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I noticed that there was no entry quotient stack, so I quickly started one, just to be able to point to it from elswhere.
The idea of quotient stack extended in a standard way using general internal groupoids in a site or topos.
In principle it works and is being used. But there may be a font issue. For me it works on Firefox on Windows.
It’s a problem with Chrome again.
added references for the common construction of quotient stacks X⫽G via prestacks of G-principal bundles equipped with G-equivariant maps to X:
Jochen Heinloth, Exp. 1.5 in: Notes on differentiable stacks (2004) [pdf]
Frank Neumann, p. 28 in: Algebraic Stacks and Moduli of Vector Bundles, impa (2011) [pdf, pdf]
(there must be more canonical references for this construction – if anyone has a good reference at hand, let’s add it!)
and pointer to this recent discussion of sufficient conditions for this construction to really yield a stack (instead of just a prestack):
added pointer to:
added pointer to:
added pointer to
for the definition of quotient stacks as stackifications of ((2,1)-presheaves of) action groupoids
also pointer to:
I have finally looked at the section here which Praphulla Koushik had announced in message #8 above (Dec. 2018), asking for comments.
Now I see that this section is trying to get at the characterization of G-quotient stacks as fibrations over BG. So I have added references which discuss this perspective (now here – these are the pertinent references that I am aware of; if there are others let’s add them in, too).
I haven’t yet touched Praphulla’s section itself (there is room to bring it into a state which would allow removing the question marks) except for prefixing it by a pointer to these references.
added publication data for this item:
15: the section looks correct to my reading and present knowledge.
I have what I think should be a very simple question but I’m not sure what the actual details are. For M a space and H a group acting on it, one can form its weak quotient, the stack M//H. In particular, for M=*, then *//H=BH. Now, suppose one has a central extension of groups 1→H→𝒢→G→1. If I let H act trivially on BG, then taking the weak quotient is BG//H=BG×BH, which is almost B𝒢 but not quite. How can I get B𝒢 as a weak quotient by an H-action? I’m guessing one can define a nontrivial action of H not on BG but on the map BG→B2H but I don’t exactly see what. I know I can compute B𝒢 as the fiber but that is all related to a BH action, whereas I am interested in an H action.
The key fact to keep in mind here is that homotopy quotients X⫽H are exactly characterized (up to equivalence) as being the fibrations over BH whose homotopy fiber is X (Prop. 0.2.1 in Equivariant Principal ∞-Bundles).
This means that to exhibit B𝒢 as a homotopy quotient of some X by H, is equivalent to finding a homotopy fiber sequence of this form:
X⟶B𝒢↓BH.Now the vertical maps come from group homomorphisms (regarded as pointed maps they are equivalent to the ∞-group homomorphisms) 𝒢⟶H, and the homotopy fiber then is X≃H⫽𝒢, whence
B𝒢≃(H⫽𝒢)⫽H(Ex. 3.2.35 in Equivariant Principal ∞-Bundles).
(The elementary example of this situation, where all groups are discrete, is spelled out in great detail at induced representation in the section “Groupoid formulation”.)
But say for group, one rarely has group homomorphisms 𝒢→H for a central extension H→𝒢→G, no? Does that not exist only when the extension has a trivial cocycle, giving back BG×BH ultimately?
That may be a problem for what you are after. What I stated is the complete answer to your question how to exhibit B𝒢 as a homotopy H-quotient.
The answer implies that if in your application there is no group homomorphism 𝒢⟶H suitable for your purpose, then your wish for a suitable homotopy H-quotient presentation of Bℋ can’t be fulfilled. That’s just the way it is.
I see. What I am ultimately after is the statement that for a G gauge theory, gauging a global H symmetry can give not only a G×H gauge theory but actually a gauge group that is a central extension 𝒢. I am not taking any Lagrangians, I am just looking at the fields Σ→BG (no connection for simplicity) and replacing the target space by a weak quotient to get a B𝒢 target space which is not just BG×BH. Or is this formulation of gauging as replacing a target space with a weak quotient a special case of some other kind of limit that does allow to obtain B𝒢? One can, as I mentioned, look at pullbacks but then, on the other hand, obtaining *//G as a pullback is too artificial at best.
Not sure if the following is what you are after, but what you describe does remind me of the situation of H-equivariant G-fiber bundles, as appropriate for a G-gauge theory in the presence of a global H-symmetry.
This requires an action of H on G by group automormisms, defining a semidirect product group extension G⋊.
Then the equivariant gauge bundles are classified by maps from to in the slice over .
This is discussed on pp. 180 of Equivariant Principal -Bundles (where is called , and is called ).
That is close! But I am mostly interested in central extensions with nontrivial cocycles, which is why I mentioned H should act trivially on but maybe not on its map to … Skimming through the book you referenced, seems one is better off working with pullbacks/homotopy fibers for this kind of this, right?
Never mind, I see what I was doing wrong…
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