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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 21st 2010
   • (edited Jul 21st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am a bit stuck/puzzled with the following. Maybe you have an idea: I have a group object $G$ and a morphism $G \to Q$. I have a model for the universal $G$-bundle $\mathbf{E}G$ (an object weakly equivalent to the point with a fibration $\mathbf{E}G \to \mathbf{B}G$). I have another object $\mathbf{E}Q$ weakly equivalent to the point such that I get a commuting diagram $$ \array{ G &\to& Q \\ \downarrow && \downarrow \\ \mathbf{E}G &\to& \mathbf{E}Q } $$ Here $Q$ is not groupal and i write $\mathbf{E}Q$ 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 $\mathbf{E}G$ and the coequalizer of that action exists and is $\mathbf{B}G$ $$ G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G \to \mathbf{B}G $$ I can also consider the colimit $K$ of the diagram $$ G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G \to \mathbf{E}Q \,. $$ That gives me a canonical morphism $\mathbf{B}G \to K$ fitting in total into a diagram $$ \array{ G &\to& Q \\ \downarrow && \downarrow \\ \mathbf{E}G &\to& \mathbf{E}Q \\ \downarrow && \downarrow \\ \mathbf{B}G &\to& K } \,. $$ Now here comes finally the question: I know that the coequalizer of $G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}G$ is a model for the _homotopy colimit_ over the diagram $$ \cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} * $$ as you can imagine. But I am stuck: what intrinsic $(\infty,1)$-categorical operation is $K$ a model of? I must be being dense....

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

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

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

   Here is not groupal and i write 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 acting on and the coequalizer of that action exists and is

   I can also consider the colimit of the diagram

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

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

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

   I must be being dense….

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJul 21st 2010
   • (edited Jul 21st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexNot 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) <a href="http://ncatlab.org/schreiber/show/differential+cohomology+in+an+(%E2%88%9E%2C1)-topos#UniversalConn">here</a>.

   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.

3. • CommentRowNumber3.
   • CommentAuthorMike Shulman
   • CommentTimeJul 21st 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexI 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"?

   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”?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJul 21st 2010
   • (edited Jul 21st 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItex$K$ was supposed to be the coequalizer of the two composite maps $$ G \times \mathbf{E}G \stackrel{\to}{\to} \mathbf{E}Q $$ But wait, I guess I am being stupid....

   was supposed to be the coequalizer of the two composite maps

   But wait, I guess I am being stupid….

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJul 21st 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexRight, so I guess i simply mean that the square $$ \array{ \mathbf{E}G &\to& \mathbf{E}Q \\ \downarrow && \downarrow \\ \mathbf{B}G &\to& K } $$ is a pushout.

   Right, so I guess i simply mean that the square

   is a pushout.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJul 22nd 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexIs $EG \to EQ$ 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 $BG \to K$ 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.

   Is 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 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.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeJul 22nd 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexYeah, it looks puzzling. I should turn the question around: given a morphism in an $(\infty,1)$-category $G \to Q$, where $G$ happens to be a group object. Does this induce any canonical morphism out of the delooping $\mathbf{B}G$? (Feel free to assume some extra properties if that helps make you think of something.)

   Yeah, it looks puzzling.

   I should turn the question around:

   given a morphism in an -category , where happens to be a group object. Does this induce any canonical morphism out of the delooping ?

   (Feel free to assume some extra properties if that helps make you think of something.)

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 22nd 2010
   • (edited Jul 22nd 2010)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWell, I should maybe add the following: that morphsm $G \to Q$ is part of a fiber sequence $$ G \to Q \to L \to \mathbf{B}G \,. $$ In the case that $G$ happens to be twice deloopable, this continues as $$ G \to Q \to L \to \mathbf{B}G \to \mathbf{B}Q $$ and that $\mathbf{B}G \to \mathbf{B}Q$ 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 $\mathbf{B}G \to \mathbf{B}Q$ 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.)

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

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

   and that is what I need.

   So one way to ask what I am asking is: in the case that is not twice deloopable, what’s a reaonable universal approximation to the non-existent 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.)

9. • CommentRowNumber9.
   • CommentAuthorMike Shulman
   • CommentTimeJul 23rd 2010
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItexShould 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?

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJul 23rd 2010
    • (edited Jul 23rd 2010)
    • PermaLink
    Author: Urs
    Format: MarkdownItex> 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 $$ LConst \Gamma \to Id $$ of the terminal geometric morphism. I am working in an "locally $\infty$-connected" situation, so that $\mathbf{\flat} := LConst \Gamma$ is a right adjoint. As such it preserves looping, and hence I get a fiber sequence $$ G \to \mathbf{\flat}_{dR} \mathbf{B}G \to \mathbf{\flat}\mathbf{B}G \to \mathbf{B}G \to \mathbf{\flat}_{dR} \mathbf{B}^2 G \to \mathbf{\flat}\mathbf{B}^2 G \to \mathbf{B}^2 G $$

    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

    of the terminal geometric morphism. I am working in an “locally -connected” situation, so that is a right adjoint. As such it preserves looping, and hence I get a fiber sequence

11. • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeJul 23rd 2010
    • PermaLink
    Author: Mike Shulman
    Format: MarkdownItexOk. I think I'm not going to be able to help you without spending a lot more time to understand your situation; sorry.

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