Questions tagged [stacks]

In mathematics a stack or 2-sheaf is a sheaf that takes values in categories rather than sets.

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connections on Lie groupoids/differentiable stacks

Let $(\Gamma_1\rightrightarrows \Gamma_0)$ be a Lie groupoid. There are many places which define the notion of connection on a Lie groupoid. As far as I have seen, there is no mention of these ...
Praphulla Koushik's user avatar
3 votes
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Reference request: Derived structure on the moduli stack of Higgs bundles

I am reading arXiv:1708.08124. When talking about the moduli stack of Higgs bundles on a projective curve $X$. It is said on page 59, first paragraph that It is often better to put derived ...
chan kifung's user avatar
3 votes
2 answers
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Do disjoint unions of stacks commute with finite fibre products?

Choose a big $\mathit{fppf}$-site $(\mathbf{Sch})_{\mathit{fppf}}$ and let $S$ be a scheme in that site. Let $\{\mathcal{X}_i\mid i\in I\}$ be a family of stacks in groupoids over $S$ and let $\...
sdigr's user avatar
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Integrality of Atkin-Lehner operator for $\Gamma_1(N)$

A result due to B. Conrad (http://math.stanford.edu/~conrad/papers/prasanna-inv.pdf, Theorem A.1) states that the Atkin-Lehner operator $w_{Q,k}$ is $\mathbb{Z}[1/Q]$-integral on $M_k(\Gamma_0(N))$. ...
Daniel Johnston's user avatar
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basic question on quotient stacks

Let $X$ be a scheme over $S$, and $G$ be an affine group scheme over $S$ acting on $X$. This Wikipedia article (or also this related MO question) defines a quotient stack $[X/G]$ as a category of ...
quasi-mathematician's user avatar
5 votes
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Limit of quotient stacks

Let $k$ be a field (we can set it to be either perfect or algebraically closed if necessary), let $G$ be a (split) reductive group over $k$. Let $(X_i)$ be a filtered projective system of finite type $...
Arnaud ETEVE's user avatar
3 votes
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Smoothness of the stack of Shtukas without modifications

It is a well-known fact, that the stack of shtukas $\mathop{\mathcal{S}\!\mathit{ht}_{\pm 1}^1}$ with two legs and elementary modifications (i.e. those of type $(1,0,\dotsc,0)$ or $(0,\dotsc,0,-1)$) ...
sdigr's user avatar
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Does $\text{Bun}_G$ have the homotopy type of a classifying space in positive characteristic?

In these lecture notes by Jacob Lurie, he identifies the homotopy type of $\text{Bun}_G$ with that of a certain classifying space $B\mathcal{P}_{sm}$ when the group scheme $G$ is over $\mathbb{C}$. ...
xir's user avatar
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Compactly supported cohomology of a topological DM stack

Consider a separated topological Deligne-Mumford stack $\mathfrak X$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently, $\mathfrak X$ is locally ...
Mohan Swaminathan's user avatar
4 votes
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What is the local structure of a general Artin stack?

Let $X$ be an Artin stack over the complex numbers. What can one say about the local structure of $X$, i.e. what is the simplest class of stacks by which were can always find a cover of $X$ by open ...
John Pardon's user avatar
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Stacks as local quotients or via atlases

If one looks up the definition of a Deligne--Mumford stack or an Artin stack, one usually finds something like: A DM (resp. Artin) stack is a stack $X$ satisfying [insert condition in the diagonal ...
John Pardon's user avatar
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3 votes
3 answers
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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\...
Praphulla Koushik's user avatar
2 votes
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Reference for calculating the dimension of algebraic stacks

I am interested in the dimension of algebraic stacks.But, I’m in trouble because I can’t calculate it.Are there any good reference or tools for calculating it? I use the “Champs algébriques” by ...
Walter field's user avatar
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Spaces of Bordered (Ordinary, Spin- or Super-) Riemann Surfaces

It is known from the work of Deligne and Mumford that the "space" of punctured/marked Riemann surfaces is a Deligne-Mumford stack. I have few questions regarding similar statement for the spaces of ...
QGravity's user avatar
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$C^*$-algebras appearance in study of Lie groupoids and differentiable stacks

I am reading Differentiable stacks, gerbes, and twisted K-Theory by Ping Xu. To talk about (twisted) K-theory of differentiable stacks, author introduced (page $41$) the set up of $C^*$-algebras. All ...
Praphulla Koushik's user avatar
3 votes
0 answers
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Connectedness for stacks

Let $X$ be a stack for the Zariski (or etale) site over an arbitrary field $k$. The functor $\pi_0(X) : Alg_k \to Sets$ of path-components of $X$ is defined as the composition $$Alg_k \overset{X}{\...
Exit path's user avatar
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4 votes
1 answer
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Reducing the stack condition (descent condition) over an fpqc site to the case of single coverings

