All Questions
8 questions
6
votes
1
answer
373
views
How to differentiate natural transformations?
Let $G_{\bullet} = (G_1 \rightrightarrows G_0)$ and $H_{\bullet} = (H_1 \rightrightarrows H_0)$ be Lie groupoids and $\varphi_\bullet, \psi_\bullet : G_\bullet \to H_\bullet$ be Lie groupoid morphisms,...
- 325
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 ...
- 6,023
3
votes
0
answers
84
views
stack (in groupoids) over a site $\mathcal{C}$
Question : What is a stack (in groupoids) over a site $\mathcal{C}$ for you?
There are two a ways to think about it.
A stack over a site $\mathcal{C}$ is a category $\mathcal{D}$ with a functor $\...
- 6,023
8
votes
1
answer
281
views
Stack associated to Lie group and manifold
Given a Lie group $G$, we have a Lie groupoid $(G\rightrightarrows *)$ and stack $BG=B\mathcal{G}$ of principal $G$ bundles.
Given a smooth manifold $M$, we have Lie groupoid $(M\rightrightarrows M)...
- 6,023
3
votes
0
answers
156
views
Criterion for a sheaf $\mathfrak{S}^{op}\rightarrow (Set)$ to be representable
I am reading Differentiable stacks and gerbes by Kai Behrend and Ping Xu.
Let $\mathfrak{S}$ denote the category of smooth manifolds and smooth maps. Consider Grothendieck topology given by open ...
- 6,023
12
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 ...
- 6,023
2
votes
0
answers
192
views
Diagonal is representable then composition is representable
Let $\mathcal{X}$ be a stack over $S$ i.e., a stack over category of schemes over $S$ (which we denote by $Sch/S$) which comes with a functor $\mathcal{X}\rightarrow Sch/S$. Consider the diagonal map ...
- 6,023
3
votes
1
answer
1k
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 (...
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