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I asked this question on MSE here

This question was inspired by: The influence of conjugacy class sizes on the structure of finite groups.

My question is as follows: Is there a way to study the relationship between the category of finite groups and their conjugacy classes categorically? Could you please provide references for this?

Clarification of the question: we know the impact of conjugacy classes in the classification of finite groups, but open questions remain, so my main goal is to find a way to connect category theory with questions related to finite groups and conjugacy classes, to solve them.

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    $\begingroup$ This is way too vague to admit a reasonable answer. $\endgroup$
    – Andy Putman
    Commented Mar 22 at 19:01
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    $\begingroup$ @AndyPutman what vague about it? $\endgroup$
    – Naif
    Commented Mar 22 at 19:12

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This question will likely be closed, but I'll try to give the OP some references before it is. First, let me answer the OP's question in the comments, regarding why this is vague and why the question will likely be closed:

  1. It's full of typos (I will try to fix them, but their presence suggests the OP didn't put much effort into writing the question).

  2. It was cross-posted simultaneously. It would have been better to post to MSE, engage with answers there for a week, think deeply, and only post to MO if there was still confusion that couldn't be answered at the level of MO.

  3. The MSE version has an extra part of the question that shows the OP is a relative novice, putting this far below research level.

  4. It's vague, because actually there is a HUGE research area on this connection, as you can see below.

The short answer to the question is "yes, of course." There are SO MANY ways to get at finite groups and conjugacy classes using category theory. Here are just a few off the top of my head.

  1. Fusion systems.

https://web.mat.bham.ac.uk/C.W.Parker/Fusion/fusion-intro.pdf

https://www.math.univ-paris13.fr/~bobol/ako.pdf

https://maths.nuigalway.ie/~park/papers/intro-fusion-systems.pdf

  1. Mackey functors

https://ncatlab.org/nlab/show/Mackey+functor

https://people.math.rochester.edu/faculty/doug/otherpapers/WebbMF.pdf

Relatedly, there is a history of Japanese mathematicians using Mackey and Tambara functors on questions related to finite simple groups. I learned about this from Hiroyuki Nakaoka. I recommend checking out the references in his early papers on the subject.

  1. Transfer systems, indexing systems, and homotopical combinatorics. Getting at the poset of conjugacy data.

https://www.ams.org/notices/202402/rnoti-p260.pdf

https://kyleormsby.github.io/posts/2021/09/homotopical-combinatorics/

  1. The following book takes a categorical lens in some places: http://homepages.math.uic.edu/~smiths/book.pdf

Based on the MSE thread, I think the OP might like even more basic ways of getting at conjugacy data using categories, and will get such answers over there. I wanted to post about some higher level approaches that are active research areas at the moment.

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