Questions tagged [lie-groupoids]
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81
questions
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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
$$G(M) := \text{...
- 163
4
votes
0
answers
95
views
Averaging over a Weinstein groupoid?
(Not sure if this question belongs here or on m.SE)
For a Lie group, $G$ (of dimension $n$), one can average over the group:
$$
\Gamma = \int_{G} d\mu(g) ~g
$$
(where $d\mu(g)$ is the left-Haar ...
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2
votes
1
answer
195
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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 ...
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1
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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 ...
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8
votes
1
answer
369
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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 $\...
- 1,073
3
votes
1
answer
171
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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 ...
- 337
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votes
0
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89
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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 $...
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2
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0
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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 ...
- 2,139
4
votes
1
answer
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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 ...
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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 ...
- 1,073
3
votes
2
answers
352
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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 ...
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2
votes
1
answer
187
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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
...
- 61
2
votes
1
answer
137
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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 $...
2
votes
1
answer
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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 ...
- 1,243
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votes
2
answers
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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+...
- 1,185
5
votes
1
answer
170
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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{...
- 1,946
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votes
2
answers
310
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 ...
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3
votes
0
answers
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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 ...
user108998
2
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1
answer
107
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"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 ...
- 1,243
3
votes
0
answers
79
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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 ...
- 1,243
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0
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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 ...
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2
votes
1
answer
135
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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 ...
- 1,243
7
votes
1
answer
209
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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,...
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1
vote
1
answer
244
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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 ...
- 1,185
6
votes
3
answers
568
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 ...
- 1,185
2
votes
0
answers
79
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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 ...
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6
votes
1
answer
268
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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 ...
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3
votes
1
answer
148
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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 ...
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8
votes
0
answers
195
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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 ...
- 10.2k
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3
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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 ...
- 1,185
3
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0
answers
210
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 ...
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0
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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 ...
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4
votes
1
answer
168
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 ...
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votes
3
answers
431
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\...
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0
answers
237
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 ...
- 10.2k
3
votes
0
answers
236
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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 ...
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6
votes
1
answer
323
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 ...
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2
votes
1
answer
237
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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 $\...
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2
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284
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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 "...
- 5,807
5
votes
1
answer
263
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 ...
- 1,898
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votes
0
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148
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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 ...
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3
votes
1
answer
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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}\...
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votes
1
answer
152
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Requirement that source and target maps are surjective submersions
Definition I am aware of for Lie groupoid is that (among other things) the source and target maps $s,t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ are submersions.
On page 9 of Du Li's thesis Higher ...
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2
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636
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Yoneda Embedding and pull back
Given a manifold $M$ we have a geometric stack associated to it namely $\underline{M}$ whose objects are smooth maps to $M$. For the sake of consistency I am writing $BM$ for $\underline{M}$.
Given a ...
- 5,807
2
votes
2
answers
487
views
Fibered product of stacks comes from a Lie groupoid
I am adding some context here. I am reading Introduction to Differentiable Stacks by Gregory Ginot.
In page no $7$, just before the remark $2.2$ he says the following.
One shall be careful that ...
- 5,807
4
votes
1
answer
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unit element under map of stacks $B\mathcal{G}\rightarrow B\mathcal{H}$
Let $\mathcal{G}$ be a Lie groupoid. The target map $t:\mathcal{G}_1\rightarrow \mathcal{G}_0$ is a principal $\mathcal{G}$ bundle.
This article Orbifolds as Stacks? by Eugene Lerman calls (in page $...
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vote
1
answer
308
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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 ...
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5
votes
1
answer
357
views
In what sense bibundles are called as generalized morphisms
Definition : Let $\mathcal{G}$ and $\mathcal{H}$ be Lie groupoids. A bibundle from $\mathcal{G}$ to $\mathcal{H}$ is a manifold $P$ together with two maps $a_L:P\rightarrow \mathcal{G}_0,a_R:P\...
- 5,807
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0
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Idea behind definition of classifying space over an orbifold
Today I was explaining to some one the notion of $\mathcal{G}$ spaces, covering spaces over orbifolds from Orbifolds as Groupoids: an Introduction.
Definition : Let $X$ be a locally compact ...
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1
answer
316
views
Proper and etale groupoid is locally a translation groupoid
I am reading Orbifolds as Groupoids: an Introduction by Ieke Moerdijk.
In page $8$ when explaining local charts, it says the following :
Let $\mathcal{G}$ be a Lie groupoid. For an open set $U\...
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