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Given an $\infty$-topos $\mathbf{H}$ with a comonad $\flat$ on it and given a pointed connected object $\mathbf{B}G$, write $\theta_G \colon G \to \flat_{dR} \mathbf{B}G$ for the homotopy fiber of the homotopy fiber of the counit $\flat \mathbf{B}G \to \mathbf{B}G$.
I’d like to characterize the internal automorphism group $\mathbf{Aut}(\theta_G)\in Grp(\mathbf{H})$ of $\theta_G$ regarded as an object in $\mathbf{H}^{\Delta^1}$, hence the group whose global points are diagrams in $\mathbf{H}$ of the form
$\array{ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR} \mathbf{B}G \\ {}^{\mathllap{\simeq}}\downarrow &\swArrow^{\simeq}& \downarrow^{\mathrlap{\simeq}} \\ G &\stackrel{\theta_G}{\longrightarrow}& \flat_{dR} \mathbf{B}G }$There will be a map from $\mathbf{Aut}^{\ast/}(\mathbf{B}G) = \mathbf{Aut}_{Grp}(G)$ to this group in question. Is this an equivalence?
I was thinking this should be easy, but now maybe I am being dense.
Well, in the special case $\flat=Id$, we have $\flat_{dR}\mathbf{B}G = \ast$, so I think $Aut_{\Delta^1}(\theta_G) = Aut(G)$ is the automorphisms of $G$ 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 $\flat \mathbf{B}G$ to be pointed.)
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
$\array{ G && && \Omega^1_{flat}(-,G) \\ & \searrow^{\mathrlap{\theta_G}} && \swarrow \\ && \flat_{dR}\mathbf{B}G }$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 $\mathbf{Aut}_{Grp}(G)$. 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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