simple-group probing functor

 algebra
Notation F:Grp→SetF: \Grp \to \Set Domain category of groupsCodomain category of sets
This functor maps a group GG to the collection F(G)≔∏κHom⁡(Lκ,G),F(G) \coloneqq \textstyle\prod_{\kappa} \Hom(L_\kappa,G), where κ\kappa ranges over all infinite cardinals and LκL_\kappa is an infinite simple group of cardinality κ\kappa; we choose Lκ≔Alt⁡fin(κ)L_\kappa \coloneqq \Alt_{\fin}(\kappa) to make things concrete. For each GG, the collection F(G)F(G) is essentially small, since for κ>card⁡(G)\kappa > \card(G) every homomorphism Lκ→GL_\kappa \to G is trivial. Thus, a more precise definition would be F(G)≔∏κ≤card⁡(G)Hom⁡(Lκ,G),F(G) \coloneqq \textstyle\prod_{\kappa \leq \card(G)} \Hom(L_\kappa,G), since only then F(G)F(G) is actually a set, but the first definition makes it more apparent that FF is a functor. This is the canonical example of a continuous functor Grp→Set\Grp \to \Set that is not representable, and not a right adjoint.

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*This also uses the deduced satisfied properties.

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