# Visualization of an algebraic stack

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!)

• I do not know why do you think (or where it is mentioned) that $\mathcal{M}$ is parametrized by $U$... It only says that there exists a scheme $U$ and a morphism of stacks $U\rightarrow \mathcal{M}$, satisfying some properties... By Yoneda Lemma (I can say more here if you want), this morphism $U\rightarrow \mathcal{M}$ corresponds to the category $\mathcal{M}(U)$, which by definition is the collection of vector bundles over the scheme $U$... – Praphulla Koushik Jul 22 '19 at 13:10
• I am trying to understand your question... Are you asking why $u:U\rightarrow \mathcal{M}$ is not considered to be surjective? I understand you want to visualize stack.. – Praphulla Koushik Jul 22 '19 at 13:24
• I understand that the name "surjective" is misleading... Actual name is "representable surjective (submersion)"... Experts use surjective to mean representable surjective (submersion)... It means, the map $u:U\rightarrow \mathcal{M}$ is such that, for any map $v:V\rightarrow \mathcal{M}$, the product $U\times_{\mathcal{M}}V$ is representable by a scheme and that the projection map $U\times_{\mathcal{M}}V\rightarrow V$ is surjective (submersion) of schemes... Does this partially answer your question? – Praphulla Koushik Jul 22 '19 at 13:38
• I do not have hope for some one saying about picture of $\mathcal{M}$.. I will be more than happy to see if there is any.. :) – Praphulla Koushik Jul 22 '19 at 14:08
• "Nice" stacks over an algebraically closed field $k$ are locally of the form $[X/G]$ for an affine scheme $X$ and a linearly reductive $k$-group scheme $G$: see arxiv.org/abs/1504.06467. You can picture this by drawing the $G$-orbits on $X$ and decorating points with the stabilizers. – dorebell Jul 23 '19 at 5:46