What is the correct definition of semisimple linear category?

Question

I have been using the notion of "semisimple linear category" for a while now, but I never bothered to write down a definition. I've always just waved my hands and said "Schur's lemma holds and every object is the direct sum of simple objects". I find myself writing some notes on the subject, and it occurred to me that I had some problems coming up with the correct definition.

The natural reference is nLab. Semisimple linear categories are definition 2.2 of their article "semisimple category". However, I have an issue with their (and by proxy, Müger's) approach. Namely, they require that the category has finite biproducts (direct sums). Next, they require that for any pair of objects $V$, $W$ the natural map

$$\sum_{i\in I}\mathrm{Hom}(V,X_i)\otimes \mathrm{Hom}(X_i,W)\to\mathrm{Hom}(V,W)$$

is an isomorphism, where $\{X_i,\,\, i\in I\}$ is a collection of simple objects chosen to make this isomorphism work. My understanding is that these $X_i$ are supposed to be representatives of the different isomorphism classes of simple objects in $\mathscr{C}$. However, in general there might be infinitely many different such isomorphism classes. This would mean that the direct sum is infinite. This is a problem, since we only required finite biproducts. How do we fix this?

My guess is that if every object is the direct sum of (finitely many) simple objects, then this sum will always be finite, and hence we don't have any problems. If we replace this hom-space-symmetry axiom with the simpler "Every object is the direct sum of finitely many simple objects" do we recover the correct definition?

Answer 1

If I were to try to define "semisimple linear category", and this is indeed something I do try to define, I would say: it is a category, linear over your base commutative ring, which has finite direct sums and splittings of idempotents, and such that for every object $X$, the ring $\mathrm{End}(X)$ is semisimple.

I would then remark that "has finite direct sums and splittings of idempotents" is an extremely natural condition for a linear category, as it is precisely asking that the linear category be Cauchy complete (i.e. it should contain [(co)weighted] (co)limits for all [linear] diagrams such that (co)limits of those shapes are preserved by all linear functors).

Then I would point out that these conditions imply that the category is abelian and that every object is a finite direct sum of simple objects. Also the converse: an abelian category comes with a notion of "simple object", and if every object is a finite direct sum of simples, then it is semisimple in the above sense.

Credit where it's due: I learned this definition from David Reutter, and a version is in his paper with Chris Douglas on semisimple 2-categories.