# what is $\mathrm{Bun}(G)$?

I don't even know where to begin. There's a discussion of stacks and they talk about $\mathrm{Bun}(G)$. I don't know what it is, or what it's elements are or why it is important. Google and wikipedia don't really help since they pre-suppose.

One resource says $\mathrm{Bun}(G)$ is the moduli stack of $G$-bundles where $G$ is an affine algebraic group over a field $k$.

• the embedding $G \to GL_n$ induces a morphism of stacks $\mathrm{Bun}_G \to \mathrm{Bun}_{{GL}_n}$
• $\mathrm{Bun}_G$ is depends on the space $X$, $\mathrm{Bun}_G(X)$ is a groupoid
• $\mathrm{Bun}_G$ is a functor, meaning that it is well behaved under maps of spaces. A map $Y \to X$ yields another "induced" map: $$\mathrm{Bun}_G(X) \to \mathrm{Bun}_G(Y)$$

I believe $X$ is a scheme; projective, connected smooth curve over a field $k$, not necessarily algebraically complete. $G$ is an affine algebraic group over $k$.

What are the étale or fppf topologies here?

The only projective curves I can think of are Riemann surfaces, which are often smooth and connected. The field $k$ could be $\mathbb{C}$ but possibly $\mathbb{Q}$ or a number field. My idea of a "bundle" is something with local trivialization and around each loop you assign an element of $G$; gluing copies of $U_i \times G$ with transition maps. Over a number field, perhaps we lose the analytic topology.

The embedding theorem maybe says we can encode the gluing information using $n \times n$ matrices (a representation) and functorality says a map between schemes should "lift" to a map between bundles.

So these moduli stacks should be spaces of $n \times n$ matrices with various restrictions. They may themselves form a variety of some kind (certainly a scheme).

As you can see I am totally clueless.

• If you are working over a space $X$, then the points should be principal $G$-bundles over $X$ and the 1-cells should be isomorphisms of principal $G$-bundles. The way to present this object very much depends on which field you are working in / how you want to use it. – Daniel Barter Apr 26 '17 at 18:03
• As a special case, consider the bundle $\mathbb{R} \times G$. The automorphism group of this thing is the loop group ${\rm maps}(\mathbb{R},G)$ – Daniel Barter Apr 26 '17 at 18:06
• Why are you interested in this topic? As you have noticed, there is a lot to say about this. It might help to understand what perspective you are coming from. For instance you might only need the case where the base field is $\mathbb C$, or you might only need $G=GL_n$, where one can work with vector bundles, which are conceptually and practically simpler. – Will Sawin Apr 26 '17 at 18:29
• I think this question is not for the site since it does not fall within the remit. But I'd like to see it stay because I do not know, and would like to see a list of good sources for learning more about it (I have looked previously and had the same problem, namely that the literature is enormous). – Stiofáin Fordham Apr 26 '17 at 19:05
• For me personally it was helpful to view this $\mathrm{Bun}_G$ as an algebro-geometric counterpart/model of the classifying space of $G$ that is so important in homotopy theory. So these are various forms/analogs of the Grassmanian. In some cases I've seen (e. g. in Beilinson-Drinfeld or Ben Zwi-Frenkel) it really is a(n affine) Grassmanian and can be realized as an ind-scheme. For $G=GL_n$ one, probably oldest and simplest possibility is to consider it as something "almost equal to" $GL_{n+N}/(GL_n\times GL_N)$ or something similar, for "very large" $N$. – მამუკა ჯიბლაძე Apr 26 '17 at 19:25

I am surprised that nobody has mentioned (yet) that an essential point of the algebro-geometric notions of (algebraic) stack and algebraic space is to completely shift the burden of construction problems: one gives up on trying to make any kind of actual ringed space at all, and in fact the whole point is to create a kind of "geometry for functors". In particular, Bun$_G$ is demoted to the status of a definition (there's no "space" to be constructed: one just defines a certain groupoid-valued functor) and the real content is to show that it is "near enough" to representable functors that one can import to it many concepts from algebraic geometry.

