# Uniquenes of decompositions in abelian semisimple categories

Basing my intuition on the semisimple Lie algebra case, I have a question about semisimple abelian categories. Let $\mathcal{C}$ be such a category, and let $X$ be an object in $\mathcal{C}$. By definition of semisimplicity we can decompose $X \simeq S_1 \oplus \cdots \oplus S_k$ into a direct sum of simple objects. Is it true, and if so why, that for any other decomposition $X \simeq T_1 \oplus \cdots \oplus T_l$ of $X$ into simple objects, we must have $k=l$ and a map $i:\{1,\dots,k\} \to \{1,\dots,k\}$ such that $$T_a \simeq S_{i(a)}, \text{ for all } a = 1, \dots, k?$$

• I'm sure your favourite proof for semisimple modules will adapt readily to this more general case. – Lord Shark the Unknown Sep 7 '18 at 19:17