## A puzzle about operations

June 10, 2013 by Qiaochu Yuan

Previously we described -ary operations on (the underlying sets of the objects of) a concrete category , which we defined as the natural transformations .

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

Note that 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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on June 12, 2013 at 12:50 pm |OmarThe 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.

on June 12, 2013 at 12:51 pm |Qiaochu YuanThis 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.

on June 23, 2013 at 12:40 pm |Operations, pro-objects, and Grothendieck’s Galois theory | Annoying Precision[…] « A puzzle about operations […]