All Questions
Tagged with stacks at.algebraic-topology
24 questions
1
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0
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377
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G-local systems via the classifying stack BG
First, let $BG$ be the classifying stack of a Lie group $G$ in either Top or Diff (compactly generated topological spaces or differentiable manifolds). A map $f: X \to BG$ determines a principal $G$-...
user501319
25
votes
1
answer
3k
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Is there a ring stacky approach to $\ell$-adic or rigid cohomology?
Ever since Simpson's paper [Sim], it was observed that many different cohomology theories arise in the following way: we begin with our space $X$, we associate to it a stack $X_\text{stk}$ (which ...
- 711
5
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0
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190
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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 ...
- 1,761
3
votes
1
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576
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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 ...
- 18.7k
5
votes
0
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222
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"Strict" homotopy theory of topological stacks/orbifolds
If we fix a finite group $G$, there are two different useful homotopy theories on the set of $G$-equivariant topological spaces (which are CW complexes, say). One, the "weak" homotopy theory, is given ...
- 7,952
14
votes
2
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1k
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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 ...
- 928
10
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0
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414
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Segal-Freed-Hopkins-Teleman = Atiyah-Hirzebruch/Leray-Serre?
Freed-Hopkins-Teleman (section 3.7) generalise Segal's (Proposition 5.3) spectral sequence for equivariant K-theory to more general local quotient groupoids (that is, topological groupoids locally ...
- 35.4k
41
votes
10
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4k
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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 ...
- 1,720
3
votes
0
answers
227
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Mayer-Vietoris sequence for orbifolds
Is there a version of the Mayer-Vietoris long exact sequence for orbifolds? I am interested in orbifold homology as opposed to the homology of the underlying topological space.
- 965
7
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0
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312
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Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?
We are taught that, in general:
A type of objects that has nontrivial automorphisms cannot have a fine moduli space.
The proof generally goes along the lines of:
Take an object $X$ with a non-...
- 907
2
votes
1
answer
245
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Given a map of classifying spaces, can the target be described as a groupoid quotient of the source mod some action of some (co)kernel?
Let $H \to G$ be a homomorphism of affine algebraic groups (over characteristic $0$, if it matters). The case I care most about is when $H \to G$ is an inclusion. There is a corresponding map $f: \...
- 54.6k
3
votes
0
answers
303
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Finiteness of the connected components of a stack
Let $X$ be an algebaic stack over a scheme $S$, for any $S$-scheme $Y$ we can consider the groupoid $X(Y)$ of $Y$-points. Denote by $\pi_0(X(Y))$ the set of isomorphism classes of the groupoid.
Are ...
- 845
4
votes
2
answers
633
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Based loop groups as stacks?
I have been stuck for some time, thinking about the following question.
Let $G$ be a Lie group. Its classifying space $BG$ can be seen as the differentiable stack $[pt/G]$, which is of dimension $-...
- 123
9
votes
1
answer
1k
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Relation between $BG$ in topology and in algebraic geometry
This could as well have been asked in the comments to this question, but I prefer to open a new one for the sake of clarity.
Say $G$ is a reductive group over the complex numbers, with compact real ...
- 23.3k
6
votes
0
answers
529
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Nowhere vanishing section of vector bundles over varieties as connectivity of morphism of stacks
The following is, amongst others, a Hartshorne exercise:
Let $V$ be a $k$-variety of dimension $n$ and $\mathcal{E}$ a vector bundle of rank greater than $n$, then, generically, a generating section ...
- 2,297
3
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0
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502
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Analysis of Eilenberg-MacLane Stacks
In a series of three papers from the fifties, Eilenberg and MacLane did a pretty exhaustive study of what we now call "Eilenberg-MacLane spaces" and used a lot of machinery to do it, e.g. Whitehead's $...
- 10.4k
5
votes
2
answers
502
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How to specify a finite group up to inner automorphism?
I want some finite set of data to which I can canoically associate a "group up to inner automorphism", and which can be constructed canoically from a "group up to inner automorphism". I have a few ...
- 18.7k
17
votes
1
answer
2k
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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 ...
- 40.2k
7
votes
0
answers
433
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Quasi-coherent sheaves on $M_{FG}$ and the exact functor theorem
I'm struggling with these notes, and one of the things I don't really understand is the following. The notes consider the stack $M_{FG}$ of formal groups; this is the stack associated to the prestack ...
- 25.6k
10
votes
1
answer
2k
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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 $[X/G]$ ...
- 4,070
22
votes
3
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1k
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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 ...
- 15.5k
14
votes
2
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2k
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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 ...
- 18.2k
20
votes
2
answers
3k
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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....
- 23.3k
5
votes
1
answer
283
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how good an approximation to the equivariant derived category is given by the Grassmannian filtration of the classifying space?
So, let's say one has an action of $GL_n$ on an algebraic variety $X$ over a field $k$, and two objects $F,G$ in the equivariant derived category (i.e., the derived category of constructible sheaves ...
- 44.7k