Basic questions about stacks

Question

I'm trying to understand some basics of stacks in algebraic geometry and have three questions:

1) As far as I understand, the moduli stack of vector bundles over a scheme $X$ is a replacement for the non-existent moduli space of vector bundles over $X$. Is this the only reason for the study of this stack or isn't it actually important to remember the isomorphisms between vector bundles?

2) What about replacing schemes by manifolds and define moduli stacks in a similar way? Does it make sense to talk about about stacks in the "euclidean" topology on the site of open subsets of euclidean spaces? Of course it makes sense, but I'm wondering if there is any literature about it. For example, is there a "generalized manifold" $BGL_n$ such that for every manifold $X$ we have an equivalence of categories between $Hom(X,BGL_n)$ and the category of vector bundles on $X$?

3) The objects of algebraic geometry have made a great evolution in the 20th century. Projective varieties, schemes, algebraic spaces, stacks. Is there an "upper bound" of this process of abstraction? I think that each abstraction was motivated by concrete geometric problems, but it might be argued if we actually solve these problems just by enlarging the category of geometric objects in consideration. This leads to the vague question: What are the fundamental ideas of algebraic geometry which will hopefully survive the next abstraction?

Answer 1

1. The notion of fine moduli space requires the existence of a universal family. In this case, you want a scheme $M$ equipped with a rank $n$ vector bundle $V$ on $M \times X$, such that pullback induces a natural bijection between the set of maps from any other scheme $Y$ to $M$ and the set of rank $n$ vector bundles on $X \times Y$. You can view $V$ as a family of vector bundles on $X$, parametrized by $M$. Vector bundles of positive rank do not admit universal families, in part due to the existence of automorphisms (and the existence of schemes with nontrivial fundamental group that can act as bases of nontrivial families). I don't have a precise grasp on what your question is asking, but depending on the application of choice, one can sometimes work with a coarse moduli space (which is roughly a way to ignore automorphisms), and one can sometimes rigidify the moduli problem to get a natural cover of the stack by a scheme. If $X$ is a general scheme (instead of, e.g., a point or a projective curve) the stack of vector bundles is unlikely to be algebraic, and nether simplifying option looks promising.

2. My wild guess is that you intend the Euclidean site to be an analogue of the category of affine schemes of finite type. You can form a notion of stack in topological spaces, by following the usual fibered category route, and you can certainly restrict to the subcategory of open subsets of Euclidean space. As Johannes Ebert mentioned in the comments, Noohi has some papers online that describe topological stacks. Some names that show up in the smooth setting include Alan Weinstein, Cristian Blohmann, and Chenchang Zhu (but I am relatively unfamiliar with this area).

3. Right now, there is an upper bound on the information content in mathematical abstraction given by the finite size of the human brain. Even if the robots take over, there is the finite size of the observable universe. More to the point at hand, objects more abstract than stacks were already considered in algebraic geometry during the 20th century. For example Grothendieck's Pursuing Stacks is one of the early attempts to apply homotopy theory techniques to work with more abstract objects like $n$-categories and $n$-stacks. I am not qualified to answer your revised question about fundamental ideas.

Answer 2

2) I'm not sure what you are asking with the first half of the question. Are you talking about the category of subsets of a fixed Euclidean space with unique inclusions, or are you talking about a more general category, where 'the same' subset can be included multiple ways? In the second example, you need to specify when two subsets of Euclidean space are 'the same', usually by picking a structure group which acts on Euclidean space (and hence its subsets). In either case, I believe that there is a Grothendieck topology and stacks exist. In the first case (unique inclusions), I think that stacks are the same as set-valued sheaves on Euclidean space, since the lack of interesting automorphisms means any fiber category can be replaced by its set of equivalence classes. In the second case, you should get the category of orbifolds with that structure group, and maybe some other objects as well.

The second half of your question is correct. There is a classifying stack $BGL_n$ such that $Hom(X,BGL_n)$ is the same as the category of vector bundles. Tautologically, this is the stack which assigns to every scheme $X$ the groupoid of vector bundles on that scheme, or equivalently the groupoid of principal $GL_n$-bundles. The latter statement is better for generalizing to an arbitrary group. Classifying stacks are neat examples of stacks, because they can be covered by a single closed point. In fact, once you have the terminology to make it precise $$ BG = pt/G $$ where $pt$ is a closed point with trivial $G$ action (here, I am assuming we are working over $\mathbb{C}$-schemes or differentiable manifolds, so that a closed point makes sense).

3) It's likely that any attempt to say that some mathematical gadget is the ultimate version of a long line of development will look silly in a few years. However, there is a sense in which stacks are the upper-bound, because of how stacks are defined. Stacks are the universal kind of object which contains the moduli space of every moduli problem.

Contrast this with the other geometric objects you list, which are all constructions. When something is defined as a construction, you hope you got all examples, but you have to check. Stacks don't have this problem, because in a sense, they are just a formal language in which to write down the questions. When studying a moduli problem with stacks, you immediately get a stack and the real work is in saying anything meaningful about that stack (representability, algebricity, etc).