Dualities for self–small groups

Abstract

We construct a family of dualities on some subcategories of the quasi-category S \mathcal {S} of self-small groups of finite torsion-free rank which cover the class S \mathcal {S} . These dualities extend several of those in the literature. As an application, we show that a group A ∈ S A\in \mathcal {S} is determined up to quasi–isomorphism by the Q \mathbb {Q} –algebras { Q Hom ⁡ ( C , A ) : C ∈ S } \{\mathbb {Q}\operatorname {Hom}(C,A):\,C\in \mathcal {S}\} and { Q Hom ⁡ ( A , C ) : C ∈ S } \{\mathbb {Q}\operatorname {Hom}(A,C):\,C\in \mathcal {S}\} . We also generalize Butler’s Theorem to self-small mixed groups of finite torsion-free rank.