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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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 ...
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Categories of descent data
Let us work over the etale site $\mbox{Aff}/S$ (for the sake of definiteness) for some fixed base scheme $S$, where the covers are jointly surjective etale maps $\{ U_i \rightarrow U\}_{i\in I}$ (and $...
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What about stacks of categories in algebraic geometry? II
I've made this a new question, rather than expanding the first one.
Torsten gives a good answer, and it partially illustrates in practice the 'second approach' I outlined in my other question. (You ...
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Local structure of Deligne-Mumford stacks
Let $\mathcal{X}$ be a separated Deligne-Mumford stack over an algebraically closed field $k$ and let $X$ be the corresponding coarse moduli space, which we assume to exist. There is a map $p:\mathcal{...
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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 ...
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Is there a non-quotient stack with affine stabilizers whose good moduli space is a geometric point?
Definitions: One says that a map $\pi\colon\mathcal X\to X$ from an algebraic stack to an algebraic space is a good moduli space if $\pi$ is cohomologically affine and universal for maps to schemes. (...
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Maximal algebraic sub-groupoids
By a theorem of Ehresmann, topological and Lie categories (by which I mean categories internal to $Top$ and $Diff$ respectively, with the condition that the source and target in the latter case are ...
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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 ...
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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 $...
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$G_m$-cohomology of a motif (that corresponds to a stack?)
As in the question For a G-variety, what could one say about the motif of the corresponding simplicial variety
I am in the following situation: $G$ is an algerbraic group, and X is a smooth $G$-...
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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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Sheaves on stacks and interesting functors
Let $G$ be a finite group and $H \subset G$ a normal subgroup. Consider $G$, $H$, and $X=G/H$ as affine algebraic groups over some algebraically closed base field $k$.
I hear that there is an ...
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Rational points of an algebraic space over finite field
If $X$ is an algebraic space of finite type over a finite field $k$, then I think it is true that the set of $k$ rational points of $X$ is finite.
This is of course true for $X$ is a scheme. I wish ...
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Proving that a map is a morphism
Example : Consider the (open, not compactified) moduli space of stable maps $ \mathcal M_g(1,d)$ of maps of smooth curves of genus $ g$ to $\mathbb P^1$. To each map
we associate its branch divisor, ...
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Two questions on algebraic stacks
Question 1: The main reference on algebraic stacks (Laumon and Moret--Bailly) defines a separable algebraic stack as one having universally closed diagonal. For schemes separability is simply defined ...
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The Grothendieck plus construction for stacks of n-types
In Jacob Lurie's Higher Topos Theory, Section 6.5.3, he briefly mentions that to stackify a presheaf of $n$-groupoids, one needs to apply the "+"-construction $\left(n+2\right)$ times, and ...
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What is the signficance of the existence of a moduli stack to a moduli problem?
This is a question in pretty unfamiliar territory for me, so if I have conceptual mistakes please correct me.
Let's say we begin with a naive moduli problem: we want a moduli space (whatever space ...
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closed substacks of cartesian powers of a stack
Let $\mathbb{Z}/2\mathbb{Z}$ act on $\mathbb{A}^1$ as $x \mapsto -x$, and let $\mathscr{X}$ be the quotient stack. It has coarse moduli space $\mathbb{A}^1$ and a residual gerbe $B\mathbb{Z}/2\mathbb{...
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When do maps of ineffective orbifolds descend to their effective part?
If $$f:\mathscr{X} \to \mathscr{Y}$$ is a map between (possibly ineffective) orbifolds (in the sense of differentiable stacks, or orbifold groupoids), does it follow that $f$ induces a map between ...
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When is a category of groupoid schemes fibred over schemes?
The category of topological categories $Cat(Top)$ is fibred over $Top$ - the functor sending a groupoid $X_1 \rightrightarrows X_0$ to its object space $X_0$ is a Grothendieck fibration. Now one can ...
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What condition on a "bibundle between categories" generalizes "right-principal bibundle between groupoids"?
My question is long on background and motivation, and almost but not quite answered over at the nLab. I'll write up a bunch before asking my question (feel free to skip to the end or look at the ...
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Applications of Stacks
I've been aware of stacks since grad school, and I can usually follow in rough lines a discussion about stacks, but I've often wondered what particular (purely!) scheme-theoretic argument or theorem ...
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Weak colimits of weak and strict presheaves in groupoids
Let $C$ be a small category, and for this question, let groupoid mean an (essentially small) groupoid. There are two 2-categories in question: the 2-category of strict presheaves in groupoids and ...
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Is every (Artin/DM) algebraic stack fibered in sets an algebraic space?
If $X$ is an algebraic stack fibered in sets (and therefore essentially a sheaf), is it an algebraic space? It seems conceivable that at least when $X$ is Deligne-Mumford, it is actually an algebraic ...
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Can Deligne-Mumford stacks be characterized by their restriction to a small subcategory?
If I have a Deligne-Mumford stack $\Pi : X \to (\mathrm{Sch}/k)$ for some field $k$, can it be reconstructed from $\Pi^{-1}(C) \subset X$ for some "small" subcategory $C \subset (\mathrm{Sch}/k)$? For ...
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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 $[X/G]$ ...
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Who first came up with the idea of essential/Morita equivalence of internal groupoids/categories?
