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How much is known/has been studied about symmetric monoidal closed structures on the category of abelian groups $\mathbf{Ab}$ up to equivalence? Is there any nice characterization of the usual tensor product of abelian groups among these?

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  • $\begingroup$ Do you know of a closed symmetric monoidal structure that is not $\otimes$? $\endgroup$ Jun 12, 2023 at 20:53
  • $\begingroup$ @JeroenvanderMeer No, but i wasn't be able to prove or find any reference proving or disproving that. $\endgroup$
    – Carla_
    Jun 12, 2023 at 20:58

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There is exactly one (symmetric) closed monoidal structure on the category of abelian groups (up to equivalence). This is Proposition 3 of Foltz–Lair–Kelly's Algebraic categories with few monoidal biclosed structures or none.

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