Questions tagged [stacks]

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

Filter by
Sorted by
Tagged with
65 votes
17 answers
16k views

Good introductory references on algebraic stacks?

Are there any good introductory texts on algebraic stacks? I have found some readable half-finsished texts on the net, but the authors always seem to give up before they are finished. I have also ...
Daniel Bergh's user avatar
  • 1,538
37 votes
1 answer
3k views

What about stacks of categories in algebraic geometry?

Stacks qua moduli spaces were introduced to keep track of nontrivial automorphisms of the objects they parameterize. In essence they are groupoids of objects with some form geometric cohesion. The ...
David Roberts's user avatar
  • 33.4k
16 votes
2 answers
3k views

Is the Torelli map an immersion?

The Torelli map $\tau\colon M_g \to A_g$ sends a curve C to its Jacobian (along with the canonical principal polarization associated to C); see this question for a description which works for families....
David Zureick-Brown'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
3 votes
1 answer
876 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
36 votes
3 answers
6k views

Conjectures in Grothendieck's "Pursuing stacks"

I read on the nLab that in "Pursuing stacks" Grothendieck made several interesting conjectures, some of which have been proved since then. For example, as David Roberts wrote in answer to ...
AAK's user avatar
  • 5,841
13 votes
1 answer
954 views

How is a Stack the generalisation of a sheaf from a 2-category point of view?

A stack is usually given in terms of: -A category $F$ fibered over another $C$ such that the functor $Hom(x,y), x,y \in F(\alpha), \alpha \in C$ is a sheaf -The descent data are effective. There ...
HaroldF's user avatar
  • 433
4 votes
1 answer
1k views

Pushout schemes/stacks

I would like to know in what generality do we have pushouts for schemes/stacks/derived stacks. More precisely, let $f : X \rightarrow Y$ be a proper flat surjective morphism of schemes of finite type ...
Libli's user avatar
  • 7,210
3 votes
2 answers
926 views

Understanding the definition of atlas of a stack over the category of manifolds

I am reading https://arxiv.org/abs/0806.4160 to understand orbifolds as stacks. Definition : Let $D\rightarrow Man$ be a stack over category of manifolds. An atlas for $D$ is a manifold $X$ and a ...
Praphulla Koushik's user avatar
2 votes
2 answers
509 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
40 votes
10 answers
4k views

Phenomena of gerbes

What is your favourite example of Gerbes? I would like to know Where do we find Gerbes in "nature"? The examples could vary from String theory to Galois theory. For example my favourite examples of ...
tttbase's user avatar
  • 1,700
31 votes
7 answers
3k views

Categorical construction of the category of schemes?

The answer to the following question is probably well known or the question itself is well known not to have a reasonable answer. In the latter case could you please let me know what the "right" ...
algori's user avatar
  • 23.2k
28 votes
2 answers
2k views

morphisms representable by algebraic spaces vs morphisms representable by schemes

So I've been working with moduli stacks in algebraic geometry for a while now, with no formal training in the technicalities of the theory of algebraic stacks (ie, I've read a few articles and I learn ...
stupid_question_bot's user avatar
20 votes
7 answers
3k views

What are the occurrences of stacks outside algebraic geometry, differential geometry, and general topology?

What are the occurrences of the notion of a stack outside algebraic geometry, differential geometry, and general topology? In most of the references, the introduction of the notion of a stack takes ...
Praphulla Koushik's user avatar
19 votes
2 answers
3k views

Why do gerbes live in H^2?

Line bundles on a scheme $X$ live in $H^1(X,O_X^*)$, where $O_{X}^{*}$ is the sheaf of invertible functions. If $X$ is noetherian separated, then we can think of this $H^1$ to be Čech cohmology w.r.t....
Qfwfq's user avatar
  • 22.7k
17 votes
1 answer
2k views

Homotopy theory of topological stacks/orbifolds

Motivation $\newcommand{\T}{\mathscr{T}}$ I have many times found myself saying some variant of the following. Let $\T_g$ be the Teichmüller space of a surface of genus $g$, and $\Gamma_g$ its ...
Dan Petersen's user avatar
  • 39.1k
17 votes
1 answer
3k views

In what topology DM stacks are stacks

Background/motivation One of the main reason to introduce (algebraic) stacks is build "fine moduli spaces" for functors which, strictly speaking, are not representable. The yoga is more or less as ...
Andrea Ferretti's user avatar
16 votes
3 answers
2k views

Is every algebraic space the quotient of a scheme by a finite group?

