# Visualization of an algebraic stack

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"

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 $$U$$

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.

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).
• 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 [...] Jul 22, 2019 at 11:23