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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2010
    • (edited Jul 21st 2010)

    I am a bit stuck/puzzled with the following. Maybe you have an idea:

    I have a group object G and a morphism GQ. I have a model for the universal G-bundle EG (an object weakly equivalent to the point with a fibration EGBG).

    I have another object EQ weakly equivalent to the point such that I get a commuting diagram

    GQEGEQ

    Here 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×EGEGBG

    I can also consider the colimit K of the diagram

    G×EGEGEQ.

    That gives me a canonical morphism BGK fitting in total into a diagram

    GQEGEQBGK.

    Now here comes finally the question: I know that the coequalizer of G×EGEG is a model for the homotopy colimit over the diagram

    G×GG*

    as you can imagine. But I am stuck: what intrinsic (,1)-categorical operation is K a model of?

    I must be being dense….

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2010
    • (edited Jul 21st 2010)

    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.

    • CommentRowNumber3.
    • CommentAuthorMike Shulman
    • CommentTimeJul 21st 2010

    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”?

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2010
    • (edited Jul 21st 2010)

    K was supposed to be the coequalizer of the two composite maps

    G×EGEQ

    But wait, I guess I am being stupid….

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJul 21st 2010

    Right, so I guess i simply mean that the square

    EGEQBGK

    is a pushout.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeJul 22nd 2010

    Is EGEQ 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 BGK 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.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2010

    Yeah, it looks puzzling.

    I should turn the question around:

    given a morphism in an (,1)-category GQ, 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.)

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJul 22nd 2010
    • (edited Jul 22nd 2010)

    Well, I should maybe add the following: that morphsm GQ is part of a fiber sequence

    GQLBG.

    In the case that G happens to be twice deloopable, this continues as

    GQLBGBQ

    and that BGBQ 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 BGBQ 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.)

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeJul 23rd 2010

    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?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 23rd 2010
    • (edited Jul 23rd 2010)

    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

    LConstΓId

    of 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

    GdRBGBGBGdRB2GB2GB2G
    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJul 23rd 2010

    Ok. I think I’m not going to be able to help you without spending a lot more time to understand your situation; sorry.