The idea that stacks can be identified with groupoids internal to the base site $S$ up to what is variously called essential/Morita equivalence is well known. The basic idea is that one takes the 2-...
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Fibered category with an adjoint inclusion
Suppose $X:D \to C$ is a fibered category (I do not assume the fibers to be groupoids). Suppose that $X$ is actually left adjoint to a fully faithful embedding $C \hookrightarrow D$. Is there a ...
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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 ...
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When is the K-theory presheaf a sheaf?
Let $F$ be a Deligne-Mumford stack that is of finite type, smooth and proper over $\mathrm{Spec~}k$ for a perfect field $k$. Consider $K_m$, the presheaf of $m$-th $K$-groups on $F_{et}$, the etale ...
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stacks as Morita equivalence classes
I have often encountered definitions of the kind "stacks are equivalence classes of groupoids under Morita equivalence" in topological or differentiable context, with the notion of Morita equivalence ...
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Are root stacks characterized by their divisor multiplicities?
Definitions/Background
Suppose $S$ is a scheme and $D\subseteq S$ is an irreducible effective Cartier divisor. Then $D$ induces a morphism from $S$ to the stack $[\mathbb A^1/\mathbb G_m]$ (a ...
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When is the base change morphism an isomorphism?
This is a rewrite of a previous question, which was in turn a follow up question to Leray-Hirsch principle for étale cohomology The motivation is to clarify some points in Torsten Ekedahl's ...
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Question about global quotient stacks
In "Brauer groups and quotient stacks", Edidin et. al prove the following theorem:
Theorem 2.7. Let $\mathcal{X}$ be an algebraic stack over a Noetherian base (of finite type). Then the diagonal $\...
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Noether-style isomorphism theorem for stacks?
Let $G$ be a group, actiong on a set $X$ and $H$ a normal subgroup.
Then we have a canonical isomorphism
$$(X/H)/(G/H)\rightarrow X/G$$
I would like to have a statement like this for stacks, more ...
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Seeing stacks in the Calculus of Functors
Recently I was told (by an algebraic geometer) that when algebraic geometers look at the Calculus of Functors, they think of stacks.
When I look at the Calculus of Functors, I see a categorification ...
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Twisted curves, admissible covers, and an algebraic analogue of a specific monodromy computation
This problem arose when trying to understand the stack of twisted stable maps into a stack (specifically BG), as introduced by Dan Abramovich, Angelo Vistoli and several co-authors (Olsson, Graber, ...
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Notion of stack fibered in monoidal categories?
This can be seen as a follow up my question here:
Is there a notion of "fibered category with boxproducts"?
Given a monoidal fibration $f:E\rightarrow B$
(i.e. a strict monoidal functor ...
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Applications of topological and diferentiable stacks
What are some examples of theorems about topology or differential geometry that have been proven using topological/differentiable stacks, or, some examples of proofs made easier by them? I'm well ...
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Stacks in the fpqc topology
This is related to Matt Satriano's earlier question about an analog of Artin's theorem for stacks with an fpqc cover by a scheme.
Suppose one developed the theory of stacks in the fpqc topology and ...
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Does the concept of a basis for a topology on a category exist?
If we want to define a sheaf F on a topological space X and we have a basis B for the topology of X, what we can do is to define objects and restrictions for guys in B, check that they satisfy the "B-...
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Universal property of X//G?
Given an operation of say a topological group on a topological space, one can form the quotient stack X//G:
the stack associated to the action groupoid.
Does this stack satisfy some kind of universal ...
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When is a stack (NOT) geometric?
Following the terminology of $n$-Lab, a geometric stack $\mathcal{X}$ on a site $\mathcal{(C,J)}$ is a stack for which there exists a representable epimorphism $X \to \mathcal{X}$ from an object $X$ ...
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Double Category of Topological Stacks
There are two equivalent ways of describing topological stacks.
One is the "stacky" definition, that is, a topological stack is a stack $\mathbb{X}$ on $Top$ (a Grothendieck universe thereof, if you'...
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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" ...
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Cotangent bundle of a differentiable stack
If you ever wanted to construct the tangent bundle of a differentiable stack, it's relatively simple:
First, if $\mathbf{X}$ is a stack coming from a Lie groupoid $\mathcal{G}$, you could just say $\...
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Ordinary cohomology of stacks
Let $\mathbf{X}$ be a stack over $Top$ (a lax sheaf of groupoids, or some such thing). If it admits a surjective representable map $F \to \mathbf{X}$ then one can form the iterated fibre product to ...
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Connections on principal bundles via stacks?
Let G be a Lie group and M a smooth manifold. Suppose that P is a principal G-bundle over M. Then by Yoneda, this corresponds to a smooth map $p:M \to [G]$, where $[G]$ is the differentiable stack ...
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finite etale covering of stacks
If $Y \to X$ is a finite etale map of schemes, then there exists a finite Galois morphism $Z \to X$ (i.e. it's a $Aut(Z/X)$-torsor) that factors as $Z \to Y \to X.$ The case when $X$ is normal is easy ...
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Stacks in the Zariski topology?
I have two naive questions about stacks.
1) Is it possible to define stacks in the Zariski topology?
Presuming you can:
2) If a stack has a coarse moduli, and the coarse moduli space is a ...
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