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On the one hand, we have BG the space, or the BG the “delooping” one-object groupoid with morphisms G, which we might write G => *. These are closely related; the former is the geometric realization of the nerve of the latter.
On the other hand we have BG the classifying topos or stack, which is (I think) the category of all principal G-bundles.
The notation and similar role played by those objects suggest they are versions of the same thing. On nLab, we find in classifying topos it reads says that the correspondence of toposes GBund(X) = Topos(Sh(X), G) is analogous to the correspondence pi0 GBund(X) = pi Top(X,BG).
Ok but is it just an analogy, or is there some kind of stackification or Yoneda process that turns BG the space/groupoid into BG the topos/stack? Or is there some kind of truncation or geometric realization process that turns BG the stack back into BG the space?
In the article moduli stack it says the moduli stack *//G is the base of the universal principal bundle. Does that mean in the category of stacks? What’s the “total space” over *//G? Is it a stackified version of EG?
It’s hard to believe that these theories are so utterly parallel just by coincidence without a literal connection.
The category of principal $G$-bundles over is the stackification (over the category of base manifolds or spaces) of the one-object (Lie, topological, or plain) groupoid associated to $G$.
Does that mean in the category of stacks? What’s the “total space” over $*//G$?
Yes. The total space is the stack associated to the action groupoid $G//G$ (using the multiplication action of $G$ on itself), the projection is the equivariant map $G\to *$.
Thanks David. Is there a reverse or adjoint process to “stackification”? Is there a way to recover a space or groupoid from a stack, to recover BG the category?
Thanks Dmitri. So if I do something like described at shape modality, it will turn the category of line bundles on topological spaces into the topological space RP^infinity?
Well, there’s the 2-category of presheaves of groupoids, and stackification goes from that to stacks, and it has a right adjoint, but it’s just the inclusion of a sub-2-category. In this instance, the presheaf is represented by an actual internal groupoid, and I don’t believe there’s a functorial way to go back that far. There’s a noncanonical way, namely if the stack is presentable/geometric, then it admits a nice sort of epimorphism from a representable sheaf (a chart/atlas/cover) and then one can cook up a groupoid from this. Different choices of cover will give rise to different, but Morita/weakly equivalent groupoids. If you are working in say topological spaces then you can geometrically realise the nerve of the groupoid to get a classifying space. Similarly with manifolds, but the result will be at best a diffeological space, in general.
…, to recover BG the category?
I’m not sure what you mean by this one.
(My post was directly after #3, but there was an error and it didn’t go up, and I didn’t notice until now).
As far as I know, the shape construction more gets you a space representing a homotopy type, so I doubt you will get $\mathbb{RP}^\infty$ on the nose, rather just some $K(\mathbb{Z}/2,1)$ space (assuming by “line bundles” you mean real line bundles).
Re #6
Is there a way to recover a space or groupoid from a stack, to recover BG the category?
I’m not sure what you mean by this one.
Yeah I think just saying “BG the category” is too ambiguous, right? I meant “recover the category BG with one object and morphisms G (or the space which is geometric realization of same) from the category/stack of all principal G-bundles”.
Because the former category is connected and the latter category is not, so it’s clear they are not equivalent as categories or have homotopy equivalent geometric realizations, so it must be a different notion of equivalence.
But I heard you say Morita equivalence of groupoids, is this how these two groupoids are equivalent?
Re #8:
Thank you Dmitri. This answer looks very helpful, thank you.
OK, given the category $G\mathbf{Bun}$ of all principal $G$-bundles, on manifolds, say, then you can take the terminal stack on the category of manifolds (i.e. $\mathbf{Mfld}$ itself), and take the functor $\mathbf{Mfld} \to G\mathbf{Bun}$ sending a manifold to the trivial bundle on it. This is a epimorphism of stacks from a representable stack (represented by the terminal manifold) that additionally is representable, and a submersion. Such a functor is called a cover. Take the comma object of this functor with itself, and the result is a representable stack, represented by $G$. Then the (2-)Čech nerve of this truncated simplicial stack becomes a levelwise representable simplicial object, which is the nerve of $BG$.
This story holds more generally for arbitrary differentiable stacks, but it’s not so often they come with a more-or-less canonical cover like $G\mathbf{Bun}$.
One level up, there’s an equivalence between the 2-category of differentiable stacks and the 2-category of Lie groupoids, anafunctors and transformations (a combination of results of Pronk and myself). Equivalence in this latter 2-category is the same as Morita equivalence.
An analogous story holds in the topological category, or the algebraic category etc.
Thank you David.
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