It is easiest to explain this with an example. Let's contrast the approaches to the "Hilbert scheme" for a projective scheme $X$ over a field $k$ (to fix ideas). In the original approach of Grothendieck, he defined a functor $\underline{\rm{Hilb}}_{X/k}$ on the category $k$-schemes, namely $\underline{\rm{Hilb}}_{X/k}(S)$ is the set of finitely presented closed subschemes $Y \subset X \times S$ such that the structure map $Y \rightarrow S$ is flat; this is a contravariant functor of $S$ via pullback (i.e., for any $f:S' \rightarrow S$ over $k$, the map $$\underline{\rm{Hilb}}_{X/k}(f): \underline{\rm{Hilb}}_{X/k}(S) \rightarrow \underline{\rm{Hilb}}_{X/k}(S')$$ carries $Y \subset X \times S$ to $Y \times_S S' \subset (X \times S) \times_S S' = X \times S'$). The real substance in this approach is to show $\underline{\rm{Hilb}}_{X/k}$ is a representable functor. This amounts to constructing a "universal structure": a distinguished $k$-scheme $H$ equipped with an $H$-flat finitely presented closed subscheme $Z \subset X \times H$ such that for any $k$-scheme $S$ and any $Y \in \underline{\rm{Hilb}}_{X/k}(S)$ whatsoever there exists a unique $k$-map $g:S \rightarrow H$ for which the $S$-flat closed subscheme $$Z \times_{H,g} S \subset (X \times H) \times_{H, g} S = X \times S$$ coincides with $Y$. In effect, $k$-maps to $H$ "classify" the functor $\underline{\rm{Hilb}}_{X/k}$.

But Grothendieck's representability result was proved via methods of projective geometry, and there were strong indications that one should seek a result like this more widely for proper $k$-schemes (beyond the projective case) yet nobody could find such a way to do this in the framework of schemes. Artin brilliantly solved the problem by widening the scope of algebraic geometry via his introduction of algebraic spaces, but it must be appreciated that in so doing one gives up on constructing a "representing space" in any sense akin to what Grothendieck does. In particular, algebraic spaces are not ringed spaces at all. Instead, they are Set-valued functors which are "near enough" to representable functors that it becomes possible to make sense of many concepts from algebraic geometry for these structures (and there is an associated topological space permitting one to speak of irreducibility, connectedness, etc.).

The crucial point is that in the Artin approach, the functor is the algebraic space. That is, in Artin's approach it is a complete misnomer to speak of "constructing the moduli space" for the Hilbert functor $\underline{\rm{Hilb}}_{X/k}$ for a proper $k$-scheme: the functor one has defined above literally is to be the algebraic space, so there is nothing to be constructed there. Instead, the real work is to build the representable functors "near enough" to this so that one can conclude that the functor one has already defined is indeed an algebraic space (and hence one can do meaningful geometry with it, after getting acclimated to the new setting).

Artin's breakthrough was to identify checkable criteria with which one could show (entailing some real work) that many functors of interest are "near enough" to representable functors to be algebraic spaces (and his PhD student Donald Knutson devoted his thesis to working out very many concepts of algebraic geometry in the wider setting of algebraic spaces, often entailing new kinds of proofs from what had been done for schemes). But even with $\underline{\rm{Hilb}}_{X/k}$ for projective $X$ there is a fundamental difference in the nature of the results: Grothendieck built the Hilbert scheme ${\rm{Hilb}}_{X/k}$ as a countably infinite disjoint union of quasi-projective schemes, so one knows the connected components are quasi-compact, whereas the Artin approach gives no information whatsoever on quasi-compactness properties for the algebraic space $\underline{\rm{Hilb}}_{X/k}$ (but of course it has the huge merit of being applicable far more widely, allowing to "do algebraic geometry" with many many more functors of interest; nonetheless, Grotendieck's quasi-compactness results for Hilbert schemes remain vital as a tool for proving quasi-compactness features of rather abstract algebraic spaces).

