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## A puzzle about operations

Previously we described $n$-ary operations on (the underlying sets of the objects of) a concrete category $(C, U)$, which we defined as the natural transformations $U^n \to U$.

Puzzle: What are the $n$-ary operations on finite groups?

Note that $U$ is not representable here. The next post will answer this question, but for those who don’t already know the answer it should make a nice puzzle.

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### 3 Responses

1. The obvious guess, since the forgetful functor is “sort of representable”, i.e., representable by a (non-finite) group, is that the n-ary operations are elements of the profinite completion of the free group on n letters, but I haven’t thought it through to see if this is correct or not.

• This is correct! The n-th power of the forgetful functor is pro-representable by the diagram of all finite quotients of the free group on n letters.

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