This is the lemma 4.25 of Vistoli's note Let $S$ be a scheme, $\mathscr{F} \to \mathscr{S}ch/S$ a fibred category. Then $\mathscr{F}$ is a stack over the fpqc site on $S$ iff (1) $\mathscr{F}$ ...
k.j.'s user avatar
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2 votes
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Significance of some expected results when defining Grothendieck topology

Let $\mathcal{C}$ be a category. Fixing an object $U$ of $\mathcal{C}$, there are some obvious functors we can associate to it, for example: $h_U:\mathcal{C}^{op}\rightarrow \text{Set}$ given by $V\...
Praphulla Koushik's user avatar
2 votes
1 answer
471 views

Stack associated to Groupoid object in category $\text{Sch}/S$

Consider the category of manifolds $\text{Man}$. A groupoid object in the category of manifolds is called a Lie groupoid, denoted by $\mathcal{G}$. There is a way to associate a stack (over the ...
Praphulla Koushik's user avatar
3 votes
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stack (in groupoids) over a site $\mathcal{C}$

Question : What is a stack (in groupoids) over a site $\mathcal{C}$ for you? There are two a ways to think about it. A stack over a site $\mathcal{C}$ is a category $\mathcal{D}$ with a functor $\...
Praphulla Koushik's user avatar
1 vote
1 answer
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Reference requence: scheme of complete homomorphisms of rank $r$ via blowups

I'm reading these notes where it states in section $3$: (transcribed because I can't post image) Step 1. Introduce the stacks of degenerated and iterated shtukas which extends that of shtukas. This ...
edgarlorp's user avatar
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Prerequisites for understanding algebraic geometry of “algebraic gerbes”

I am trying to learn about algebraic geometry of gerbes. I am familiar with set up of gerbes in the case of differential geometry. Though there is some similarity between differentiable gerbes and ...
Praphulla Koushik's user avatar
8 votes
0 answers
173 views

Smooth sub-orbifolds in the language of stacks

In most geometric categories, "monomorphism" is too general to describe useful notions of "embedding". This is the case e.g. for schemes, complex manifolds, and differentiable manifolds. So "embedding"...
Qfwfq's user avatar
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Road map for moduli space/moduli problem/moduli stack

I am familiar with (most of the) contents of Angelo Vistoli's notes on Descent theory (Stacks). I am also comfortable with basics of Schemes, their Cohomology (Cech), from Hartshorne's Algebraic ...
Praphulla Koushik's user avatar
2 votes
1 answer
353 views

Local question and descent category for a quasi-coherent sheaf on $\mathbb{G}_m$-gerbe

Update: I removed what I thought was unecessary and tried to be more straightforward in the hope to get an answer. Context: Suppose I have a $\mathbb{G}_m$-gerbe $\mathcal{G}$ over a scheme $X$ with ...
FelixBB's user avatar
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Stacks in moduli spaces of sheaves research

I want to get some practice and build more appreciation for the use of stacks in the context of classical moduli spaces of sheaves. Here by classical I vaguely mean hands-on description of the ...
Bananeen's user avatar
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Surjectivity of pushforward on Chow rings for stacks

Let $f:X\rightarrow Y$ be a proper morphism of smooth Deligne-Mumford stacks of finite type over $\mathbb{C}$ that is birational, but not flat. The coarse spaces of $X$ and $Y$ are both not smooth. Is ...
Samir Canning's user avatar
15 votes
2 answers
2k views

Which definition of "proper" is better?

It is well known that topology and algebraic geometry assign different meanings to the word "proper". Let us recall the relevant definitions from topology (and we work in the context of topological ...
John Pardon's user avatar
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6 votes
1 answer
385 views

Etale sheaves on algebraic spaces vs. Etale sheaves on affines

Let's fix a field $k$. First, consider $Aff_k$ to be the category of affine finite type $k$ schemes. On this category, one can define the etale topology and thus consider the site $Aff_k^{et}$, then ...
Anette's user avatar
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11 votes
0 answers
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Visualization and new geometry in higher stacks

I am trying to develop a geometrical intuition for "higher spaces", i.e. both in the sense of higher dimensional spaces (more than three dimensions) and in the sense of abstractions beyond ...
Martin Hurtado's user avatar
4 votes
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279 views

MSRI Workshop videos and lecture notes

I am referring to MSRI workshop "Introductory Workshop on Algebraic Stacks, Intersection Theory, and Non-Abelian Hodge Theory". There are videos and lecture notes available which can be seen on ...
Praphulla Koushik's user avatar
8 votes
1 answer
274 views

Stack associated to Lie group and manifold

Given a Lie group $G$, we have a Lie groupoid $(G\rightrightarrows *)$ and stack $BG=B\mathcal{G}$ of principal $G$ bundles. Given a smooth manifold $M$, we have Lie groupoid $(M\rightrightarrows M)...
Praphulla Koushik's user avatar
1 vote
2 answers
657 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 ...
Praphulla Koushik's user avatar
6 votes
1 answer
309 views