As another illustration, consider the functor $M_{g,n}$ of $n$-pointed smooth genus-$g$ curves. In the Mumford approach via GIT (say with $n$ big enough to get rid of non-trivial automorphisms), there is tremendous effort done to build a universal structure. In the Artin approach via stacks (which allows $n$ to be quite small too), the moduli problem is the stack: it is incorrect to speak of "constructing the moduli space" in this approach, since one has literally defined $M_{g,n}$ and there's all there is to it (granting that one is sufficiently fluent with descent theory and coherent duality to see at a glance that $M_{g,n}$ enjoys nice descent properties). But instead the real effort is to build appropriate "scheme charts" over $M_{g,n}$ to give precise meaning to a sense in which $M_{g.n}$ is "near enough" to representable functors that we can make sense of many concepts of algebraic geometry for this functor or for its groupoid-valued version when $n$ is small.

The situation with Bun$_G$ is similar: the groupoid assignment is the geometric object. Techniques of Artin provide a precise sense in which some schemes can be regarded as "smooth" over this gadget, with those used to define geometric concepts for Bun$_G$ in the same spirit as one uses open balls to define concepts for manifolds, but it is rather a misnomer to speak of "constructing Bun$_G$"! That is, one has to clearly distinguish the serious task of describing some "charts" on Bun$_G$ (with which to make computations and prove theorems) from the much more mundane matter of simply defining what Bun$_G$ is: it is the assignment to any $X$ of the groupoid of $G$-bundles on $X$, and the descent-like properties are easy to verify (so it is a "stack in groupoids"), and there is nothing more to do as far as "making the moduli space" is concerned. Of course, one cannot do anything serious with this until having carried out the real effort with Artin's criteria to affirm that this stack admits enough "scheme charts" to be an Artin stack and hence admit the rich array of meaningful concepts for it as in algebraic geometry (irreducible components, coherent sheaf theory, cohomology, smoothness properties, dimension, etc.) There is no gluing to be done: we define the global moduli problem at the outset. We have to do work to show this admits meaningful geometric concepts, and in practice there can be especially useful "scheme charts" or techniques or "local description" with which one can explore it. However, one really should not regard these explicit descriptions as "constructing Bun$_G$"; the construction is just the initial basic definition mentioned above. Another wrinkle is that in the algebro-geometric setting one has to use some serious Grothendieck topologies to make this all work, so it isn't as simple as with making manifolds by gluing open balls (likewise for the notion of $G$-bundle).

Of course, there are interesting and instructive analogies with constructions in homotopy theory, but to make coherent sense of actual algebro-geometric proofs involving Bun$_G$ one should recognize that this "space" is demoted to the status of a definition (in complete contrast with the setting for Hilbert schemes by Grothendieck, where he had to really build a representing scheme before geometry could be done) and the substantial effort is to build many kinds of interesting "scheme charts" over it with which to explore the geometric features of this stack. The framework of Artin stacks gives a systematic way to make sense of this process for many kinds of moduli problems encountered in practice. But at the end of the day stacks and their cousins are not ringed spaces (even though their structure can be explored using many auxiliary ringed spaces); e.g., even though there is an associated topological space with which to define various topological notions, in no sense is a map determined in terms of some kind of map of ringed spaces resting on those associated topological spaces (in contrast with the cast of schemes). The way one gets around the loss of contact with a specific ringed space is by a huge amount of descent theory. I think it is very important to always keep that in mind or else a lot of relevant issues in definitions and proofs will be rather confusing.

