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 schemes, 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).

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).

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 principle $GL_n$-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 schemes, 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).

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 schemes, 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).