Mathematics

Questions tagged [topological-stacks]

21 questions
7
votes
2 answers

Relation between two notions of $BG$

The following is something that's always niggled me a little bit. I usually think about stacks over schemes, so I'm a bit out of my element—I apologize if I say anything silly below. Let $G$ be a sufficiently topological group (e.g. you can assume a…
Alex Youcis
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6
votes
0 answers

Stackification of finite categories

Assume I have a base category $\mathscr S$ with finite limits and a geometric morphism $\gamma :\mathscr S\to Fin$ into the category of finite sets (for example because $\mathscr S$ is positive coherent). Say I have also a topology $J$ on $\mathscr…
Nico
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4
votes
0 answers

Understanding Sheaves and Stacks

I am trying to understand the notion of sheaves and stacks. Intuitively, the former, sheaves are bit easy to understand as a gluing of compatible families of sets assign to opens sets of a topological space. In other words, it is a contravariant…
Bumblebee
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4
votes
1 answer

Proposition 4.1 in Vistoli's notes on descent

Proposition 4.1 of Vistoli's notes says $\mathsf{Top}^2$ is a stack over $\mathsf{Top}$. The proof starts by looking at the fibered product $\coprod_iU_i \times _U\coprod_iU_i$, however this object never comes up again in the whole proof. Why is it…
popo
  • 49
3
votes
1 answer

Grothendieck topology on stacks/fibred category

Let $p : \mathcal{S} \rightarrow \mathcal{C}$ be a stack (or fibred category, I do not know if just being a fibred category is enough) over the site $\mathcal{C}$ . I have read somewhere (I forgot where but I think on "Stacks project") that the site…
Mathemagician
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2
votes
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The stack classifying non-oriented triangles is equivalent to a quotient stack $[\widetilde{T}/S_3]$

I come across problems when I try to compute the stack classifying non-oriented triangles. My reference is the book Algebriac Stacks ( K. Behrend, B. Conrad, D. Edidin, B. Fantechi, W. Fulton, L. Göttsche & A. Kresch). Sorry I can't find the online…
Yining Chen
  • 95
2
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0 answers

"Category" of (pre)stacks over a fixed category/site

I am wondering if prestacks over a fixed category, or stacks over a fixed site, form an (interesting) category, equipped with prestacks morphisms . First of all, do they form a proper class or a set? Also, since sheaf categories on a topological…
W.Rether
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2
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Higher stacks and BG

I am wondering how one should think of the higher stacks $B^n(G)$? Here is what I mean: The stack $BG$ can be thought of as the quotient of a point by the group $G$ ($BG = */G$) in the proper category. It can also be represented by a groupoid $G…
unknownymous
  • 1,109
2
votes
1 answer

basic reference for stacks

I'd like a reference for a very basic definition of stacks Kashiwara's paper almost does it except for example writing such as for every open set U in the cover there is a cat C(U) so U --> C(U) is a functor to the category of categories?
Jim Stasheff
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2
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1 answer

How many sequences possible in stack , if the input(1,2,3,...,n) is in order?

How many permutations can be obtained in the output (in the same order) using a stack assuming that the input is the sequence $(1, 2, 3, 4, 5,\dots, n)$ in that order? Example If $n=5$, then outputs $3,4,5,2,1$ $1,2,3,4,5$ $5,4,3,2,1$ etc. are…
hululu
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2
votes
1 answer

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-trivial automorphism $\phi$ . Form a twisted family on…
John Gowers
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1
vote
1 answer

How to view a pullback of differentiable stack as representing isomorphisms of objects?

I am currently try to understand some notes on topological and differentiable stacks by J. Heineloth. Here, the pullback of stacks is defined at definition 2.1. Let $\mathcal{M} \stackrel{F}{\to} \mathcal{N} \stackrel{F'}{\leftarrow} \mathcal{M}'$…
Yadeses
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1
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1 answer

Why model categories in theory of stacks?

This question is bit related to my previous question about stacks. After understanding the definition of a stack (to be precise $(2,1)$ -sheaf), now I am wondering about $\infty$ -stacks. According to my understanding, next generalization of this…
Bumblebee
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1 answer

Minimal site that induces a stack from a psuedofunctor

I'm working with Vistoli's definition of stacks [ 1 ] Let $\Phi:C^{op}\to {Cat}$ be a psudeofunctor. Is there always a minimal Grothendieck topology on $C$ such that it induces a stack? If not, are the conditions on $C$ or $\Phi$ that one can impose…
Omer Rosler
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1
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$\mathsf{Top}^2$ is a stack over $\mathsf{Top}$?

In section 4.1 of Vistoli's notes he starts by showing we may locally construct arrows in $\mathsf{Top}^2$, i.e arrows over $U$. Then, the author says that we can moreover do the same for spaces , although this is more complicated. This is followed…
pop
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