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

I am interested in thinking visually about algebraic stacks (also higher and derived stacks, but let´s start from the beginning) and I find quite suprising the fact that there isn´t a single "picture" of an algebraic stack in the literature. I would expect some analogue to Mumford´s illustration of schemes for some example of algebraic stack but I cannot find it.

To be concrete, take moduli stack of vector bundles $\mathcal{M}$ (of fixed rank and chern class) on a scheme $S$. **How would you represent it graphically?**
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 bundles related by isomorphims. However the condition that makes the stack to be algebraic (and consequently a *geometric space*) is roughly (for an Artin stack) that the diagonal is representable, quasi-compact and separated and that it has a smooth cover by an scheme (the atlas). Following T. Gomez "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 $U$

It would be great if someone can clarify this and specially to offer some way to imagine visually an algebraic stak (a picture itself would be even better!)

representableby a scheme and that the projection map $U\times_{\mathcal{M}}V\rightarrow V$ issurjective (submersion)of schemes... Does this partially answer your question? $\endgroup$ – Praphulla Koushik Jul 22 '19 at 13:388more comments