Questions tagged [lie-groupoids]

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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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72 views

Algebraic K-theory of enveloping algebras and PBW-algebras - a reference request

Let for simplicity $k$ be a field of characteristic zero, let $A$ be a finitely generated $k$-algebra which is regular and let $\alpha: L\rightarrow \operatorname{Der}_k(A)$ be a Lie-Rinehart algebra (...
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232 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+...
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1answer
143 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{...
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2answers
201 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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47 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 ...
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1answer
89 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 ...
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64 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 ...
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157 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 ...
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1answer
98 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 ...
4
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1answer
115 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,...
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1answer
186 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 ...
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3answers
342 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 ...
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55 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 ...
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1answer
146 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 ...
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68 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 ...
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169 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 ...
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3answers
318 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 ...
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103 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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59 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 ...
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1answer
121 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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3answers
306 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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195 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 ...
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212 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 ...
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1answer
243 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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1answer
213 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 $\...
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2answers
265 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 "...
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1answer
185 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 ...
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135 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 ...
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1answer
319 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}\...
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1answer
112 views

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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2answers
483 views

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 ...
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2answers
396 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 ...
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1answer
212 views

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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1answer
240 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 ...
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1answer
271 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\...
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142 views

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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1answer
256 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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2answers
635 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. ...
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243 views

Necessity/Motivation for generalised homomorpisms

I am reading Ieke Moerdijk's article "Orbifolds as Groupoids : an Introduction". In that notes author defines a notion of generalized map between Lie groupoids. Let $\mathcal{G}$ and $\mathcal{H}$...
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1answer
274 views

Isotropy group of a Lie groupoid is a Lie group

I am trying to see that Isotropy group/object group/vertex group of a Lie groupoid is a Lie group. Let $\mathcal{G}$ be a Lie groupoid and $x$ be an object in $\mathcal{G}$ i.e., $x\in \mathcal{G}_0$...
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0answers
499 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}...
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5answers
1k 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 ...
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1answer
154 views

Lie Groupoid of a Transitive Lie algebroid

If A is an integrable and transitive Lie Algebroid, and G is a corresponding Lie groupoid, then: is G necessarily transitive too? I guess it is not generally true, but I wonder under which conditions ...
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846 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 ...
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131 views

Action of a Lie groupoid on a Lie Algebroid?

Let $\pi:E\longrightarrow M$ be a vector bundle. Then we can associate a Lie groupoid $\mathsf{Gl}(E)\rightrightarrows M$ where $$\mathsf{Gl}(E):=\{E_x\stackrel{lin. isom.}{\longrightarrow} E_y: x, y\...
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1answer
325 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?
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1answer
212 views

Gluing together together differentiable stacks

I am trying to figure out the conditions under which you can glue together a collection of (differentiable) stacks by equivalences, and get a differentiable stack. More precisely, I have a collection ...
4
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1answer
188 views

A-Paths as morphisms of Lie Algebroids $TI\longrightarrow A$?

In the paper Integrability of Lie Brackets Marius Crainic and Rui Fernandes describe obstructions to integrate a Lie algebroid to a Lie groupoid. The process of integration relies on the construction ...
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171 views

On Holonomy in (regular) Riemannian Foliations

Right now, I am trying to understanding the role of holonomy fields on Riemannian foliations, which lead me to the following (probably topological) groupoid: Let $\mathcal{F}\subset M$ be a ...