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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 18th 2015
   • (edited Apr 18th 2015)
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   Author: Urs
   Format: MarkdownItexGiven 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.

   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.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeApr 18th 2015
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   Author: Mike Shulman
   Format: MarkdownItexWell, 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.)

   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.)

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 20th 2015
   • (edited Apr 20th 2015)
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   Author: Urs
   Format: MarkdownItexThanks, 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...

   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…