoids intuitively?

I am trying to understand about Lie groupoids but not able to get feeling for what it actually is. So, question here is, What are Lie groupoids? How similar are they to Lie groups, Groupoids and ...
2 votes
0 answers
83 views

Examples of strictification of a weak category obtained from a generalisation of a strict category

I have made the following observation (hopefully a correct one) when reading the paper Orbifolds as stacks: They start with the strict $2$-category category of Lie groupoids, functors, natural ...
7 votes
1 answer
322 views

Notions of Lie 2-groupoids

The term Lie $2$-groupoid is used in the literature in more than one context. Some examples are given below: Ginot and Stiénon's paper $G$-gerbes, principal $2$-group bundles and characteristic ...
8 votes
0 answers
199 views

What are the Newton groupoids from Drinfeld's paper on the Grinberg-Kazhdan theorem?

The paper the Grinberg-Kazhdan formal arc theorem and the Newton groupoids by Drinfeld seems to contain many interesting things which are beyond me. For now, I am trying to get some intuition for the ...
7 votes
3 answers
443 views

Why the third stage of Cech nerve a Lie 2-groupoid?

In the page https://ncatlab.org/nlab/show/Lie+2-groupoid the Lie 2-groupoid is defined as the 2 truncated $\infty$-Lie groupoid. I am not much comfortable with the language of higher category theory ...
4 votes
0 answers
226 views

Reference request : Quotient manifold theorem for Lie groupoid action on a manifold

Let $G$ be a Lie group and $M$ be a smooth manifold. Let $G\times M\rightarrow M$ be a smooth map giving a free, proper action of $G$ On $M$. Then, by quotient manifold theorem, we see that there ...
4 votes
1 answer
169 views

Lie groupoids being homotopy equivalent

Let $M,N$be two smooth manifolds. Let $f,g:M\rightarrow N$ be two smooth maps. We have the notion of a homotopy (smooth homotopy) from the maps $f$ to the map $g$. Is there a similar concept for ...
4 votes
1 answer
450 views

Requirement for weak pullback to be a Lie groupoid (Moerdijk)

Let $\phi:\mathcal{G}\rightarrow \mathcal{K}$ and $\psi:\mathcal{H}\rightarrow \mathcal{K}$ be morphisms of Lie groupoids. We define weak pullback/2-fibre product corresponding to $\phi:\mathcal{G}\...
9 votes
0 answers
246 views

Holonomy as a right adjoint, monodromy as a left adjoint

This question about the difference between holonomy and monodromy has an interesting answer by Ronnie Brown. An excerpt: So holonomy comes out as a kind of right adjoint, and monodromy as a kind ...
6 votes
1 answer
406 views

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

Given a foliation, there is a holonomy groupoid and a classifying map to the Haefliger classifying space via the Haefliger groupid. What is the relation between these groupids?
3 votes
0 answers
243 views

First thoughts about fundamental group of a topological (Lie) groupoid

I am reading the paper Chern-Weil map for principal bundles over groupoids. In page number $13$, authors say let us recall the definition of fundamental group of a topological groupoid. But, they ...
6 votes
1 answer
348 views

De Rham cohomology of Lie groupoid

Let $G$ be a Lie group acting on a manifold $M$. Consider the transformation groupoid $\mathcal{G}=(G\times M\rightrightarrows M)$. We have the notion of de Rham cohomology of a Lie groupoid by ...
2 votes
1 answer
237 views

Simplicial manifold associated to Lie groupoid

Let $\Gamma=(\Gamma_1\rightrightarrows \Gamma_0), \Gamma’=(\Gamma’_1\rightrightarrows \Gamma’_0)$ be Lie groupoids and $\Gamma_{\bullet} ,\Gamma’_{\bullet}$ be the simplicial manifolds associated to $\...
4 votes
2 answers
296 views

Automorphisms of which structure form a Lie groupoid

Given a manifold $M$, the collection of all automorphisms of $M$, denoted by $\text{Aut}(M)$ forms a Lie group. Do we have similar setting in case of Lie groupoid? Is there "a structure" whose "...
5 votes
1 answer
275 views

Isotropy subgroupoid of a regular Lie groupoid

Let $(G\rightrightarrows M)$ be a Lie groupoid (i.e. a groupoid with source map $s$ and target map $t$ such that $G,M$ are smooth manifolds and the structural maps are all smooth (and $s$,$t$ are ...
0 votes
2 answers
794 views

Why study orbifolds? [closed]

Question is as in the title. Why study orbifolds? I study orbifolds as locally compact Hausdorff spaces $X$ having an orbifold structure, i.e., there exists an orbifold groupoid (proper foliatio. ...