# Tag Info

Accepted

### Understanding the definition of stacks

A canonical example of a sheaf of sets on a topological space $X$ is the sheaf that sends an open subset $U$ of $X$ to the set of continuous real-valued functions on $U$. The gluing property then says ...
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### Stacks as local quotients or via atlases

I'm not entirely sure if this qualifies as an answer, but it is certainly too long for a comment. I hope that someone else will give a better answer. If you want to define an algebraic stack as ...
Accepted

### Stack being represented by a scheme/manifold

If all objects of a stack have trivial automorphism groups then it is equivalent to a sheaf, as pointed out by Daniel Litt in the comments. Pick your favourite non-representable sheaf as a ...
• 34.7k
Accepted

### Morita equivalence of Lie groupoids and isomorphism of differentiable stacks

The "well-known fact" is simply not true if you assume "isomorphic stacks" means literally isomorphic (say as fibred categories). My impression is that people who work in certain ...
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Accepted

### What does it mean for a space to be a differentiable stack?

Stacks form an (∞,1)-category. The latter informal notion has many equivalent implementations: simplicial category, topological category, quasicategory (also known as ∞-category), Segal category, ...
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Accepted

### Anafunctors vs the plus construction

The long-expected answer. $\DeclareMathOperator{\op}{op} \DeclareMathOperator{\Cat}{\mathbf{Cat}}\DeclareMathOperator{\Gpd}{\mathbf{Gpd}} \DeclareMathOperator{\disc}{disc}\DeclareMathOperator{\pr}{pr}$...
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### What is the local structure of a general Artin stack?

The stack of curves of genus 0 with at most one node is a quotient stack (see The integral Chow ring of the stack of at most 1-nodal rational curves, but Edidin and Fulghesu), so you are fine in this ...
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### Categorifying the definition of a principal $G$ bundle

A modern presentation that fully covers the indicated cases can be found in the work of Nikolaus–Schreiber–Stevenson: Principal ∞-bundles – General theory. Principal ∞-bundles – Presentations. In ...
• 36.7k
Accepted

### Examples of of gerbe over stacks in terms of manifolds

There are no other such gerbes. If $M$ and $N$ are manifolds, and $p\colon \underline{M}\to \underline{N}$ is a gerbe, then the corresponding map of manifolds is a diffeomorphism. The same holds if ...
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Accepted

• 34.7k
1 vote
Accepted

### Understanding definition of gerbe over a stack

I am trying to write down what does it mean to say those two maps $\mathcal{D}\rightarrow \mathcal{C}$ and $\mathcal{D}\rightarrow \mathcal{D}\times_{\mathcal{C}}\mathcal{D}$ to be epimorphisms. I am ...
• 5,983

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