# Equivalences of semisimple abelian categories

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.

• So in particular, if $\mathcal{C}$ and $\mathcal{D}$ are $\mathbb{C}$-linear, then cardinality of their classes of simple objects is sufficient? – Max Schattman Feb 26 at 17:06
• @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. – Qiaochu Yuan Feb 26 at 21:24
• @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. – Qiaochu Yuan Feb 27 at 2:01