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I am a bit stuck/puzzled with the following. Maybe you have an idea:
I have a group object G and a morphism G→Q. I have a model for the universal G-bundle EG (an object weakly equivalent to the point with a fibration EG→BG).
I have another object EQ weakly equivalent to the point such that I get a commuting diagram
G→Q↓↓EG→EQHere Q is not groupal and i write EQ only for the heck of it and to indicate that this is contractible and the vertical morphisms above are monic (cofibrations if due care is taken).
So I have G acting on EG and the coequalizer of that action exists and is BG
G×EG→→EG→BGI can also consider the colimit K of the diagram
G×EG→→EG→EQ.That gives me a canonical morphism BG→K fitting in total into a diagram
G→Q↓↓EG→EQ↓↓BG→K.Now here comes finally the question: I know that the coequalizer of G×EG→→EG is a model for the homotopy colimit over the diagram
⋯G×G→→→G→→*as you can imagine. But I am stuck: what intrinsic (∞,1)-categorical operation is K a model of?
I must be being dense….
Not sure if it helps to see the pattern or distract from its general structure: but my detailed setup is described in a bit more detail (though still in a rough fashion) here.
I don’t understand the definition of K; is there a typo in the displayed equation after “I can also consider the colimit K of the diagram”?
K was supposed to be the coequalizer of the two composite maps
G×EG→→EQBut wait, I guess I am being stupid….
Right, so I guess i simply mean that the square
EG→EQ↓↓BG→Kis a pushout.
Is EG→EQ a cofibration? If not, then it seems that that pushout has no homotopical meaning. If so, then it’s an acyclic cofibration, since both EG and EQ are contractible, hence BG→K is also an acyclic cofibration and thus a weak equivalence, so K is the same as BG. Regardless, it doesn’t seem that K can contain any information about Q, since that pushout doesn’t contain Q anywhere.
Yeah, it looks puzzling.
I should turn the question around:
given a morphism in an (∞,1)-category G→Q, where G happens to be a group object. Does this induce any canonical morphism out of the delooping BG?
(Feel free to assume some extra properties if that helps make you think of something.)
Well, I should maybe add the following: that morphsm G→Q is part of a fiber sequence
G→Q→L→BG.In the case that G happens to be twice deloopable, this continues as
G→Q→L→BG→BQand that BG→BQ is what I need.
So one way to ask what I am asking is: in the case that G is not twice deloopable, what’s a reaonable universal approximation to the non-existent BG→BQ here?
(Actually, I have a guess for that, too, using some extra structure I have availabke, but I can’t show that my guess reproduces the above ordinary pushout construciton.)
Should it be obvious that such a fiber sequence can be continued if G is twice deloopable? Or is this a special characteristic of your situation?
Should it be obvious that such a fiber sequence can be continued if G is twice deloopable? Or is this a special characteristic of your situation?
This is special for my situation.
The fiber sequence that i am considering is that induced by the counit
LConstΓ→Idof the terminal geometric morphism. I am working in an “locally ∞-connected” situation, so that ♭:=LConstΓ is a right adjoint. As such it preserves looping, and hence I get a fiber sequence
G→♭dRBG→♭BG→BG→♭dRB2G→♭B2G→B2GOk. I think I’m not going to be able to help you without spending a lot more time to understand your situation; sorry.
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