Let $\mathcal{A}$ and $\mathcal{B}$ be two tensor categories whose Grothendieck semirings are isomorphic. Does it follow that the categories $\mathcal{A}$ and $\mathcal{B}$ are isomorphic (i) as categories? (ii) as monoidal categories.
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6$\begingroup$ No to (i) and (ii): consider free abelian groups of finite type and finite dimensional vector spaces on some prime field. $\endgroup$– D.-C. CisinskiMar 12, 2022 at 13:26
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$\begingroup$ Some questions: What is an "free abelian group of finite type"? Do you mean finitely generated? What is the Groth. ring in this case? Why are the cats not isomorphic? Why can we not map simples to simples and then extend to an equivalence? $\endgroup$– Adam BondalMar 12, 2022 at 15:35
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$\begingroup$ Yes, I mean "of finite type". In all the example I mention, the Grothendieck semi-ring is the one of natural number (this is the theory of rank of finitely generated free modules). $\endgroup$– D.-C. CisinskiMar 13, 2022 at 9:01
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2$\begingroup$ @AdamBondal We can’t get an equivalence by mapping simples to simples because the endomorphism ring of $\mathbb{Z}$ is not isomorphic to the endomorphism ring of $\mathbb{F}_2$ (for instance). $\endgroup$– Robert FurberMar 14, 2022 at 12:34
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