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I am interested in thinking visually about algebraic stacks (also higher and derived stacks, but let´s start from the beginning).

As the visuallization of an algebraic stack is virtually impossible I warn about this is a soft question!

I do not really know if I have understood correctly the definition of an algebraic stack. Take for example the moduli stack of vector bundles $\mathcal{M}$ on a scheme $S$. As it is an algebraic stack there should exist a scheme $U$ and a smooth and surjective morphism $u:U \rightarrow \mathcal{M}$. This scheme $U$ is the atlas and it is supposed to be a smooth cover of $\mathcal{M}$.

1) Is this atlas $U$ and unique scheme or a union of schemes?

2) The way I usually imagine $\mathcal{M}$ visually is that the fiber over a point of the base scheme is not a single vector bundle but a set of vector bunldes related by isomorphims. Is this true? However following the following "Introduction to Algebraic Stacks" atlas of a stack

I understand from this that if you have a family of isomorphic vector bundles $V_{ki}$ in each point $k$ (where $i=1,2,3,...$ denotes different isomorphic vector bundles over the same point $k$) the scheme $U$ is going to contain at least one $V_{ki}$ for at least one value of $i$ in each point of $U$ enter image description here

But, how can you say that $\mathcal{M}$ is parametrized by $U$? You cannot have all the possible different (although isomorphic) vector bundles $V_{ki}$ just moving along $U$! It would be great if someone can provide some kind of picture or graphical explanation.

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1) No the atlas is not unique. After all the defn of an atlas is simply a scheme with a smooth (or etale if you want a DM stack) surjective map to your stack. You can for instance pick an atlas, and take another scheme with a smooth surjective map to your scheme.

2) I'm not sure if that's how I would 'picture' $\mathcal{M}$ - how would your picture work for - say $BG = [pt/G]$? Personally I picture $\mathcal{M}$ as a black box which carries a universal family, and where I can 'probe' its geometry by considering maps of schemes $S\rightarrow \mathcal{M}$.

Alternatively if it's given by a quotient stack (which tends to be the case), say $[X/G]$, then I can also work $G$-equivariantly on $X$. One can also take the coarse moduli space and remember where the stacky locus is (parametrizing objects with automorphisms).

Maybe this isn't too satisfying for geometric intuition..

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  • $\begingroup$ Thank you! I do not really see why my picture couldn´t work for $BG$... You would substitute $k_{i}$ for $\left \{ pt \right \}$ and $V_{ij}$ for $G_{k}$ for each topological group $G_{k}$. The "picture" you propose in 2) is certainly "not geometric enough" to my taste (mine is not either!) because it is more like a function than like a "space". Maybe this is a key aspect of the discussion. Algebraic stacks are supposed to be geometric objects because some properties like smoothness, irreducibility, separatedness, properness etc could be studied on them [...] $\endgroup$ Jul 22, 2019 at 11:23
  • $\begingroup$ [...] but I would expect that due to this geometricity they could be kind of drawn, not in a unique figure, because basically each of its points carries structure and we cannot visualize such a "dense" space, but maybe with a set of pictures. It is remarkable that I haven´t seen any picture like the ones of neverendingbooks.org/mumfords-treasure-map for an algebraic stack $\endgroup$ Jul 22, 2019 at 11:31
  • $\begingroup$ PS: I am going to ask the same question in Math Overflow and see if I find any other insight... $\endgroup$ Jul 22, 2019 at 11:34

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