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Let $\mathcal{C}$ and $\mathcal{D}$ be two semisimple abelian categories. Is it true that $\mathcal{C}$ and $\mathcal{D}$ are equivalent IFF they the cardinality of their classes of simple objects is the same.

This seems like it should be obviously true, but I am afraid I miss something obvious.

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No; you also need the isomorphism classes of the endomorphism rings to match.

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  • $\begingroup$ Agreed, I don't see anything missing here, other than maybe what might have caused the OP to believe this (basically if the category looks like finite dimensional modules for an algebra over an algebraically closed field). $\endgroup$ – Tobias Kildetoft Feb 26 at 7:56
  • $\begingroup$ So in particular, if $\mathcal{C}$ and $\mathcal{D}$ are $\mathbb{C}$-linear, then cardinality of their classes of simple objects is sufficient? $\endgroup$ – Max Schattman Feb 26 at 17:06
  • $\begingroup$ @Max: no, that is still not enough to guarantee that the endomorphism rings match. For example, $C$ might be the category of vector spaces over $\mathbb{C}$ and $D$ might be the category of vector spaces over $\mathbb{C}(t)$. Schur's lemma tells you that the endomorphism ring of a simple object is a division ring and that's the only guarantee you get with these hypotheses. $\endgroup$ – Qiaochu Yuan Feb 26 at 21:24
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    $\begingroup$ @Max: that's right. It would suffice to require both $\mathbb{C}$-linearity and that the endomorphism rings of the simples are finite-dimensional over $\mathbb{C}$. This would be true, for example, if the categories had faithful linear functors to $\text{Vect}(\mathbb{C})$ such that the simples are sent to finite-dimensional vector spaces, which, as Tobias observes, would be true if they were categories of modules over finite-dimensional $\mathbb{C}$-algebras. $\endgroup$ – Qiaochu Yuan Feb 27 at 2:01
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    $\begingroup$ In the future, I recommend that instead of saying "this seems like it should be obviously true," even to yourself, you try to write down the proof. Then if you show it to someone else they can explain which step of the proof, if any, is incorrect. $\endgroup$ – Qiaochu Yuan Feb 27 at 2:03

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