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?
2026-05-16 04:07:08.1778904428
Is there a classification of symmetric monoidal closed structures on $\mathbf{Ab}$?
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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.