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Let $G$ be a group, and consider the action of $G$ on itself by conjugation. If we think of $G$ as a one object category, then the conjugation action can be realised as automorphisms of this category, and we may build the associated $2$ category with one object, with additional $2$ morphisms given by elements of $G$, acting as conjugation.

The question is whether one can extend this construction to take into account that the $2$ morphisms also have a notion of "equivalence", whether they are conjugate when viewed as elements of $G$. These relations should be witnessed by elements of $G$, expressing when two $2$ morphisms are conjugate, which then have more relations witnessed by elements of $G$ (by conjugation on these witnesses), and so on.

Ideally this whole package would respect the underlying group structure in some sense, since we are considering automorphisms of all the data in the previous stage to obtain the next stage.

One can give silly ways of describing this heap of data, so as a test/benchmark, one could ask for some general categorical object, which is built only out $G$ as a one object category, which observes "categorically" the following fact about finite groups: For $G$ finite, with $p\in \mathbb{Z}$ coprime to $|G|$, then for any $x,y\in G$, there exists $g,h\in G$ with $(xy)^p=gx^p g^{-1}h y^p h^{-1}$.

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    $\begingroup$ As you probably known, $G\to Aut(G)$ is a crossed module, i.e. a category in groups. You can go one step further and consider the crossed square of a crossed module. It should be possible to go ahead using $cat^n$-groups for instance. $\endgroup$
    – Fernando Muro
    Commented May 8, 2021 at 22:34
  • $\begingroup$ Pity I cannot say more, but at some point I was trying to unravel the structure that begins with the set of objects $G$ and morphisms generated by $[x,y]:yx\to xy$. I believe I managed to show that this much comprises a braiding and that the conjugation action is not more and not less than that. Most likely for topological or simplicial groups one can build on top of that all the higher structure but I never came to looking into that. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Apr 7, 2022 at 16:45

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Following up a bit on Fernando's comment, you look to be trying to define things a bit like the tensor square of a group (acting on itself). Look at the work by Ronnie Brown, et al, Some computations of non-abelian tensor products of groups, J. Algebra, 111, (1987), 177 – 202. This would give you a crossed square. There are variants using an exterior product, which may also be of interest (see work by Graham Ellis)

As Fernando mentions there are $cat^n$-groups / crossed n-cubes of groups that might encode higher commutator data but I think you would be needing more than just a single group, for instance a group with a family of normal subgroups.

There is a separate response, which says that if G is thought of as a one object groupoid, then there is the functor category $G^G$, which encodes the conjugation. It has an associated crossed module which is exactly the $G\to Aut(G)$ one that was mentioned by Fernando.

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  • $\begingroup$ Hi, Tim! I meant, given a $cat^n$-group $G$, is it possible to define an automorphism $cat^n$-group $Aut(G)$ and merge them to an `inner automorphisms' $cat^{n+1}$-group? Like with $G\to Aut(G)$ when $G$ is a group. We can take it one step further using Norrie's paper since $G\to Aut(G)$ is a crossed module. I was giving for granted this had already been done. $\endgroup$
    – Fernando Muro
    Commented May 10, 2021 at 9:20
  • $\begingroup$ To be honest, I'm not sure. It may be that one gets a $cat^{2n}$-group. I know that there are abstract versions of the $G\to Aut(G)$ construction in certain types of category which include the category of crossed modules, but I cannot recall the details. $\endgroup$
    – Tim Porter
    Commented May 11, 2021 at 10:34
  • $\begingroup$ After a bit of ferreting, and perhaps thinking, one could try and replace the $cat^n$-group by an equivalent simplicial group, thought of as an $\infty$-group, $G$,now take the auto-equivalences of that $G$, this gives an $\infty$-groupoid analogue of the automorphism group of the original object, ... but that merely says there should be something there it does not produce it for you as a construction on the original object. Working with simplicial groups however may give a slightly different perspective on the original question. $\endgroup$
    – Tim Porter
    Commented May 11, 2021 at 11:00
  • $\begingroup$ As the technicalities of the simplicial question would not really fit here, I have gone over to contacting people who may be interested by e-mail. (If there are several people who would like to join in, we could start a discussion on the Cateogy zulip site, or you can contact me directly.) $\endgroup$
    – Tim Porter
    Commented May 11, 2021 at 11:52

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