Sheaves over a sheaf

Everything I write I mean in the in the sense of Lurie's HTT. Suppose that $ \mathcal{C}$ be a site and let $ F \in Fun( \mathcal{C}^{op} , \mathcal{S})$. Is it always/ever true that $ Sh(\mathcal{C}...
Anette's user avatar
  • 585
2 votes
1 answer
212 views

Central extension gives a gerbe over stack

Consider a central extension of Lie groups $1\rightarrow S^1\rightarrow \hat{G}\xrightarrow{\pi} G\rightarrow 1$. I understand that this mean $\pi:\hat{G}\rightarrow G$ is a surjective homomorphism ...
Praphulla Koushik's user avatar
9 votes
0 answers
195 views

Does the category of $G$-equivariant sheaves have enough injectives?

The question is related to this one. Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$. Let $G$ be a topological group which ...
Zhaoting Wei's user avatar
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7 votes
1 answer
947 views

Understanding the definition of $G$-gerbe

In Introduction to Differentiable Stacks Gregory Ginot defines a $G$-gerbe as the following. Let $G$ be a Lie group. A $G$-gerbe over a stack $\mathcal{C}$ is a gerbe over stack $\mathcal{D}\...
Praphulla Koushik's user avatar
2 votes
2 answers
511 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 ...
Praphulla Koushik's user avatar
3 votes
0 answers
150 views

Criterion for a sheaf $\mathfrak{S}^{op}\rightarrow (Set)$ to be representable

I am reading Differentiable stacks and gerbes by Kai Behrend and Ping Xu. Let $\mathfrak{S}$ denote the category of smooth manifolds and smooth maps. Consider Grothendieck topology given by open ...
Praphulla Koushik's user avatar
2 votes
1 answer
279 views

Extend Group Action of $\mathbb{A}^1 /G$ to Projective Line

My question refers to an argument used in Torsten Ekedahl's paper: https://arxiv.org/abs/0903.3148 in Example ii) (page 8): We consider a finite subgroup of affine transformation of $\mathbb{A}^1$. ...
user267839's user avatar
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2 votes
0 answers
152 views

Is there a definition of an unpointed schematic homotopy type?

In the paper Champs Affines (http://www.math.univ-toulouse.fr/~btoen/chaff.pdf) Toen introduces pointed schematic homotopy types (SHTs) to solve Grothendieck's schematization problem (described in ...
Patrick Elliott's user avatar
13 votes
4 answers
2k views

Motivation for definition of Quotient stack

I am reading "Some notes on Differentiable stacks" by J. Heinloth. In that paper, the notion of quotient stack is defined as follows. Let $G$ be a Lie group action on a manifold $X$ (left ...
Praphulla Koushik's user avatar
16 votes
1 answer
1k views

GAGA for stacks

I am curious about stacky generalizations of the following GAGA theorem: If $X, U$ are complex algebraic varieties of finite type, $X$ is proper and $f:X\to U$ is an analytic map then $f$ is ...
Dmitry Vaintrob's user avatar
4 votes
1 answer
230 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 $...
Praphulla Koushik's user avatar
4 votes
1 answer
520 views

Stack being represented by a scheme/manifold

On page $10$ of the survey article Algebraic stacks, by T. Gomez (arXiv:math/9911199), we have following result If a stack has an object with an automorphism other than the identity, then the ...
Praphulla Koushik's user avatar
8 votes
2 answers
553 views

$2$-fiber product is a scheme then map of stacks is representable

Ariyan Javanpeykar said here in comments that, $X\times_{\mathcal{X}}X$ being a scheme is equivalent to representability of $X\rightarrow \mathcal{X}$. Context is as in this question. Suppose $p:...
Praphulla Koushik's user avatar
2 votes
0 answers
186 views

Diagonal is representable then composition is representable

Let $\mathcal{X}$ be a stack over $S$ i.e., a stack over category of schemes over $S$ (which we denote by $Sch/S$) which comes with a functor $\mathcal{X}\rightarrow Sch/S$. Consider the diagonal map ...
Praphulla Koushik's user avatar
3 votes
1 answer
881 views

Diagonal is representable then any morphism is representable

Ariyan Javanpeykar said here in comments that, If the diagonal is representable, then isn't any morphism $S\rightarrow \mathcal{X}$ with $S$ a scheme representable? I could not find the statement (...
Praphulla Koushik's user avatar
4 votes
0 answers
1k views

English translation of G.Laumon, L.Moret-Bailly book Champs algébriques

Is there an English translation of G.Laumon, L.Moret-Bailly book Champs algébriques. Most questions on this site on stacks received this book as reference in comments/answers. So, I want to ask if ...
Praphulla Koushik's user avatar
5 votes
1 answer
717 views

$BG$ the stack, $BG$ the simplicial presheaf

I have a theoretical question about comparing two objects that I have recently come across. For concreteness, let us work over the category $C$ of schemes over $k$. Let $G$ be an algebraic group over ...
HuynA's user avatar
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