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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 simple 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?

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?

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, Muger's) approach. Namely, they require that the category has finite biproducts (direct sums). Next, they require that for any pair of simple 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?

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 simple 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?