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1. • CommentRowNumber1.
   • CommentAuthorperezl.alonso
   • CommentTimeApr 15th 2025
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexNot an edit, but is there anything concrete known about this kind of automorphism group for an [[infinity-group]]? Say its homotopy type? <a href="https://ncatlab.org/nlab/revision/diff/automorphism+2-group/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/automorphism+2-group/6">v6</a>, <a href="https://ncatlab.org/nlab/show/automorphism+2-group">current</a>

   Not an edit, but is there anything concrete known about this kind of automorphism group for an infinity-group? Say its homotopy type?

   diff, v6, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 15th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexNot a concrete answer, but just to amplify that for a general $\infty$-group $\mathcal{G}$ its automorphism $\infty$-group is $\mathrm{Aut}(\mathbf{B} \mathcal{G})$, but that as the homotopy type of $\mathcal{G}$ ranges, its deloopings $\mathbf{B}\mathcal{G}$ range equivalently over all pointed connected homotopy types. Therefore the question of understanding automorphism $\infty$-groups is, up to the evident/trivial permutation symmetry of connected components, equivalent to computing automorphism groups of connected homotopy types. I don't think there is anything very general one can say about this, but there will be discussion for various classes of homotopy types. So I guess to make progress you will have to narrow in on some class of $\infty$-groups that you are interested in.

   Not a concrete answer, but just to amplify that for a general -group its automorphism -group is , but that as the homotopy type of ranges, its deloopings range equivalently over all pointed connected homotopy types.

   Therefore the question of understanding automorphism -groups is, up to the evident/trivial permutation symmetry of connected components, equivalent to computing automorphism groups of connected homotopy types.

   I don’t think there is anything very general one can say about this, but there will be discussion for various classes of homotopy types.

   So I guess to make progress you will have to narrow in on some class of -groups that you are interested in.

3. • CommentRowNumber3.
   • CommentAuthorperezl.alonso
   • CommentTimeApr 15th 2025
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexI'm mainly thinking of loop spaces, though I guess that doesn't do much as far as narrowing down is concerned.

   I’m mainly thinking of loop spaces, though I guess that doesn’t do much as far as narrowing down is concerned.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 15th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexRight, it doesn't :-). $\infty$-groups $\mathcal{G}$ are equivalently loop $\infty$-groups, namely of their delooping: $\mathcal{G} \simeq \Omega B \mathcal{G}$. On the nLab this is recorded at *[[May recognition theorem]]*.

   Right, it doesn’t :-). -groups are equivalently loop -groups, namely of their delooping: .

   On the nLab this is recorded at May recognition theorem.

5. • CommentRowNumber5.
   • CommentAuthorperezl.alonso
   • CommentTimeApr 15th 2025
   • (edited Apr 15th 2025)
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexYeah, and that is exactly why I am looking at those in the first place... What if I first $n$-truncate the loop space?

   Yeah, and that is exactly why I am looking at those in the first place… What if I first -truncate the loop space?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 15th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexThat will sure increase the chance to say anything of generality, but it will still be hard, I think. The keywords to search for are "self-equivalences" or "self-homotopy equivalences", "homotopy automorphisms" and "hAut(-)".

   That will sure increase the chance to say anything of generality, but it will still be hard, I think.

   The keywords to search for are “self-equivalences” or “self-homotopy equivalences”, “homotopy automorphisms” and “hAut(-)”.

7. • CommentRowNumber7.
   • CommentAuthorperezl.alonso
   • CommentTimeApr 15th 2025
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexAlright, thanks, Urs.

   Alright, thanks, Urs.