In this MO question it is claimed that a catchphrase for "algebraic spaces" could be that they are "the result of looking at the orbit space of the action of a finite group on a scheme". Hence my ...
Qfwfq's user avatar
  • 22.7k
15 votes
2 answers
5k views

Understanding the definition of the quotient stack $[X/G]$

I'm trying to understand the definition of the quotient stack $[X/G]$ as defined in Frank Neumann's Algebraic Stacks and Moduli of Vector Bundles. Explicitly, let $G$ be an affine smooth group $S$-...
Brian Fitzpatrick's user avatar
15 votes
2 answers
2k views

Why is lack of functoriality of the Lisse-Etale topology specific to the Lisse-Etale topology?

I'm trying to follow the explanation given in Olsson's "Sheaves on Artin stacks" for the lack of functoriality for lisse-étale topology: Let $f:Y \to X$ be a morphism of algebraic stacks. The functor $...
ykm's user avatar
  • 702
13 votes
1 answer
927 views

Several simple questions on the geometry of higher stacks

I'm trying to understand definition/work out some examples. So, there are some simple questions about higher stacks. For the simplicity assume that we are working with higher DM (Deligne-Mumford) ...
Klim Puhov's user avatar
13 votes
2 answers
1k views

Are non-algebraic stacks useful in algebraic geometry?

The title is a bit vague. What I want to know is if there is any geometric application of non-algebraic stacks. I know e.g. the category of coherent sheaves is an example. But I want to ask if people ...
36min's user avatar
  • 3,748
12 votes
1 answer
436 views

Why are non-singleton covering families often ignored?

It seems to me that frequently when discussing stack conditions and descent, people consider only singleton covering families, i.e. there is some single covering map $U\to X$, for which one constructs ...
Mike Shulman's user avatar
  • 64.8k
10 votes
1 answer
1k views

Exactly how is 'the diagonal is representable' used for algebraic stacks...

...apart from stating properties of $(s,t):X_1 \to X_0\times X_0$ for the a presenting algebraic groupoid $X_1 \rightrightarrows X_0$? Once we know that given a stack $\mathcal{X}$ we have a smooth ...
David Roberts's user avatar
  • 33.4k
10 votes
1 answer
484 views

What is the total space of a stack after all?

From my general experience I think for myself of what follows as some kind of taboo question for some reason: in my imagination, everybody wants an answer to this but somehow thinks it shall not be ...
მამუკა ჯიბლაძე's user avatar
10 votes
1 answer
2k views

Classifying stacks and homotopy type of a point

Suppose we are working in a category of schemes over a scheme $S$. The scheme $S$ itself is geometrically a ``point''. Let $G$ be a group scheme that acts on a scheme $X$. The quotient stack $[X/G]$ ...
Dima Sustretov's user avatar
10 votes
2 answers
2k views

Reference for Weighted Projective Stacks

For a sequence of positive integers $a_0, \ldots, a_n$ and a ring $R$, there is a graded ring $R[x_0,\ldots, x_n]$ where $x_i$ is in degree $a_i$. There is a corresponding $\mathbb{G}_m$-action on $...
Lennart Meier's user avatar
9 votes
1 answer
567 views

Singularities of the moduli stack of Calabi-Yau threefolds

Let $M$ be the moduli of polarized Calabi-Yau threefolds over $\mathbb C$ with fixed Euler characteristic. The coarse moduli space is singular (as usual), but what about the stack? In many cases I ...
El Nino's user avatar
  • 93
9 votes
1 answer
762 views

Sites which are stacks over themselves

A site C with pullbacks is subcanonical (all representable presheaves are sheaves) if and only if its codomain fibration $Arr(C) \to C$ is a prestack (all hom-presheaves are sheaves). Is there a ...
Mike Shulman's user avatar
  • 64.8k
9 votes
2 answers
1k views

Derived topological stacks?

I apologize for the vagueness of the following. Informally, in the site of commutative rings, one roughly get the notion of a derived stack by swapping out the commmutative rings with its subcategory ...
zzz's user avatar
  • 868
9 votes
1 answer
741 views

"Approximating" $BGL(1)$ by projective spaces

Given a representation $V$ of a group $G$, we can think of $V$ as a vector bundle over the classifying stack $BG$, and we can define its index $\chi(BG; V)$ to be the dimension of the $G$-invariant ...
Kevin H. Lin's user avatar
  • 20.7k
8 votes
1 answer
508 views

Stacks over diffeologies

Konrad Waldorf shows in his paper one may realize a Grothendieck topology on the category of diffeological spaces. Is there any work exploring stacks over the category of diffeologies?
Seth Wolbert's user avatar
8 votes
2 answers
552 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
7 votes
1 answer
428 views

Formal completion of a quotient stack

$\newcommand{\Rep}{\operatorname{Rep}}$ $\newcommand{\mo}{\operatorname{-mod}}$ $\renewcommand{\hat}{\widehat}$ I apologize in advance if this is a naive question but my background in algebraic ...
Adrien's user avatar
  • 8,234
7 votes
1 answer
2k views

The fibre product of two quotient stacks

My question is to know whether the fibre product of $[X/G]$ by $[Y/H]$ over a base scheme is $S$ is $[X \times_S Y/G \times H]$? And if yes, do you have any reference for it? Thank you.
Kimra's user avatar
  • 131
7 votes
0 answers
500 views

Is the category of quasi-coherent sheaves on a concentrated stack locally finitely presentable?

