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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 18th 2015
    • (edited Apr 18th 2015)

    Given an \infty-topos H\mathbf{H} with a comonad \flat on it and given a pointed connected object BG\mathbf{B}G, write θ G:G dRBG\theta_G \colon G \to \flat_{dR} \mathbf{B}G for the homotopy fiber of the homotopy fiber of the counit BGBG\flat \mathbf{B}G \to \mathbf{B}G.

    I’d like to characterize the internal automorphism group Aut(θ G)Grp(H)\mathbf{Aut}(\theta_G)\in Grp(\mathbf{H}) of θ G\theta_G regarded as an object in H Δ 1\mathbf{H}^{\Delta^1}, hence the group whose global points are diagrams in H\mathbf{H} of the form

    G θ G dRBG G θ G dRBG \array{ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR} \mathbf{B}G \\ {}^{\mathllap{\simeq}}\downarrow &\swArrow^{\simeq}& \downarrow^{\mathrlap{\simeq}} \\ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR} \mathbf{B}G }

    There will be a map from Aut */(BG)=Aut Grp(G)\mathbf{Aut}^{\ast/}(\mathbf{B}G) = \mathbf{Aut}_{Grp}(G) to this group in question. Is this an equivalence?

    I was thinking this should be easy, but now maybe I am being dense.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeApr 18th 2015

    Well, in the special case =Id\flat=Id, we have dRBG=*\flat_{dR}\mathbf{B}G = \ast, so I think Aut Δ 1(θ G)=Aut(G)Aut_{\Delta^1}(\theta_G) = Aut(G) is the automorphisms of GG as a type, not necessarily preserving the group structure. So maybe we need to assume more about \flat? (One assumption we’re already making is that *=*\flat\ast=\ast, in order for BG\flat \mathbf{B}G to be pointed.)

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 20th 2015
    • (edited Apr 20th 2015)

    Thanks, Mike. Yes, I realized after posting that as stated the question was too naive. I’ll try to think of a better question.

    I am fishing a bit in the dark. I am looking at the auto-equivalences of homotopy fiber products over cospans

    G Ω flat 1(,G) θ G dRBG \array{ G && && \Omega^1_{flat}(-,G) \\ & \searrow^{\mathrlap{\theta_G}} && \swarrow \\ && \flat_{dR}\mathbf{B}G }

    where the top right object is 0-truncated, and I seem to have indication that for some applications I should be restricting attention to just Aut Grp(G)\mathbf{Aut}_{Grp}(G). So I am trying to see if there is some good general abstract reason to consider this restriction.

    Well, that’s vague and possibly useless. I’ll try to think of a better question to ask here…

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