• This would also serve as an answer to some much more general question about the relationship between schemes (which I think of as "level n" of AG abstraction for some small n, affine algebraic varieties over C being level 0) and stacks, algebraic spaces, and the functorial viewpoint on AG ("level n+1")...and it's great. I'd up vote multiple times if I could! – Joshua Grochow Apr 28 '17 at 4:20
• @JoshuaGrochow: 2) yes, that was my point. - 1) I agree, one important feature of the FOP is that when you have a moduli problem what Nature gives you is indeed a functor/pseudofunctor/fiberedcategory, and you can already think of it "tautologically" as some sort of space (no need of representability proofs!). (...) – Qfwfq Apr 30 '17 at 19:46
• (...) On the other hand: A) you can also say an analogous thing about, say, $\mathrm{Spec}(R)\mapsto \mathrm{GL}_n(R)$, or $S\mapsto\mathrm{Hilb}_X^n(S)$, B) as soon as you want to prove something is algebraic, you have to construct an atlas i.e. a presentation as a groupoid in schemes, C) also, many moduli problems admit a (not very intrinsic) description as global quotient stacks: here you don't need the F.o.P.. – Qfwfq Apr 30 '17 at 19:46
• @JoshuaGrachow: see also the very good answer by Angelo here mathoverflow.net/questions/77110/… (if you only have a FOP -i.e. don't have algebraicity- then it's more difficult to make "geometry" with it) and by TorstenEkedahl here: mathoverflow.net/questions/56962/… . Sorry for the long chain of comments! – Qfwfq Apr 30 '17 at 20:25
• @JoshuaGrochow: The purpose of Artin's criteria is to precisely to provide a list of checkable conditions sufficient to ensure existence of the required "scheme charts" (e.g., etale or smooth cover by a representable functor, in a suitable sense of such kind of "cover"). For this, the geometric substance of the definition of the given functor/groupoid needs to be understood. One can also sometimes get by without Artin's criteria, using that other things are known to be algebraic spaces/stacks. But it is technically important to recognize that one is not trying to make a ringed space. – nfdc23 May 1 '17 at 2:01

If $G$ is an affine algebraic group, there is a classifying stack $BG$. For any scheme $S$, you have a groupoid $BG(S)$ whose objects are principal $G$-bundles $P \to S$ over $S$ (locally trivial with respect to some chosen topology). That is, they are schemes $P$ equipped with a $G$-action, admitting a cover $\{S_i \to S \}$ of $S$ such that the base change of $P$ to each $S_i$ is equivariantly isomorphic to $G \times S_i$.

If you fix some variety $X$, you can define the stack $Bun_G(X)$ as the Hom stack $\underline{\operatorname{Hom}}(X, BG)$. For any scheme $S$, the objects of the groupoid $Bun_G(X)(S)$ are principal $G$-bundles on $X \times S$.

Sometimes, the variety $X$ is fixed in the beginning of a discussion, and it is left out of the notation. Then you will see $Bun_G$ instead of $Bun_G(X)$. Naturally, this may be a source of confusion.

One way to construct $Bun_G(X)$ for $X$ a projective curve is to choose a finite set of points on $X$ such that any $G$-bundle can be trivialized on the complement of those points. Then, choose trivializations of your $G$-bundles on formal neighborhoods of those points and on the complement. In some cases (e.g., $G$ semisimple), this is possible with one point. Then the gluing data give you a double quotient $Bun_G(X) \cong G(X-p) \backslash G((t)) / G[[t]]$.

You can think of the points of $\mathrm{Bun}_G(X)$ as the set of maps $$X\longrightarrow \mathrm{B}G$$

where the $\mathrm B G$ is the classifying "space" (stack) for principal $G$-bundles in the algebraic category (i.e. it classifies algebraic principal $G$-bundles over schemes) or, if you don't like schemes, you can do this story in the category of analytic spaces, as we will do in the following paragraphs.

Actually, in the formalism of stacks, the points of $\mathrm B G$ do not form just a set of points but a groupoid: specifically, you think of $\mathrm B G$ as the action groupoid associated to the trivial action of $G$ on the point $*$. That's a perfectly legitimate groupoid internal to the category of analytic spaces, which means its set of objects and set of morphisms are actually analytic spaces - even complex manifolds in this case. You can also think of $X$ as a groupoid: the action groupoid of the trivial group acting on $X$. Since $X$ and $G$ are nice smooth complex manifolds, you don't even really need to think about non-reduced analytic spaces until you need to describe infinitesimal properties of $\mathrm{Bun}_G(X)$ (for example, maps from non-reduced irreducible zero-dimensional spaces -"fat points"- are enough to determine if $\mathrm{Bun}_G(X)$ is smooth).

