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?
Best Answer
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.
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).
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.