Skip to main content

Questions tagged [lie-groupoids]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
3 votes
1 answer
157 views

Morita equivalence of Lie groupoids and isomorphism of differentiable stacks

It's a well known fact two Lie groupoids are Morita-equivalent iff they induce isomorphic differentiable stacks (I'll call this statement "(1)"). It's also well known that there is a ...
Kandinskij's user avatar
3 votes
0 answers
128 views

Bibundle induced by a morphism of stacks

[This is a repost, because I've written the wrong page number in the previous version of this question. I'm sorry] I'm currently reading "Orbifolds as stacks" by Eugene Lerman and I'm stuck ...
Kandinskij's user avatar
3 votes
0 answers
92 views

Cohomology of differentiable stacks: should the sheaf be fine?

I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fifth page. Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its ...
Kandinskij's user avatar
4 votes
0 answers
187 views

Cohomology of a differentiable stack: evaluation at a point

I'm reading these Behrend's notes on cohomology of stacks, and I can't get over a detail in the fourth page. Let $X_\bullet=(X_1\rightrightarrows X_0)$ be a Lie groupoid and let $\mathcal{N}$ be its ...
Kandinskij's user avatar
5 votes
1 answer
257 views

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

(I'd like to premise that I'm not an expert about these topics (just a student), so many of my doubts and perplexities are probably symptoms of my mathematical immaturity) I'm currently studying ...
Kandinskij's user avatar
15 votes
0 answers
185 views

Are Lie groupoids just groupoids internal to smooth manifolds?

It seems to be common to say "no" - but is this true? Two weeks ago I asked for a counterexample, but received no replies. To give some background, let's recall that the difference between ...
Konrad Waldorf's user avatar
7 votes
0 answers
114 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, $G_1 \;{}_s\!\times_t G_1$, ...
Konrad Waldorf's user avatar
1 vote
0 answers
29 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 ...
zxcv's user avatar
  • 131
6 votes
1 answer
386 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{...
Connor Grady's user avatar
4 votes
0 answers
119 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 ...
dennis's user avatar
  • 423
2 votes
1 answer
224 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 ...
Praphulla Koushik's user avatar
1 vote
1 answer
275 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 ...
Praphulla Koushik's user avatar
8 votes
1 answer
412 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 $\...
Josh Lackman's user avatar
  • 1,188
4 votes
1 answer
189 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 ...
J_P's user avatar
  • 439
0 votes
0 answers
111 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 $...
Doron Grossman-Naples's user avatar
2 votes
0 answers
146 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 ...
user40276's user avatar
  • 2,209
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 ...
Hao Yu's user avatar
  • 781
1 vote
0 answers
67 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 ...
Josh Lackman's user avatar
  • 1,188
3 votes
2 answers
535 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 ...
Praphulla Koushik's user avatar
2 votes
1 answer
214 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 ...
Žan Grad's user avatar
2 votes
1 answer
155 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 $...
Emilio Minichiello's user avatar
2 votes
1 answer
116 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 ...
Ben MacAdam's user avatar
  • 1,253
7 votes
2 answers
325 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+...
Adittya Chaudhuri's user avatar
5 votes
1 answer
174 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{...
Rylee Lyman's user avatar
  • 1,996
3 votes
2 answers
378 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 ...
Praphulla Koushik's user avatar
3 votes
0 answers
83 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 ...
user avatar
2 votes
1 answer
110 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 ...
Ben MacAdam's user avatar
  • 1,253
3 votes
0 answers
84 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 ...
Ben MacAdam's user avatar
  • 1,253
4 votes
0 answers
181 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 ...
Praphulla Koushik's user avatar
2 votes
1 answer
141 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 ...
Ben MacAdam's user avatar
  • 1,253
7 votes
1 answer
231 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,...
Dry Bones's user avatar
  • 321
1 vote
1 answer
278 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 ...
Adittya Chaudhuri's user avatar
6 votes
3 answers
629 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 ...
Adittya Chaudhuri's user avatar
2 votes
0 answers
86 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 ...
Praphulla Koushik's user avatar
7 votes
1 answer
350 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 ...
Praphulla Koushik's user avatar
3 votes
1 answer
159 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 ...
Praphulla Koushik's user avatar
8 votes
0 answers
202 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 ...
Arrow's user avatar
  • 10.4k
7 votes
3 answers
453 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 ...
Adittya Chaudhuri's user avatar
5 votes
0 answers
233 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 ...
Praphulla Koushik's user avatar
3 votes
0 answers
176 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 ...
Praphulla Koushik's user avatar
5 votes
1 answer
173 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 ...
Praphulla Koushik's user avatar
3 votes
3 answers
503 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\...
Praphulla Koushik's user avatar
9 votes
0 answers
255 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 ...
Arrow's user avatar
  • 10.4k
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 ...
Praphulla Koushik's user avatar
6 votes
1 answer
360 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 ...
Praphulla Koushik's user avatar
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 $\...
Praphulla Koushik's user avatar
4 votes
2 answers
302 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 "...
Praphulla Koushik's user avatar
5 votes
1 answer
281 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 ...
Alexander Schmeding's user avatar
2 votes
0 answers
154 views

Condition on a Lie groupoid to be represented by manifold/group or an action groupoid

Let $\mathcal{G}$ be a Lie groupoid. I am thinking of following questions. When do we know $\mathcal{G}$ is weakly/Morita equivalent to a Lie groupoid of the form $(G\rightrightarrows *)$ for some ...
Praphulla Koushik's user avatar
4 votes
1 answer
470 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}\...
Praphulla Koushik's user avatar