Maps $X\to \mathrm B G$ are just maps of (analytic) groupoids, and they form themselves (the objects of) a (set theoretic) groupoid.

To really "know" what a stack $\mathcal M$ (e.g. $\mathrm{Bun}_G(X)$) "is", it's not enough to know its points: you also have to know which are the morphisms to it and from it involving another space or, more generally, groupoid (actually, by Yoneda's lemma it's enough to know the morphisms to it from spaces). Morphisms involving $\mathcal M$ to or from another analytic groupoid $\mathcal N$, are by definition Morita morphisms of groupoids. More succintly: the category of analytic stacks is the category with objects analytic groupoids and morphisms Morita morphisms of analytic groupoids. So, in particular, Morita equivalent groupoids are to be thought of as isomorphic stacks.

(There is a little sublety though: what I called the "category" of analytic stacks is actually a $2$-category in that maps of stacks are to be considered somehow "homotopic" when there is a $2$-morphism between them, pretty much in the same way as two functors between categories are to be regarded as "homotopic" if there is a natural isomorphism between them).

For $\mathrm{Bun}_G(X)$ there is another, equivalent but probably more expressive, characterization (or definition?) of morphisms $S\rightarrow\mathrm{Bun}_G(X)$ for a space $S$. A map $$\mathcal P : S\to\mathrm{Bun}_G(X)$$ is a (flat) family of principal $G$-bundles on $X$ parametrized by $S$, which means essentially a principal $G$-bundle $\mathcal P$ over $X\times S$. This is a "modular" interpretation of $\mathrm{Bun}_G(X)$; indeed, for $S=\mathrm{point}$, you retrieve the set (actually, groupoid) of principal $G$-bundles over $X$.

What about the étale, fppf, and other Grothendieck topologies? Well, I'm not 100% confident, but I think that if you work in the analytic setting you could disregard these abstractions and just work with the analytic (classical) topology, which I think is finer than each of the others (when seen as a Grothendieck topology, and compared appropriately with the others, working over $\mathbb C$).

• In the complex-analytic setting the flat topology is not subsumed by the etale topology; the relation between the two involves the same issues as in the algebro-geometric context (and even there, the fppf topology has merits over the etale topology when one is studying some maps with "bad" fibers); admittedly the flat topology is less important in characteristic 0 than in characteristic $>0$ (for a variety of reasons), but it does have uses there too and as such cannot be replaced with the etale topology. The distinction already appears for branched covers of Riemann surfaces. – nfdc23 Apr 27 '17 at 14:32
• @nfdc23: thanks for the comment. What about the comparison between the (Grothendieck topology induced by the) "classical" analytic topology (I mean, not the analytic-Zariski one) and the other topologies (at least from the point of view of principal $G$-bundles)? – Qfwfq Apr 27 '17 at 15:39
• The classical and etale topologies are very closely related, ultimately due to a combination of the Artin approximation theorem and various more basic features of etale morphisms (e.g., a map between finite type $\mathbf{C}$-schemes is etale if and only if the associated map of analytifications is a local analytic isomorphism, and more deeply via Artin approximation if $(X,x)$ and $(X',x')$ are such $\mathbf{C}$-schemes equipped with $\mathbf{C}$-points then there is a common etale neighborhood if and only if the analytifications have isomorphic actual neighborhoods of $x$ and $x'$). – nfdc23 Apr 27 '17 at 15:47
• So, technically speaking, can we say that classical and étale are equivalent? If not, which of the two is the finest? – Qfwfq Apr 27 '17 at 16:03
• @Qfwfq As nfdc23 mentioned, ramified covers provide examples of flat maps without analytic slices. – Will Sawin Apr 27 '17 at 16:21