Let's call an Artin stack $X$ concentrated iff it is quasi-compact and quasi-separated (the latter usually being included in the definition of an Artin stack). The category of quasi-coherent sheaves $\...
Martin Brandenburg's user avatar
7 votes
0 answers
582 views

Understanding the higher stack of perfect complexes

One of the most famous examples of higher Artin stacks is the stack of perfect complexes. I recall here the basic stuff: We fix a function $b: \mathbb{Z} \rightarrow \mathbb{N}$ which is zero ...
Martin Hurtado's user avatar
6 votes
1 answer
542 views

Commutative group algebraic spaces

Is the category of commutative group algebraic spaces (commutative group objects in algebraic spaces) locally of finite type over a field, an abelian category? I would benefit from a reference
user avatar
6 votes
0 answers
290 views

Is there a Geometric/Smooth version of Homotopy Hypothesis using the path $\infty$-Groupoid of a Smooth Space?

A version of Homotopy Hypothesis says that the Fundamental $n$-grupoids model Homotopy $n$-types... and if we continue upto $\infty$, then the Fundamental $\infty$- groupoids or Kan Complexes model ...
Adittya Chaudhuri's user avatar
6 votes
1 answer
249 views

Question regarding 2-mathematics: Can you stackify a 2-functor without prestackifying it first?

Let $C$ be a site and $CAT$ the 2-category of categories. Given a contravariant 2-functor $A:C\rightarrow CAT$, we can of course consider the associated stack. This is done by first considering the ...
Modnar's user avatar
  • 63
4 votes
1 answer
533 views

Intrinsic Characterization of when an orbifold (or more general stack) is effective?

Recall that an orbifold is an etale and proper differentiable stack $X$. Etale means that it admits an etale atlas $M \to X$ from a manifold $M$ (which is to say it is represented by an etale Lie ...
David Carchedi's user avatar
4 votes
2 answers
563 views

Algebraic stacks as (étale) groupoid algebraic spaces/schemes

Assume given an algebraic stack(*) $\mathcal{X}$ with presentation $X_0 \to \mathcal{X}$, and the corresponding groupoid $X = (X_0\times_\mathcal{X} X_0 \rightrightarrows X_0)$ in algebraic spaces (or ...
David Roberts's user avatar
  • 33.4k
4 votes
1 answer
352 views

Is a gerbe over a manifold is a special case of a gerbe over a stack?

There is a notion of Gerbe over a Manifold and a notion of Gerbe over a stack. Given a manifold $M$, there is a way to associate a stack $\underline{M}$ with it and this gives an embedding of ...
Praphulla Koushik's user avatar
4 votes
0 answers
522 views

Questions about the algebraic space $\mathbb{A}^1/\mathbb{Z}$

Let $X = \mathbb{A}^1_{\mathbb{C}}/\mathbb{Z}$, where $\mathbb{Z}$ acts on $\mathbb{A}^1$ via translation. [To clarify, $X$ is an \'etale sheaf with a smooth presentation $\mathbb{A}^1_{\mathbb{C}}\to ...
Bernd's user avatar
  • 161
4 votes
1 answer
912 views

Do algebraic stacks satisfy fpqc descent?

It is known, thanks to Gabber, that algebraic spaces are sheaves in the fpqc topology: Stacks project 03W8 Is the analogous statement for algebraic (Artin) stacks true? If not, is it true under ...
beginner's user avatar
3 votes
0 answers
176 views

"standard limit arguments" involved in showing that roughly every DM stack is locally a quotient stack

I am trying to understand proposition 3.6 of this paper (perhaps I am in over my head): https://arxiv.org/pdf/math/0703310.pdf If we denote the stack $\mathcal{M}$ and its coarse moduli space as $M$ ...
Fred's user avatar
  • 121
3 votes
1 answer
530 views

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
  • 18.3k
3 votes
1 answer
670 views

References for constructible sheaves on complex analytic stacks

I'm looking for references on constructible sheaves and the six operation formalism on analytic stacks (stacks fibered over complex analytic spaces). Does anyone have some suggestions? Basically I ...
shenghao's user avatar
  • 4,195
2 votes
2 answers
381 views

Cohomological description of gerbes over stacks

When understanding about gerbe over a manifold $X$ from Hitchin - Lectures on special Lagrangian submanifolds it is said that We are basically in gerbe territory (for smooth manifolds) if any one ...
Praphulla Koushik's user avatar
2 votes
1 answer
470 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