Questions tagged [lie-groupoids]
The lie-groupoids tag has no usage guidance.
83
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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, $G_1 \;{}_s\!\times_t G_1$, ...
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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
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1
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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 ...
6
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1
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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
$$G(M) := \text{...
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1
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references to learn the general theory Lie $\infty$-groupoids and Lie $\infty$-algebroids
Kirill Mackenzie has a book on the general theory of Lie groupoids and Lie algebroids.
Is there such a reference for the general theory of Lie $\infty$-groupoids and Lie $\infty$-algebroids; that ...
3
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2
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Lie algebroid associated to a vector bundle
Let $E\rightarrow M$ be a vector bundle.
Kirill Mackenzie in the book General theory of Lie groupoids and Lie algebroids associates a Lie algebroid to $E\rightarrow M$ in the following steps:
talk ...
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Is every singular foliation induced by a Lie algebroid?
Let $M$ be a smooth manifold.
A smooth distribution $D$ on $M$ is the union of a family $\{D_p \leq T_p M : p\in M\}$ of vector spaces such that there is a family $\mathcal C $ of smooth vector fields ...
4
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Averaging over a Weinstein groupoid?
(Not sure if this question belongs here or on m.SE)
For a Lie group, $G$ (of dimension $n$), one can average over the group:
$$
\Gamma = \int_{G} d\mu(g) ~g
$$
(where $d\mu(g)$ is the left-Haar ...
3
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3
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475
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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
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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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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
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1
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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
...
7
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3
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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, ...
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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
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1
answer
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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 ...
8
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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 $\...
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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 ...
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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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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 ...
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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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0
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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 ...
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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 ...
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149
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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 $...
6
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3
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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
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2
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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+...
5
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1
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173
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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{...
3
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2
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342
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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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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
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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 ...
3
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83
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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 ...
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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 ...
2
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1
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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 ...
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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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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 ...
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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 ...
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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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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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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 ...
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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 ...
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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
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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
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1
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450
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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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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 ...
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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
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243
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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 ...
6
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1
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348
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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
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1
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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 $\...
4
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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
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1
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275
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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 ...
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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. ...