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Questions tagged [topological-stacks]

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Simplicial manifolds which do not satisfy Kan condition locally?

Similar to Kan condition for simplicial sets, there are also Kan condition for simplicial manifolds, that is, we ask the horn projection $p^k_j: X_k \to Hom(\Lambda[k,j], X)$ to be a surjective ...
Chenchang Zhu's user avatar
5 votes
0 answers
96 views

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 ...
Nico's user avatar
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2 votes
0 answers
81 views

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. ...
Yining Chen's user avatar
0 votes
1 answer
35 views

Continuous map over some third space (trying to understand the stack)?

I am trying to understand the notion of stack and that is why I am reading http://homepage.sns.it/vistoli/descent.pdf whose p.70 (first paragraph of 4.1.1) contains the example about 2 mappings: $f:X \...
TomR's user avatar
  • 1,321
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1 answer
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In what sense stack (from category theory) is the (fibred) category as wikipedia asserts?

I am reading wikipedia article on the stacks https://en.wikipedia.org/wiki/Stack_(mathematics) and it contains assertion: The intuitive meaning of a stack is that it is a fibred category such that &...
TomR's user avatar
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0 votes
1 answer
74 views

Sheaf relative to a presheaf

I was looking at these notes https://arxiv.org/pdf/math/0503247.pdf and it keeps using the phrase "sheaf relative to a presheaf" (i.e. definition 3.3). What does this mean? An online search ...
Sofía Marlasca Aparicio's user avatar
1 vote
1 answer
152 views

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 ...
Bumblebee's user avatar
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4 votes
0 answers
544 views

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 ...
Bumblebee's user avatar
  • 18.3k
2 votes
1 answer
92 views

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 ...
Mathemagician's user avatar
2 votes
0 answers
97 views

"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? ...
W. Rether's user avatar
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Alexandroff compactification: continuous function extension

Let $(X, \mathcal{T})$ be a non compact topological space, $\infty \notin X$ and $(X^* := X \cup \{\infty\}, \mathcal{T}^* := \{U \subseteq X^*\mid U \cap X \in \mathcal{T} \land (\infty \in U \...
user avatar
1 vote
1 answer
39 views

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 ...
Omer Rosler's user avatar
2 votes
0 answers
255 views

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 ...
unknownymous's user avatar
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2 votes
1 answer
216 views

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 ...
Jim Stasheff's user avatar
4 votes
1 answer
183 views

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 ...
popo's user avatar
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1 vote
0 answers
50 views

$\mathsf{Top}^2$ 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, ...
pop's user avatar
  • 11
7 votes
2 answers
417 views

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 ...
Alex Youcis's user avatar
  • 54.4k
1 vote
1 answer
225 views

Differentiable stacks and morita morphism

I heard that if $[X_0/G_0]$ and $[X_1/G_1]$ are differentiable stacks, then any morphism between them is naturally equivalent to $$(G_0 \rightrightarrows X_0) \xleftarrow{\simeq} (G_2 \...
WWK's user avatar
  • 1,370
2 votes
1 answer
3k views

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 ...
Mithlesh Upadhyay's user avatar
2 votes
1 answer
474 views

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-...
John Gowers's user avatar
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