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3
votes
1answer
77 views

Understanding the stack $B\mathbb{Z}$

Here, let $\mathbb{Z}$ be the group scheme whose functor of points is the constant functor which takes a connected affine scheme to the group $\mathbb{Z}$. I'm having a bit of trouble understanding ...
4
votes
1answer
52 views

Examples of algebro-geometric moduli problems without a “natural” choice of pullback?

When I try to learn about stacks, something that is mentioned all over the place (as a reason not to define moduli problems as functors from schemes to groupoids) is the fact that the pullback of a ...
7
votes
1answer
66 views

Three meanings of étale sheaf on X

When I am studying stacky stuffs, I am always confused by the notion of ├ętale abelian sheaves on $X$, because conceivably there might be three different meanings of that: Take the global ├ętale site ...
0
votes
0answers
23 views

What are symmetric monoidal stacks?

Let $\mathcal{C}$ be a site and $\mathcal{S}$ be a stack over $\mathcal{C}$ such that every fiber $\mathcal{S}_C$ has a symmetric monoidal structure. What compatibility conditions should one impose ...
2
votes
1answer
83 views

How does a section of a stack give a sheaf?

At nLab in the article constant stack and a few other related articles, a pattern is mentioned where a section of a constant sheaf is a locally constant function, a section of a constant stack is a ...
3
votes
1answer
42 views

Stacks versus sheaves with values in categories

A (small) category is a perfectly valid algebraic structure like Groups, Rings, vector spaces, groupoids etc. So on a topological space or more generally on a site, it makes perfectly sense to ...
1
vote
0answers
33 views

Stack on commutative ring spectra?

One approach to stacks to call a stack a "sheaf of groupoids" which means a functor $$ \mathcal{C}^{\text{op}} \rightarrow \mathcal{G} $$ from a category $\mathcal{C}$ with a Grothendieck topology to ...
7
votes
1answer
88 views

Homology of stack points

This is a very basic question about how definitions in homology carry over to the easiest example of stacks. Let $G$ be a finite cyclic group. Consider the classifying stack $\mathcal{B}G$. This has a ...
3
votes
1answer
178 views

Reference for Deligne-Mumford

What is a good reference for someone new to the theory of Deligne-Mumford stacks, other than the original Deligne-Mumford paper itself? The paper itself seems readable with some effort; but the fear ...